Spin Geometry
H. BLAINE LAWSON, JR.
and
MARIE-LOUISE MICHELSOHN
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
1989
Copyright © 1989 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
Chichester, West Sussex
Library of Congress Cataloging-in-Publication Data
Lawson, H. Blaine.
Spin geometry/H. B. Lawson and M.-L. Michelsohn.
p.	cm.—(Princeton mathematical series: 38)
Includes index.
ISBN 0-691-08542-0 (alk. paper):
1. Nuclear spin—Mathematics. 2. Geometry. 3. Topology.
4. Clifford algebras. 5. Mathematical physics. I. Michelsohn,
M.-L. (Marie-Louise), 1941- II. Title. III. Series.
QC793.3.S6L39 1989
539.7'25—dc20	89-32544
Princeton University Press books are printed on acid-free paper
and meet the guidelines for permanence and durability of the
Committee on Production Guidelines for Book Longevity
of the Council on Library Resources
Printed in the United States of America
For Christie, Didi,
Michelle, and Heather

Contents
Preface	ix
Acknowledgments	xii
Introduction	3
Chapter I Clifford Algebras, Spin Groups and Their
Representations	7
§1	. Clifford algebras	7
§2	. The groups Pin and Spin	12
§3	. The algebras С€л and Cf,tS	20
§4	. The classification	25
§5	. Representations	30
§6	. Lie algebra structures	40
§7	. Some direct applications to geometry	44
§8	. Some further applications to the theory	of	Lie groups	49
§9	. К-theory and the Atiyah-Bott-Shapiro	construction	58
§10	. KR-theory and the (l,l)-Periodicity Theorem	70
Chapter II	Spin Geometry and the Dirac Operators	77
§1	. Spin structures on vector bundles	78
§2	. Spin manifolds and spin cobordism	85
§3	. Clifford and spinor bundles	93
§4	. Connections on spinor bundles	101
§5	. The Dirac operators	112
§6	. The fundamental elliptic operators	135
§7	. C£*-linear Dirac operators	139
§8	. Vanishing theorems and some applications	153
Chapter III	Index Theorems	166
§1	. Differential operators	167
§2	. Sobolev spaces and Sobolev theorems	170
§3	. Pseudodifferential operators	177
§4	. Elliptic operators and parametrices	188
viii	CONTENTS
§5	. Fundamental results for elliptic operators	192
§6	.	The	heat kernel and the index	198
§7	.	The	topological invariance of the index	201
§8	.	The	index of a family of elliptic operators	205
§9	.	The	G-index	211
§10	.	The	Clifford index	214
§11	. Multiplicative sequences and the Chem character	225
§12	. Thom isomorphisms and the Chem character defect	238
§13	. The Atiyah-Singer Index Theorem	243
§14	. Fixed-point formulas for elliptic operators	259
§15	. The Index Theorem for Families	268
§16	. Families of real operators and the C£k-index Theorem	270
§17	. Remarks on heat and supersymmetry	277
Chapter IV	Applications in Geometry and Topology	278
§1	. Integrality theorems	280
§2	. Immersions of manifolds and the vector field problem	281
§3	. Group actions on manifolds	291
§4	. Compact manifolds of positive scalar curvature	297
§5	. Positive scalar curvature and the fundamental group	302
§6	. Complete manifolds of positive scalar curvature	313
§7	. The topology of the space of positive scalar curvature metrics	326
§8	. Clifford multiplication and Kahler manifolds	330
§9	. Pure spinors, complex structures, and twistors	335
§10	. Reduced holonomy and calibrations	345
§11	. Spinor cohomology and complex manifolds with vanishing first
Chern class	357
§12	. The Positive Mass Conjecture in general relativity	368
Appendix A	Principal G-bundles	370
Appendix В	Classifying Spaces and Characteristic Classes	376
Appendix C	Orientation Classes and Thom Isomorphisms in
K-theory	384
Appendix D	Spin'-manifolds	390
Bibliography	402
Index	417
Notation Index
425
Preface
In the late 192O’s the relentless march of ideas and discoveries had carried
physics to a generally accepted relativistic theory of the electron. The
physicist P.A.M. Dirac, however, was dissatisfied with the prevailing ideas
and, somewhat in isolation, sought for a better formulation. By 1928 he
succeeded in finding a theory which accorded with his own ideas and also
fit most of the established principles of the time. Ultimately this theory
proved to be one of the great intellectual achievements of the period. It
was particularly remarkable for the internal beauty of its mathematical
structure which not only clarified much previously mysterious phenomena
but also predicted in a compelling way the existence of an electron-like
particle of negative energy. Indeed such particles were subsequently found
to exist and our understanding of nature was transformed.
Because of its compelling beauty and physical significance it is perhaps
not surprising that the ideas at the heart of Dirac’s theory have also been
discovered to play a role of great importance in modem mathematics,
particularly in the interrelations between topology, geometry and analysis.
A great part of this new understanding comes from the work of M. Atiyah
and I. Singer. It is their work and its implications which form the focus
of this book.
It seems appropriate to sketch some of the fundamental ideas here.
In searching for his theory, Dirac was faced, roughly speaking, with the
problem of finding a Lorentz-invariant wave equation Оф = zi// compat-
ible with the Klein-Gordon equation Оф = Аф where □ = (д/дх0)2 —
(d/dxj2 — (д/дх2)2 — (д/5х3)2. Causality required that D be first order in
the “time” coordinate x0. Of course by Lorentz invariance there could be
no preferred time coordinate, and so D was required to be first-order in all
variables. Thus, in essence Dirac was looking for a first-order differential
operator whose square was the laplacian. His solution was to replace the
complex-valued wave function ф with an n-tuple = (ф19... ,фп) of such
functions. The operator D then became a first-order system of the form
3 d
ц=0 OX*
X
PREFACE
where y0,... ,y3 were n x n-matrices. The requirement that
/□	\
D2 =	J
\	□/
led to the equations
УуУц УцУу	i2(5v^.
These were easily and explicitly solved for small values of n, and the
analysis was underway.
This construction of Dirac has a curious and fundamental property.
Lorentz transformations of the space-time variables (x0,... ,x3) induce
linear transformations of the n-tuples which are determined only up to
a sign. Making a consistent choice of sign amounts to passing to a non-
trivial 2-fold covering L of the Lorentz group L. That is, in transforming the
'P’s one falls upon a representation of L which does not descend to L.
The theory of Dirac had another interesting feature. In the presence of
an electromagnetic field the Dirac Hamiltonian contained an additional
term added on to what one might expect from the classical case. There
were strong formal analogies with the additional term one obtains by
introducing internal spin into the mechanical equations of an orbiting
particle. This “spin” or internal magnetic moment had observable quan-
tum effects. The n-tuples 'P were thereby called spinors and this family of
transformations was called the spin representation.
This physical theory touches upon an important and general fact con-
cerning the orthogonal groups. (We shall restrict ourselves for the moment
to the positive definite case.) In the theory of Cartan and Weyl the repre-
sentations of the Lie algebra of SO„ are essentially generated by two basic
ones. The first is the standard n-dimensional representation (and its ex-
terior powers). The second is constructed from the representations of the
algebra generated by the yM’s as above (the Clifford algebra associated to
the quadratic form defining the orthogonal group). This second represen-
tation is called the spin representation. It does not come from a represen-
tation of the orthogonal group, but only of its universal covering group,
called Spin„. It plays a key role in an astounding variety of questions
in geometry and topology: questions involving vector fields on spheres,
immersions of manifolds, the integrality of certain characteristic numbers,
triality in dimension eight, the existence of complex structures, the exis-
tence of metrics of positive scalar curvature, and perhaps most basically,
the index of elliptic operators.
In the early 1960s general developments had led mathematicians to con-
sider the problem of finding a topological formula for the index of any
elliptic operator defined on a compact manifold. This formula was to gen-
PREFACE	xi
eralize the important Hirzebruch-Riemann-Roch Theorem already estab-
lished in the complex algebraic case. In considering the problem, Atiyah
and Singer noted that among all manifolds, those whose SO„-structure
could be simplified to a Spin„-structure had particularly suggestive prop-
erties. Realizing that over such spaces one could carry out the Dirac con-
struction, they produced a globally defined elliptic operator canonically
associated to the underlying riemannian metric. The index of this operator
was a basic topological invariant called the ^-genus, which was known
always to be an integer in this special class of spin manifolds. (It is not
an integer in general.) Twisting the Dirac-type operator with arbitrary
coefficient bundles led, with some sophistication, to a general formula for
the index of any elliptic operator.
Atiyah and Singer went on to understand the index in the more proper
setting of К-theory. This led in particular to the formulation of certain KO-
invariants which have profound applications in geometry and topology.
These invariants touch questions unapproachable by other means. Their
study and elucidation was a principal motivation for the writing of this
tract.
It is interesting to note in more recent years there has been another pro-
found and beautiful physical theory whose ideas have come to the core of
topology, geometry and analysis. This is the non-abelian gauge field theory
of C. N. Yang and R. L. Mills which through the work of S. Donaldson
and M. Freedman has led to astonishing results in dimension four. Yang-
Mills theory can be plausibly considered a highly non-trivial generalization
of Dirac’s theory which encompasses three fundamental forces: the weak,
strong, and electromagnetic interactions. This theory involves modern dif-
ferential geometry in an essential way. The theory of connections, Dirac-
type operators, and index theory all play an important role. We hope this
book can serve as a modest introduction to some of these concepts.
H. B. Lawson and M.-L. Michelsohn
Stony Brook
Acknowledgments
This book owes much to the fundamental work of Michael Atiyah and
Iz Singer. Part of our initial motivation in writing the book was to give a
leisurely and rounded presentation of their results.
The authors would like to express particular gratitude to Peter Land-
weber, Jean-Pierre Bourguignon, Misha Katz, Haiwan Chen and Peter
Woit, each of whom read large parts of the original manuscript and made
a number of important suggestions. We are also grateful to the National
Science Foundation and the Brazilian C. N. Pq. for their support during
the writing of this book.
H. B. Lawson and M.-L. Michelsohn
This author would like to take this opportunity to express her deep
appreciation to Mark Mahowald who held out a hand when one was so
dearly needed.
M.-L. Michelsohn
Spin Geometry

Introduction
Over the past two decades the geometry of spin manifolds and Dirac
operators, and the various associated index theorems have come to play
an increasingly important role both in mathematics and in mathematical
physics. In the area of differential geometry and topology they have be-
come fundamental. Topics like spin cobordism, previously considered
exotic even by topologists, are now known to play an essential role in
such classical questions as the existence or non-existence of metrics of po-
sitive curvature. Indeed, the profound methods introduced into geometry
by Atiyah, Bott, Singer and others are now indispensible to mathemati-
cians working in the field. It is the intent of this book to set out the fun-
damental concepts and to present these methods and results in a unified
way.
A principal theme of the exposition here is the consistent use of Clifford
algebras and their representations. This reflects the observed fact that
these algebras emerge repeatedly at the very core of an astonishing variety
of problems in geometry and topology.
Even in discussing riemannian geometry, the formalism of Clifford mul-
tiplication will be used in place of the more conventional exterior tensor
calculus. There is a philosophical justification for this bias. Recall that to
any vector space V there is naturally associated the exterior algebra A*K,
and this association carries over directly to vector bundles. Applied to the
tangent bundle of a smooth manifold, it gives the de Rham bundle of ex-
terior differential forms. In a similar way, to any vector space V equipped
with a quadratic form 4, there is associated the Clifford algebra Ct(K,g),
and this association carries over directly to vector bundles equipped with
fibre metrics. In particular, applied to the tangent bundle of a smooth
riemannian manifold, it gives a canonically associated bundle of algebras,
called the Clifford bundle. As a vector bundle it is isomorphic to the
bundle of exterior forms. However, the Clifford multiplication is strictly
richer than exterior multiplication; it reflects the inner symmetries and
basic identities of the riemannian structure. In fact fundamental curvature
identities will be derived here in the formalism of Clifford multiplication
and applied to some basic problems.
4
INTRODUCTION
Another justification for our approach is that the Clifford formalism
gives a transparent unification of all the fundamental elliptic complexes
in differential geometry. It also renders many of the technical arguments
involved in applying the Index Theorem quite natural and simple.
This point of view concerning Clifford bundles and Clifford multiplica-
tion is an implicit, but rarely an explicit theme in the writing of Atiyah and
Singer. The authors feel that for anyone working in topology or geometry
it is worthwhile to develop a friendly, if not intimate relationship with
spin groups and Clifford modules. For this reason we have used them ex-
plicitly and systematically in our exposition.
The book is organized into four chapters whose successive themes are
algebra, geometry, analysis, and applications. The first chapter offers a
detailed introduction to Clifford algebras, spin groups and their represen-
tations. The concepts are illuminated by giving some direct applications
to the elementary geometry of spheres, projective spaces, and low-
dimensional Lie groups. К-theory and KR-theory are then introduced,
and the fundamental relationship between Clifford algebras and Bott pe-
riodicity is established.
In the second chapter of this book, the algebraic concepts are carried
over to define structures on differentiable manifolds. Here one enters prop-
erly into the subject of spin geometry. Spin manifolds, spin cobordism,
and spinor bundles with their canonical connections are all discussed in
detail, and a general formalism of Dirac bundles and Dirac operators is
developed. Hodge-de Rham Theory is reviewed in this formalism, and
each of the fundamental elliptic operators of riemannian geometry is
derived and examined in detail.
Special emphasis is given here to introducing the notion of a C£fc-linear
elliptic operator and discussing its index. This index lives in a certain
quotient of the Grothendieck group of Clifford modules. For the fun-
damental operators (which are discussed in detail here) it is one of the
deepest and most subtle invariants of global riemannian geometry. The
systematic discussion of Clfc-linear differential operators is one of the
important features of this book.
In the last section of Chapter II a universal identity of Bochner type is
established for any Dirac bundle, and the classical vanishing theorems of
Bochner and Lichnerowicz are derived from it.
This seems an appropriate time to make some general observations
about spin geometry. To begin it should be emphasized that spin geom-
etry is really a special topic in riemannian geometry. The central concept
of a spin manifold is often considered to be a topological one. It is just
a manifold with a simply-connected structure group. This is understood
systematically as follows. On a general differentiable n-manifold (n 3),
the tangent bundle has structure group GL„. The manifold is said to be
INTRODUCTION
5
oriented if the structure group is reduced to GLn+ (the connected com-
ponent of the identity). The manifold is said to be spin if jthe structure
group GLn+ can be “lifted” to the universal covering group GL„ -► GL„+.
This approach is perfectly correct, but there js_a hidden obstruction to
the viability of the concept: namely, the group GL„ (for n 3) has no finite
dimensional representations that do not come from GL*. This means that
in terms of standard tensor calculus, nothing has been gained by this re-
finement of the structure.
However, if one passes from GL„ to the maximal compact subgroup
O„, that is, if one introduces a riemannian metric on the manifold, the
story is quite different. An orientation corresponds to reducing the struc-
ture group to SO„, and a spin structure corresponds to then lifting the
structure group to the universal covering group Spin„ -► SO„. Maximal
compact subgroups are homotopy equivalent to the Lie groups which
contain them, and there is essentially no topological difference in viewing
spin structures this way. However, there do exist finite dimensional repre-
sentations of Spin„ which are not lifts of representations of SO„. Over a
spin manifold one can thereby construct certain new vector bundles, called
bundles of spinors, which do not exist over general manifolds. Their exis-
tence allows the introduction of certain important analytic tools which
are not generally available, and these tools play a central role in the
study of the global geometry of the space. It is, by the way, an important
fact that this construction is metric-dependent; the bundle of spinors itself
depends in an essential way on the choice of riemannian structure on the
manifold.
These observations lead one to suspect that there must exist a local
spinor calculus, like the tensor calculus, which should be an important
component of local riemannian geometry. A satisfactory formalism of this
type has not yet been developed. However, the spinors bundles have
yielded profound relations between local riemannian geometry and global
topology.
The main tools by which we access the global structure of spin mani-
folds are the various index theorems of Atiyah and Singer. These are pre-
sented and proved in Chapter III of the book. They include not just the
standard G-Index Theorem but also the Index Theorem for Families and
the Ctk-Index Theorem (for Ctk-linear elliptic operators). There are in
existence today many elegant proofs of index theorems which use the
methods of the heat equation. These do not apply to the Ctk-Index
Theorem however, because of the non-local nature of this index. For this
reason our exposition follows the “softer,” or more topological, arguments
given in the original proofs.
Chapter IV of the book is concerned with applications of the theory.
There is no attempt to be exhaustive; such an attempt would be pointless
6
INTRODUCTION
and nearly impossible. We have tried however to demonstrate the broad
range of problems in which the considerations of spin geometry can be
effectively implemented.
It is of some historical interest to note that while Dirac did essentially
use Clifford modules in the construction of his wave operator, he was not
really responsible for what is commonly called the “Dirac operator” in
riemannian geometry. The construction of this operator is due to Atiyah
and Singer and is, in our estimation, one of their great achievements. It
required for its discovery an understanding of the subtle geometry of spin
manifolds and a recognition of the central role it would play in the general
theory of elliptic operators. Even the formidable Elie Cartan, who sensed
the importance of the question and, of course, authored the general theory
of spinors and who was not unaware of the fundamentals of global anal-
ysis, never reached the point of defining this operator in the proper con-
text of spin manifolds. In keeping with historical developments we shall
call the general construction of operators from modules over the Clifford
bundle, the Dirac construction, and we shall call the specific operator so
defined on the spinor bundle, the Atiyah-Singer operator.
It is this operator which in a very specific sense generates all elliptic
operators over a spin manifold. It introduces a direct relationship between
curvature and topology which exists only under the spin hypothesis. The
Clfc-linear version of this operator carries an index in KO-theory. In fact
its index gives a basic ring homomorphism Q*pin -> KO"*(pt) which
generalizes to KO-theory the classical A-genus. The applications of this
to geometry include the fact that half the exotic spheres in dimensions
one and two (mod 8) do not carry metrics of positive scalar curvature.
The presentation in this book is aimed at readers with a knowledge of
elementary geometry and topology. Important things, such as the concept
of spin manifolds and the theory of connections, are developed from basic
definitions. The Atiyah-Singer index theorems are formulated and proved
assuming little more than a knowledge of the Fourier inversion formula.
There are several appendices in which principal bundles, classifying
spaces, Thom isomorphisms, and spin manifolds are discussed in detail.
The references to theorems and equations within each chapter are
made without reference to the chapter itself (e.g., 2.7 or (5.9)). Refer-
ences to other chapters are prefaced by the chapter number (e.g., III.2.7
or (IV.5.9)).
CHAPTER I
Clifford Algebras, Spin Groups and Their
Representations
The object of this chapter is to present the algebraic ideas which lie at the
heart of spin geometry. The central concept is that of a Clifford algebra.
This is an algebra naturally associated to a vector space which is equipped
with a quadratic form. Within the group of units of the algebra there is
a distinguished subgroup, called the spin group, which, in the case of the
positive definite form on R" (n > 2), is the universal covering group of SO„.
It is a striking (and not commonplace) fact that Clifford algebras and
their representations play an important role in many fundamental aspects
of differential geometry. These include such diverse topics as Hodge-de
Rham Theory, Bott periodicity, immersions of manifolds into spheres,
families of vector fields on spheres, curvature identities in riemannian
geometry, and Thom isomorphisms in К-theory. The effort invested in
becoming comfortable with this algebraic formalism is well worthwhile.
Our discussion begins in a very general algebraic context but soon
moves to the real case in order to keep matters simple and in the domain
of most interest. In §§7 and 8 we present some applications of the purely
algebraic theory to topology and to the appearence of exceptional phe-
nomena in the theory of Lie groups.
The last part of the chapter is devoted to К-theory. Basic definitions
are given and fundamental results are reviewed. The discussion culminates
with the Atiyah-Bott-Shapiro isomorphisms which directly relate the
periodicity phenomena in Clifford algebras to the classical Bott Period-
icity Theorems. In particular, explicit isomorphisms are given between
K"*(pt) = фиК($") (and KO-*(pt) = ©„ KO(Sn)) and a certain quotient
of the ring of Clifford modules. Section 10 is concerned with KR-theory
which later plays a role in the index theorem for families of real elliptic
operators. This is a bigraded theory and the corresponding Atiyah-Bott-
Shapiro isomorphism entails representations of Clifford algebras C£r s for
quadratic forms of indefinite signature.
§1. Clifford Algebras
Let V be a vector space over the commutative field к and suppose q is a
quadratic form on V. The Clifford algebra Ct(V,<?) associated to ’/and
8
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
q is an associative algebra with unit defined as follows. Let
^(n= I (grr
r = 0
denote the tensor algebra of V, and define ^(P) to be the ideal in
generated by all elements of the form v®v + q(V)l for ve V. Then
the Clifford algebra is defined to be the quotient
Cl(V,q) =
There is a natural embedding
V<—+Cl(V,q)	(1.1)
which is the image of V = (g)1 V under the canonical projection
---->Ct(V,q).	(1.2)
We prove that я,|г is injective as follows. We say that an element <p e
is of pure degree s if <p e (gj’K (Every element of is a finite sum of
elements of pure degree.) We want to show that any element <p e «/,(У) n V
is zero. Any such element can be written as a finite sum <p = £ a, ®
(fi ® vt + q(Vi)) ® bt where we may assume that the a.’s and ft/s are of
pure degree. Since <p e V = (g)1 V, we conclude that £ a,- ® (vr ® vr) ®
bt. = 0, where this sum is taken over those indices with deg at + deg bt
maximal. This equation implies, by contraction with q, that £ arg(vr) •
br = 0. Proceeding inductively, we prove that <p = 0.
The algebra Cl(V,q) is generated by the vector space V c CtyV,q) (and
the identity 1) subject to the relations:
v v=— q(v)l	(1.3)
for v e V. If the characteristic of к is not 2, then for all v,w e V,
v  w + w  v = — 2q(y,w)	(1.4)
where 2q(y,w) = q(v + w) — q(v) — q(w) is the polarization of q. The re-
lations (1.3) can be used to give the following universal characterization of
the algebra.
Proposition 1.1. Letf: V-* be a linear map into an associative k-algebra
with unit, such that
f(v)'f(v)=-q(v)l	(1.5)
for all ve V. Then f extends uniquely to a k-algebra homomorphism
f:Ct(V,q) -> st. Furthermore, Ct(F,q) is the unique associative k-algebra
with this property.
§1. CLIFFORD ALGEBRAS	*
Proof. Any linear map f: V -► extends to a unique algebra homo-
morphism f :^(V) stf. Property (1.5) implies that f = 0 on ^V),
and so f descends to Ct(V9q). Suppose now that is an associative
k-algebra with unit and that i: V is an embedding with the property
that any linear map f: V stf (stf as above) with property (1.5) extends
uniquely to an algebra homomorphism -► л/. Then the isomorphism
from V с: Ct(V,<?) to i(V) <= clearly induces an algebra isomorphism
Ct(V9q)^<%. 
This characterization of Clifford algebras is extremely useful. It shows,
for example, that they are functorial in the following sense. Given a
morphism f :(V9q) -► (V'9q')9 i.e., a к-linear map f: V-> V' between vector
spaces which preserves the quadratic forms (f*q' = q), there is, by Propo-
sition 1.1, an induced homomorphism f:CZ(V9q)-> CZ(V'9q'). Given
another morphism g'.(V'9q') -► (Vff9q")9 we see from the uniqueness in
Proposition 1.1, that g^f = g ° f.
A particular consequence of this is that the orthogonal group O(K,g) =
{f e GL(V): f*q = q} extends canonically to a group of automorphisms
of Ct(V,<j). We shall see later that this embedding
О(И,я) cz Aut(C£(V,<?))	(1.6)
actually lies in the subgroup of inner automorphisms.
An element here of particular importance is the automorphism
a:Ct(V9q)----> Cl(V9q)	(1.7)
which extends the map a(v) = — v on V. Since a2 = Id9 there is a decom-
position
C£(V,fl) = Cl°(V,4) © С£х(Р,0)	(1.8)
where Ct‘(K,g) = {(p e Ct(K,<?):a(<p) = (—l)fy} are the eigenspaces of a.
Clearly, since a(<Pi<p2) ~ a(<Pi) ’ a(<?2), we have that
CV(V9q) • Ctj(V9q) £ Cli+j(V9q)	(1.9)
where the indices are taken modulo 2. An algebra with a decomposition
(1.8) satisfying (1.9) is called a Z2-graded algebra. Note that Ct°(K,^) is a
subalgebra of Cl(V,<?). It is called the even part of Ct(K,</). The subspace
Ct^K,^) is called the odd part. It is an observation of Atiyah, Bott and
Shapiro that this Z2-grading plays an important role in the analysis and
application of Clifford algebras.
There exist some elementary and important relationships between the
Clifford algebra Ct(V,q) of a space and its exterior algebra Л* К (whose
definition is, of course, independent of the quadratic form q). There is a
natural filtration #° c cz #2 <= ... cz ^(V) of the tensor algebra,
10	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
which is defined by
&r = z
sgr
and has the property that ® #r' E &r+r'. If we set = nq(^1) we
obtain a filtration SF* c с 3F1 c ... c Ct(V,q) of the Clifford
algebra, which also has the property that
<^&r+r'	(1.10)
for all r,r'. This makes Cl(V,^) into a filtered algebra. It follows from
(1.10) that the multiplication map descends to a map	x
(^7^s-1)	(^r+s/^r+s'1) for all r,5. Setting $0* = фг^0 where
_ дрг[д?г~\ we obtain the associated graded algebra.
Proposition 1.2. For any quadratic form q, the associated graded algebra of
C£(V,<?) is naturally isomorphic to the exterior algebra A*V.
Proof. The map (X/ V -> ^r/^r~ \ which is given by vh ® • • • ®
vir [уй ’ ’ ’ viJ clearly descends to a map ArV -►	by property
(1.4). (When the characteristic of к is 2, we use the fact that • w +
w • v = 0.) This map is evidently surjective and is easily seen to give a
homomorphism of graded algebras A*V -► 30*.
To see that this map is injective we proceed as follows. The kernel of
(XfV -► consists of the r-homogeneous pieces of elements (peJq(V)
of degree r. Any such (p can be written as a finite sum <p = £ at ® (vf ®
vi +	where e Кand where we may assume that the a( and bt
are of pure degree with deg + deg b; ^r — 2. The r-homogeneous part
of (p is then of the form (pr — £ at ® ц- ® ® bt (where deg a, + deg bt =
r — 2 for each i). Since t>( л = 0 for all i, we see that the image of <p in
the exterior algebra is zero. Hence the map ArV -► <Sr is injective. 
Proposition 1.2 says that Clifford multiplication is an enhancement of
exterior multiplication which is determined by the form q. Note that
C€(V,0) Л* К
Proposition 1.3. There is a canonical vector space isomorphism
A*V-^Cl(V9q)	(1.11)
compatible with the fibrations.
Remark 1.4. The map (1.11) is, of course, not an isomorphism of al-
gebras unless q — 0. The point here is that the map is canonical. Thus we
may speak of the embeddings
ЛТсСЦ^) forallr^O.	(1.12)
§1. CLIFFORD ALGEBRAS	11
Proof. We define a map of the r-fold direct product f: V x • • • x V
Ct(V9q) by setting
f(vt,..., ur) = - £ sign(tf)t)a(1)  •  va(r)	(1.13)
r •	(7
where the sum is taken over the symmetric group on r elements. (If the
characteristic of к is not zero, one must drop the factor 1/r!.) Clearly f
determines a linear map /:ЛГИ -► Ct(V9q) whose image lies in SFr. The
composition of f with the projection SFr &'r!&'r~Y is easily seen to be
the map discussed in the proof of Proposition 1.2. Hence f is injective,
and the direct sum of these maps (1.11) is an isomorphism.
We now take up the question of tensor products. Recall that if and
St are algebras with unit over k, then the tensor product of the algebras
sS ® St is the algebra whose underlying vector space is the tensor product
of st and St and whose multiplication is given (on simple elements) by
the rule (a ® b) • (a' ® b') = (ad) ® (bbr). If, however,
г/ = г/° ф г/1 and St = St* ф St1
are Z2-graded algebras, then we can introduce a second “Z2-graded”
multiplication, determined by the rule
(a ® b) • (a' ® b') = (-	0	(1 14)
whenever b and a' are of pure degree (even or odd). The resulting algebra
is called the Z2-graded tensor product and is denoted л/ ® St.
The Z2-graded tensor product is again Z2-graded with
(jtf ® St)* = sd* ® St0 + rf1 ® Si1
(st ® St)1 = st1 ® St* 4- г/° ® St1.
It also carries a filtration 2F* cz cz SF2 cz... cz st ® St9 where
&r = £ &\st)® SF\St).
The importance of the Z2-graded tensor product for Clifford algebras
is evident from the following proposition.
Proposition 1.5. Let V = Vr Ф V2 be a q-orthogonal decomposition of the
vector space V (i.e.9 q(vL 4- v2) = Q(vi) + Q(Pi) fOr vi G К and v2 g
K2). Then there is a natural isomorphism of Clifford algebras
C^(V9q)-----► СЦУ^) ® Cl(V29q2)
where qt denotes the restriction ofqtoVi and where ® denotes the Z2-graded
tensor product.
12	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proof. Consider the map f : V -► Ct(V19qi) ® Ct(V29q2) given by f(v) =
t>i ® 1 + 1 ® v2 where v =	+ v2 is the decomposition of v with respect
to the splitting V = Vr Ф V2. From (1.14) and the ^-orthogonality of this
splitting we see that f(v) • f(v) = (t^ ® 1 + 1 ® v2)2 = vj ® 1 + 1 ®
v2 = “(<21(^1) + ^2(^2))! ® 1 = ~® 1- Hence, by Proposition 1.1,
f extends to an algebra homomorphism f:Ct(V9q) -►	®
CC(K2,^2). The image of f is a subalgebra which contains	® 1
and 1 ® Ct(V2,q2y Therefore, / is surjective. Injectivity follows easily by
considering a basis for CC(K,g) generated by a basis of V which is com-
patible with the splitting. 
We finish this section by introducing a second fundamental involution
on the algebra. The tensor algebra ^(V) has an involution, given on
simple elements by the reversal of order, i.e., ® • • • ® vr 1—► vr ®
• • • ® This map clearly preserves the ideal	and so descends to
a map
( r.Ct(V9q)----->Ct(V,q)	(1.15)
called the transpose. Note that ( )l is an antiautomorphism, i.e., (уф)' =
§2. The Groups Pin and Spin
We now consider the multiplicative group of units in the Clifford algebra,
which is defined to be the subset
Cex(K<7) = {феСг(К,^):Эф-1 with <p " x<p = <p<p ~1 = 1}	(2.1)
This group contains all elements v e V with q(v) =#= 0. When dim V =
n < 00, and к is either R or C, this is a Lie group of dimension 2". In
general, there is an associated Lie algebra c!x(K,g) = C£(K,g) with Lie
bracket given by
[x,y] = xy - yx.	(2.2)
The group of units always acts naturally as automorphisms of the
algebra. That is, there is a homomorphism
Ad: C€x(V,q)----> Aut(Cf(K,q))	(2.3)
called the adjoint representation, which is given by
Ad^(x) s q>x<p~1.	(2.4)
Taking the “derivative” of this gives a homomorphism
ad: cl x(V,q)---> Der(Ct(K,q))	(2.5)
§2. PIN AND SPIN
13
into the derivations of the algebra, defined by setting
ad/x) = [y,x].
Remark 2.1. Suppose Kis finite dimensional, and defined over 1R or C.
Then there is a natural exponential mapping exp:dx(K,g) -► CEx(K,g),
defined by setting
exp(v) = £ -^y".	(2.6)
Note that this series converges since for any choice of positive definite
inner product on Cl(V,q), we have ||xy||	c||x|| ||y|| for some c > 0. It is
easy to see that
Adexp(()1)(x)|,=0 = ady(x).	(2.7)
From this point on we shall assume that the characteristic of the field
к is different from 2. Under this assumption, we have the following
important facts concerning the adjoint representation:
Proposition 2.2. Let v e V cz Cl(V,q) be an element with q(v) 0. Then
Adv(F) = V. In fact, for all we V, the following equation holds:
-Ad» = w-2^v.	(2.8)
Proof, Since d"1 = — v/q(y), we have from (1.4) that
— ^(pjAd^w) = — q(v)vwv~r = vwv
— — v2w — 2q(v,w)v = q(v)w — 2q(v,w)v. 
This leads us naturally to consider the subgroup of elements (p e
Clx(K,g) such that Ad^(K) = V, By Proposition 2.2, this group contains
all elements v e К with q(v) =#= 0. Furthermore, we see from equation (2.8)
that whenever q(v) / 0, the transformation Adv preserves the quadratic
form q. That is, (AdJ^)(w) = ^(Adv(w)) = q(w) for all w e V. Therefore,
we define P(V,q) to be the subgroup of Clx (V,q) generated by the elements
v e V with qifi} 0, and observe that there is a representation
P(V,q) O(V,q)	(2.9)
where
O(V,q) = {Л e GL(K):	= q}	(2.10)
is the orthogonal group of the form q. The group P(K,q) has certain im-
portant subgroups.
14	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Definition 2.3 The Pin group of (K,q) is the subgroup Pin(K,q) of
P(K,g) generated by the elements ve V with q(v) = ± 1. The associated
Spin group of (V,q) is defined by
Spin(K,g) = Pin(K,<?) n Ct°(V9q).
We observe now that the right-hand side of equation (2.8) is just the
map pv:K-> V given by reflection across the hyperplane v1 = {w e
V: q(w9v) = 0}. That is, the map pv fixes this hyperplane and maps v to
— v. Unfortunately, there is a minus sign on the left in equation (2.8). This
means, for example, that if dim V is odd, then Ady is always orientation
preserving. This defect can be removed by considering the twisted adjoint
representation
Ad:C£x (K<?)----► GL(C£(K,g))
defined by setting
А<Цу) = а(<р)уф"!.	(2.11)
Clearly, Ad<pi<p2 = Advi ° лЗф2 and Ad^ = Ad^ for even elements q> (i.e.,
for <p g Ct°(V,q)). Furthermore, from (2.8) we have
Adv(w) = w - 2 —— v.	(2.12)
<?(”)
We then define the subgroup
IW e (pG Ct*(V,q): Ad^(K) = V}.	(2.13)
It is clear that P(K,g) cz P(K,g). Furthermore, we have the following.
Proposition 14. Suppose that V is finite dimensional and that q is nonde-
generate. Then the kernel of the homomorphism
P(y,q) gl(K)
is exactly the group k* of non-zero multiples of 1.
Proof, Choose a basis {vx, -.. .u„} for V such that qty^ 0 for allj and
q(vhVj) = 0 for all i =#= j. Suppose ф g Ct*(V,q) is in the kernel of Ad, that
is, suppose ф has the property that а(ф)г = vcp for all v g V. Write <p =
Фо + Фь where ф0 is even and (pr is odd, and observe that
-v<Pi = (pxv
for all v g V. The terms ф0 and <pY can be written as polynomial expres-
sions in Dp ... ,vn. Successive use of the fact (1.4) that v^j = — VjVj —
2g(Df,Dj) shows that ф0 can be expressed as ф0 = a0 + where a0 and
ar are polynomial expressions in d2, ... ,гл. Applying a shows that a0 is
§2. PIN AND SPIN
15
even and ax is odd. Setting v = in (2.14), we see that
+ vlai = “oyi +
=	“ ”i0i-
Hence, vja± = — ^(u1)a1 = 0, and so at = 0. This implies that <p0 does not
involve Dp Proceeding inductively, we see that <p0 does not involve any
of the terms Db ... 9vn and so <p0 = t • 1 for t g k.
The analogous argument can now be applied to <px. Write <Pi = fli 4-
d^o, where aQ and do not involve Note that is odd and a0
is even; and therefore, from (2.14), —	— vja0 = a1v1 4-	—
— v^a^ 4- Hence, aQ = 0 and so <pY is independent of dp By induc-
tion, <px is independent of . 9vn and so <px = 0.
Now we have <p = <p0 + (px = t * 1 g k. But cp 0, so <p g kx. 
Note that this proposition requires the twisted adjoint representation
and not the adjoint representation. The minus sign in (2.14) is crucial to
the proof.
Proposition 2.4 is false if we do not assume that q is non-degenerate.
To see this, consider the extreme case CC(K,0) = A*K. For all vl9v2 g V9
we have 1 4- vtv2 e Cf x(K,0). In fact, (1 4- r^)"1 = 1 ~ vir2* However,
for any v g V9 we see that a(l 4- dxd2)d(1 4- DiD2)-1 = U + rir2) ‘
d(1 — dxd2) = v. Hence, the kernel of the homomorphism includes many
non-scalar terms.
We now introduce the norm mapping N:Ct,(V9q) -> Ct(V,q) defined
by setting
N(<p) = <p • a(<p').	(2.15)
Here <pr denotes the transpose of cp introduced in (1.15). It is easy to see
that а(<рг) = (a(cp)y. Note that
N(d) = q(v) for v g V.	(2.16)
The importance of the norm is evident from the following proposition.
Proposition 2.5. Suppose that V is finite dimensional and that q is non-
degenerate. Then the restriction of N to the group P(K,g) gives a
homomorphism
N:P(V,q)------(2.17)
into the multiplicative group of non-zero multiples of the identity in CC(K,g).
Proof. To begin we observe that JV(P(K,g)) ckx. Choose cp g P(K,g) and
recall that by definition, а(<р)г<р-1 g V for all v g V. Applying the trans-
pose antiautomorphism, which is the identity on V9 we see that
(<p')_1ra(<p‘) = a(<p)c<p-1.
16	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Hence,
= а[а(ф')ф]и[а(ф’)ф]-1
= Ada^)v(t>) = v
for all v e К Hence, a(<p')<p is in the kernel of Ad. It is easy to check that
a(cpr) belongs to P(K,g), and therefore so does а(<рг)ф- Hence, by Proposi-
tion 2.4 we have a(<p')<p g кx. Applying a shows that ф'а(ф) = ek\
Since the transpose antiautomorphism preserves P(K,g), we conclude
that N((p) ekx for all <p g P(K,g).
We now observe that if g P(K,g), then N(<p<A) = ф(Да((/)а(ф‘) =
<pN(^)a(<pr) = <pa(<p')N(i/0 = N(cp)N(il/). Thus, N is a homomorphism on
p(r,q). 
Continuing to assume that dim V < oo and q is non-degenerate, we
have the following.
Corollary 2.6» The transformations Ad^: V -> V for <p g ₽(K,g) preserve
the quadratic form q. Hence, there is a homomorphism
Ad: P(V,q)----► O(V,q)	(2.18)
Proof. To begin we note that N(acp) = N(cp) for <p g P(K,g) since N(a<p) =
a(<p)<p* = aN(<p) = N(<p). Consequently, if we set
Vх = {veV:q(v)*0},	(2.19)
then for each v g Vx (cz we have N(Ad^(t>)) = N(a(<p)t><p !) =
/V(a<p)N(r)N((p) “1 = N(<p)N(<p)-1N(tO = N(r). Since N(<p) = q(p} for v g V
(cf. (2.16)), we see that Ad^ preserves all non-zero ^-lengths. Applying Ad^-1
now shows that Ad^(Kx) = Vх and so Ad^ leaves invariant the set of vec-
tors of zero ^-length. Thus, Ad^ is g-orthogonal. 
We now return to the group P(K,g) £ P(K,g) and observe that by
definition
P(K,g) = {vr • • • vr g C£(K,g):	... ,vr is a finite sequence from Vх}.
(2.20)
Recall that the twisted adjoint representation gives a homomorphism
Ad: P(V,q) -* O(V,q) such that
Ad^...^ = pVl ° • • • ° pVr	(2.21)
where
(2.22)
§2. PIN AND SPIN
17
is reflection across tA Thus the image of P(Kfl) under Ad is exactly the
group generated by reflections. It is an important and classical result that
this is always the entire orthogonal group.
Theorem 2.7 (Cartan-Dieudonne). Let q be a non-degenerate quadratic
form on a finite dimensional vector space V. Then every element g e O(V,q)
can be written as a product of r reflections
g =	° * * * ° PVr
where r dim(K).
We refer the reader to Artin’s book [1] for the general proof. In the
special case where V — R" and q(x) = ||x||2 is the standard norm, this
theorem is easily proved by putting the orthogonal matrix g in “diagonal”
form:
/s \
KI
+ 1
\ ±1/
where each R0i is a 2 x 2 rotation matrix (which can be expressed as a
product of two reflections).
Theorem 2.7 says that the homomorphism Ad: P(K,g) -► O(V\q) is sur-
jective. Furthermore, we could consider the group SP(K,g) = P(K,g) n
Cl°(V,q) and, since dim V is finite, the special orthogonal group
SO(K,g) = {2 e O(V.q): det(2) = 1}.
Theorem 2.7 also says that the homomorphism Ad:SP(K,g) -► SO(K,g)
is surjective. To see this, we first show that det(pv) = — 1 for any ve V,
To prove this, choose a basis . ,vn such that vx = v and q(v,Vj) = 0 for
j 2. It follows from the definition that p^i) = — and pv(vj) = Vj for
j 2, and so det (pv) = — 1 as claimed. Thus from Theorem 2.7 we conclude
that
SO(K,fl) = {pVl о • • • о pv \ q(Vj) ± 0 and r is even}. (2.23)
From the definition (cf. (2.20)) we see that SP(K,g) = {t^ • • • vr e P(V,q): r
is even}. The surjectivity of Xd:SP(K,g) -► SO(K,g) follows immediately
(see (2.21)).
18
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
We now return to the groups Pin and Spin. Recall that these are the
groups generated by the generalized unit sphere S = {re И: q(v) = ± 1}
in V. That is,
Pin(K^) = {vi • • • vr 6 P(P,fl): q(Vj) = ± 1 for all j} (2.24)
and
Spin(K,g) = {t?! • • • vr g Pin(K,g): r is even}. (2.25)
In light of the above it is natural to ask whether the homomorphism Ad
restricted to Pin(P,g) and Spin(K,g) maps onto O(K,g) and SO(K,g) respec-
tively. This seems quite likely since at a glance one can see that
Pw ~ Pv	(2.26)
for any non-zero scalar t ek, and so one should be able to renormalize
any ve Vх to have ^-length ± 1. Of course since q is quadratic, q(tv) =
t2q(v\ and the equation t2q(v) = ± 1, i.e., the equation t2 = ± a for a given
a, may or may not be solvable in a general field k. If к = 1R or C, of course,
it is always solvable. If к = Q (the rational numbers) it is very often not
solvable. (The group Q X/(Q x)2 is infinitely generated.) If к is a finite field
of characteristic #2, then kx/(kx)2 Z2 and —1 may or may not lie
in (kx)2. In the cases where Ad is not surjective, we still have the following
general fact, which is interesting because the group SO(K,g) is often almost
a simple group (see Artin [1]). (The reader interested only in the real and
complex cases can skip this proposition.)
Proposition 2.8. Each of the images Ad(Pin(K,g)) and Ad(Spin(K,g)) is a
normal subgroup ofO(V,q).
Proof. Recall (cf. (1.6)) that from the universal property of C£(K,g), the
action of O(V,q) on К extends to automorphisms of C£(K,g). It is easy to
see that these automorphisms commute with a. Suppose then that we
have v,w e V with q(v) 0 and choose geO(74 Then Ad^w) =
^(gv)w(gv)'1 = g(av)wg(v " *) = g(a(v)g ~ l(w)v “x) = g Ad„(g “1 w). Conse-
quently, we have that
Ad^ = £o аЗо,?-1	(2.27)
for all v g V with ^(v) 0 and for all g g O(V,q). The proposition now
follows immediately from (2.24), (2.25) and (2.27). 
We now come to the main result of this section. We are primarily in-
terested in the real and complex cases, so we shall focus on fields к that
have the property discussed above. We shall say that a field к of charac-
teristic ^2 is spin if at least one of the two equations t2 = a and t2 = — a
can be solved in к for each non-zero element a g к x. That is, к is spin if
kx = (kx)2 kJ (—(kx)2). The fields R, C and Zp for p a prime with p =
3(mod 4), are spin.
§2. PIN AND SPIN
Theorem 2.9. Let V be a finite-dimensional vector space over a spin field
k, and suppose q is a non-degenerate quadratic form on V. Then there are
short exact sequences
0----» F-----► Spin(K,q) SO(K,q)--------► 1	(2.28)
0-----------► Pin(K,g)-^* O(K,fl)------>1	(2.29)
where
f fz2 = {l.-l}
(Z4 = {± 1, ± V—T}	otherwise.
These sequences hold for general fields provided that SO(K,g) and O(V,q)
are replaced by appropriate normal subgroups of O(V,q).
Proof. Suppose <p = Di • • • vr g Pin(K,g) is in the kernel of Ad. Then (p g к x
by Proposition 2.4, and^so (p2 = N(<p) = N(dJ • • • N(vr) = ± 1. This es-
tablishes the kernel of Ad in both cases. The surjectivity of the homo-
morphisms follows from Theorem 2.7, the fact that pv = ptv, and the fact
that since к is spin, any ve Vх can be renormalized to have ^-length 1. 
It is interesting to observe that if к is a spin field, then either P(K,g) =
P(K,g) or P(K,q)/P(K,g) Z2. The proof (which the reader may skip) is
as follows. Since P(K,g) is generated by Vx, we know that t2q(y) e P(K,g)
for all t g kx and ve Vх. Since к is spin, this implies that P(K,g) contains
(kx)2 or —(kx)2 (and possibly more). In fact, if we set ко = {t g kx: t -1 g
P(K,g)}, then from the above and from the definition of a spin field, we
see that kx — k£ ( —к£). Thus, kx/ko = 0 or Z2. Now we have the
sequence
to £ P(M £ P(Kq) O(V,q)
where fcx_=ker(Ad) and where Ad(P(K,q)) = O(K,q). It follows that
O(V,q)s P(V,q)/k* s P(K<?)/ko • It then follows without difficulty that
P(P,<j)/P(P,<?) = kx/ko = 0 or Z2 as claimed.
We now examine the real case in some detail. Let V be an n-dimensional
vector space over R, and suppose q is a non-degenerate quadratic form
on V. Then we may choose a basis for V s R" so that
q(x) = xf + ... + x* - x*+1 - ... - xj+,	(2.30)
where r + s = n and 0 < r < n. It is standard notation to write: qr<s =
q, O,., = О(К^) and SO,., = SO(K<j)- In accordance we write
Pin,., = Pin(Kq) and Spinr>s = Spin(K,q). (2.31)
Similarly, it is conventional to write O„ = OBi0 S O0>B and SOB = SOBi0
SO0>B. Thus, we set
Pin„ s PinB>0 and Spin. s Spin„,0.	(2.32)
*.	1 ЧУ1Ч1У rtLULDKAi AND SPIN GROUPS
We also write Pns = Р(И,<?) and Prs = Р(К,<Д and note from the para-
graph above that
Pr,s = P.s	(2.33)
It is a classical fact (cf. Helgason [1]) that SOn is connected and that
SOrs, for r,s 1, has exactly two connected components. It is also a
classical fact that tiJSOJ Z2 for n 3 and	s rcJSOj.) x
MSOJ for all r,s. Hence, tiJSO?^) = tiJSO^) = Z2 and ^(SO^) =
Z2 x Z2 for all r,s 3. (Here SO®>s denotes the connected component
of the identity.)
The main result of this section is the following.
Theorem 2.10. There are short exact sequences
0----► 2 2----► Spinr>5--► SOr,s------►	1
0----►	► Pinr,s--► Ors------► 1
for all (r,s). Furthermore, if(r,s) (1,1), these two-sheeted coverings are non-
trivial over each component of Or s. In particular, in the special case
0-----> Z2----> Spin„ SO„---------> 1	(2.34)
the map £0 = Ad represents the universal covering homomorphism of SO„
for all n 3.
Proof. The exact sequences are a direct consequence of Theorem 2.9.
The kernel in each case is explicitly given by Z2 = {1, —1}. To prove
that the coverings are non-trivial, it suffices to join — 1 to 1 by a path in
Spinr>s. Choose orthogonal vectors e19e2 e R" with ^(ej = q(e2) = ±1.
(This is possible since (r,s) (1,1).) Then y(t) = +cos(2f) + e 1e2sin(20 =
(cjcos t + e2sin t)(e2sin t — ejcos t) does the job. 
The above argument also shows that restricting Ad to the identity
component Spin® 2 of Spinr l gives the universal covering homomorphism
0----> Z2----> Spin®! -U SOr0t----> 1	(2.35)
for all r 3.
§3. The Algebras C£M and Ctr ж
We shall now study the Clifford algebras Ctr>s = Ct(V,q) where V = Rr+s
and
q(x) = X2! + ... + x2 - x2+1 - ... - x2+s.	(3.1)
Of particular interest are the cases
Cln = С1пл and C£* s Cf0 „.	(3.2)
§3. THE ALGEBRAS С£и AND C£r.a	21
Dne reason for studying these algebras is the following. As seen in §2, the
dgebra C£r>s contains the groups Spinr s and Pinrs, and so any representa-
:ion of the algebra Ctr s restricts to a representation of these groups which
s non-trivial on the element — 1. (Such representations are therefore not
nduced from representations of Or>J or SOr>s.)
These algebras have a simple classical presentation:
Proposition 3.1. Let el9.,. 9er+s be any q-orthonormal basis of Rr+s c
Ctrs. Then Ctr s is generated (as an algebra) by e19... 9er+s subject to the
relations
e^j + ejCi =
-2ди
+ 2<5O
if i r
if i > r.
(33)
Proof, This follows easily from the discussion in §1. 
We also have a pretty decomposition in terms of the Z2-graded tensor
product.
Proposition 3.2. There is an isomorphism
Ctr,s = Clr ® • • • ® C£r ® C£* ® • ” ®	(3.4)
where appears r times and Ctf appears s times on the right in (3.4).
Proof, Decompose Rr+s into one-dimensional fl-orthogonal subspaces
and apply Proposition 1.5 inductively. 
It is not difficult to see that as algebras over R,
C and Ct? s R © R.	(3.5)
It follows immediately that dimR(Ctr>s) = 2r+s. Proposition 3.2 is, how-
ever, not so useful if we wish to represent Ctr>s as a matrix algebra. For
this it is more useful to find decompositions in terms of ungraded tensor
products. We shall do this in the next section.
For the remainder of this section we shall examine some of the general
properties of the algebras C£r,s. We begin with a discussion of the volume
element. Choose an orientation for Rr+s and let ex,... ,er+s be any posi-
tively-oriented, ^-orthonormal basis. Then the associated (oriented) vol-
ume element is defined to be
co =	• • • er+s	(3.6)
If e'19,.. 9e'r+s is any other such basis, then e- = Б for g = ((ffo)) e
SOr>s. From (3.3) we easily see that e\- - e'r+s = det^ej • • • er+, =
gj • • • er+s. Hence the definition (3.6) is independent of the choice of the
basis.
22	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proposition 3.3. The volume element (3.6) in Ctr s has the following basic
properties. Let n = r + s. Then
n(n+l)+s
<o2 = (-l) 2	>	(3.7)
vco = ( — 1)"” 1cdv for all v g R",	(3.8)
In particular, if n is odd, then the element cd is central in Ctrs. If n is even,
then
cpcD = caa(cp)	(3.9)
for all cp g Ctrs.
Proof. Choose a ^-orthonormal basis and apply the relations (3.3). 
We note that property (3.7) can be rewritten as
2=f(“1)s if n = 3 or 4 (mod 4)
W [(—l)s+1 ifn = 1 or 2 (mod 4)
We now make the following elementary but important observation.
Lemma 3.4. Suppose the volume element cd in satisfies cd2 = 1, and set
я+=|(1+со) and я”=|(1 —co).	(3.10)
Then and n~ satisfy the relations
n+ + n~ = 1	(3.11)
(я+)2 = я +	and (я-)2 = тГ	(3.12)
n+n~ = n~n+ = 0.	(3.13)
Proof. This is a trivial consequence of the fact that cd2 = 1. 
This leads to two basic but important facts:
Proposition 3.5. Suppose that the volume element cd in Clr s satisfies cd2 — 1,
and that r + s is odd. Then Ctr s can be decomposed as a direct sum
Clr,s = Cl*s ® Ct"	(3.14)
of isomorphic subalgebras, where Ct±s = n± • Ctr s = Ctr s • n± and where
a(C£*) = Ct*.
Proof. Since r + s is odd, we know from Proposition 3.3 that cd is central.
Hence 7t+ and я" are central and the decomposition (3.14) into ideals
§3. THE ALGEBRAS С£и AND C£r>s	23
bllows directly from (3.11), (3.12) and (3.13). Since co is an odd element,
1(71*) = 7iT and so a(Ct*s) = Ct*s. Since a is an automorphism, we con-
clude that these two ideals are isomorphic. 
Proposition 3.6. Suppose that the volume element co in Ctr s satisfies co2 = 1
2nd that r + s is even. Let V be any Ctr s-module (i.e., V is a real vector
space with an algebra homomorphism Clr s -► Hom(K, K)). Then there is
2 decomposition
V=v+ ®V~	(3.15)
into the 4-1 and — 1 eigenspaces for multiplication by cd. In fact,
V+= 7t+- V and V~ = 7t~ • V,
and for any e e Rr+s with q(e) 0, module multiplication by e gives
isomorphisms
e: V+------> V~ and e: V~----------------> V+.	(3.16)
Proof. The decomposition (3.15) is a direct consequence of (3.11), (3.12)
and (3.13), together with the observation that
co  71* = + 71±.
The isomorphisms (3.16) follow directly from the facts that by (3.8),
ея + = | e(l + co) = ~ (1 — co)e = n e
e7i = 7i+e
к
and e • e = — q(e) • 1. 
Remark. The above construction will prove useful when we are dealing
with vector bundles in the next chapter.
We now come to an important and basic fact. Recall the even-odd
decomposition C£rs = Ct°s ф C£r\s given in (1.8), where the subalgebra
C£°s is the fixed-point set of the automorphism a.
Theorem 3.7. There is an algebra isomorphism
cer,^c£r°+l,s
for all r,s. In particular,
Ct„^C£°+l
(3-17)
(3.18)
for all n.
24	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proof. Choose a q-orthonormal basis elf..., er+s+1 of Rr+s+1 so that
q(ef) =1 for l^i^r + 1 and q(ef) = — 1 for i > r + 1. Let Rr+I =
span{e(|i r + 1} and define a map f : Rr+s -» C£°+1>s by setting
f(ei) = er+1ei
for i r + 1, and extending linearly. For x =	x^,-, we have that:
f(x)2 = xlxJ-er+1e/er+1e;
и
= Zx>xjeiej
ij
= x  x = — q(x) • 1
since er+l  er+1 = — 1 and er+ tet = — eter+1 for i r + 1. It follows from
the universal property (Proposition 1.1) that f extends to an algebra
homomorphism
/:C£r>s----»C£r°+M.
Checking f on a linear basis shows that f is an isomorphism. 
We now specialize to the case of Ct„.
Proposition 3.8. Let L: C£„ -♦ CE„ be the linear map defined by setting
Ц<р)=-^^<ре}	(3.19)
J
where e19... ,en is any orthonormal basis of R". Set L = a о L. Then the
eigenspaces of L are the canonical images of Лр = APR" in Ctn. In fact
L\ap = (n - 2p)Id	(3.20)
for p = 0,... ,n.
Proof. It suffices to consider (p = er • ” ep. Then
P	и
U<p) = - L Cjei   • ePej - £ e/! • • • ере?
J=1
= - f (-l)”_1e?ei "ep~ £ (~^)pe]e1  • • ep
j=i	j=p+i
= (-lf1pe1 •••ep + (-l)p(n-p)e1 --ep
= (-1)₽(« - 2p)et • • • e„ = (n - 2p)a(<p) 
Under the canonical isomorphism Cln s A*R", Clifford multiplication
has a nice interpretation. Using the inner product on R" we can identify
R" with its dual. We can thereby talk about the interior product or con-
traction in A* R". For v e R", this is a linear map (v L): APR" -► Ap" 1[Rn
§4. THE CLASSIFICATION
25
;iven on simple vectors by
V A ... A Vp) = £ ( — l)i+ 1^Vb v)vt A ... A Pf A ... A V (3.21)
i= 1
vhere (•) indicates deletion. This gives a skew-derivation of the algebra,
.e., v L((p л ф) = (v l (p) л ф + (— l)p<p л (v L ф) for any (p g APR". It is not
iifficult to see that v l(p l) = 0 for any v g R". Hence, by universality the
nterior product extends to all elements of A*R", i.e., to a bilinear map
VR" x A*R" -► A*R".
Proposition 3.9. With respect to the canonical isomorphism Ctn A*R",
Jlifford multiplication between v g R" and any (p g Ctn can be written as
V ’ (p = v л (p — PL (p	(3.22)
*roof. Choose an orthonormal basis e19..., en for R" with v = tej for
ome t g	R. Let	(p = eit • • • eip for	< • •	• < ip. Then
v	= f-tei2”’4 =	(y л	~vL)(p	if	4	=	1
[te^ • ’ • eip	(v л	- v L)(p	if	>	1.
iince (3.22) holds on an additive basis of Ctn9 it	holds	in	general. 
§4. The Classification
n this section we shall give an explicit description of the algebras Ctr>s as
natrix algebras over R, C, or H (= quaternions). With little difficulty the
eader can check the first few cases:
Ct1>0 = C	C£0,i = R © R
C£2>0 = H	Cl0,2 = R(2)	(4.0)
С11Л = R(2)
vhere R(2) denotes the algebra of 2 x 2 real matrices.
The key facts to the classification are the following:
rheorem 4.1. There are isomorphisms
C£n,o ® Ct0>2 = Ct0 n + 2	(4.1)
С^О.л ® C£2,o = C£n+2,o	(4.2)
С£гл ®	= C£r+1>s+i	(4.3)
or all n9r,s 0.
>tote that here we are using the ungraded tensor product.
26
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proof. Let e19... ,en+2 be an orthonormal basis for R"+2 in the standard
inner product, and let q(x) = — ||x||2. Let e'19... 9e'n denote standard genera-
tors for C£n>0 and let еъе'2 denote standard generators for C£o 2 (in the sense
of Proposition 3.1). Define a map f:Rn+2^> Ctn 0 ® Ct0 2 by setting
e\ ® e'/ei'
1 ® 4'_„
/(*,) =
for 1 i n
for i = n + 1, n + 2
and extending linearly. Note that for 1 < ij < n9 we have /(ef)/(e;) +
/(^Ift^i) = (Ф; + ejei) ® (~ 1) = 2<5i;l ® 1; and for n + 1 <; a,/J n + 2
we have f(ejf(efi) + f(efi)f(ej = 1 ® Щ_пё^п + ё^.п) = 2^1 ® 1.
Also we see that /(е£)/(ев) + /(ee)/(^) = 0. It follows that /(x)/(x) =
||x||21 ® 1 for all x e R"*2. Hence, by the universal property (Proposition
1.1),/ extends to an algebra homomorphism f :С£0>иЧ.2 -► С£я>0 ® C£o>2.
Since f maps onto a set of generators for Ctn 0 ® C£o>2, it must be
surjective. Then, since dim С£0>я+2 — dim С£я>0 ® C£o>2, we conclude
that / must be an isomorphism. This proves (4.1). The proof of (4.2) is
entirely analogous.
For (4.3) we proceed in a similar manner. We choose a ^-orthogonal
basis el9... ,er+l9el9 ... ,£s+1 for Rr+s+2 such that q(ei) = 1 and q(£j) =
— 1 for all ij. We then let e'l9..., e',£'b ... ,£' and be corresponding
bases for Rr+s and R2, and we define a map/: Rr+s+2 -► Ctr>s ® С11Д
by setting
for 1 < i <. r
for i = r + 1,
and
for 1 < j < s
for j = s + 1,
and then extending linearly. We now apply Proposition 1.1 and complete
the argument as in the previous cases. 
To apply this basic proposition we shall need the following elementary
facts concerning the tensor products of algebras over R. For К = R, C
or H, we denote by K(n) the algebra of n x n-matrices with entries in K.
Proposition 4.2.
R(n) ® R(m) s R(n/n) for all n9m.	(4.4)
R(n) К = K(n) for К = C or H and for all n. (4.5)
C ®R С s C © C	(4.6)
§4. THE CLASSIFICATION
C ®R HI = C(2)
H ®R H = R(4).
(4.7)
(4.8)
Proof. The isomorphisms (4.4) and (4.5) are obvious. The isomorphism
C©C->C®Cis determined by sending
(1,0)---->i(l ® 1 + i® i),
(0,1)---->1(1 ® 1-i® i).
For the isomorphism (4.7) we consider HI as a C-module under left scalar
multiplication, and we define an R-bilinear map Ф : С x H -> Homc(HI,HI)
by setting Ф2,ч(х) = zxq. This extends (by the universal property of ®) to
an R-linear map Ф:С ®R H -► Homc(Hl,Hl) = C(2). Since Фг></ о ф^, =
Фгг .ад', we have that Ф is an algebra homomorphism. Checking Ф on a
natural basis shows that it is injective. Hence, since dimR(C ®R HI) =
dimR(C(2)), Ф is an isomorphism.
The isomorphism (4.8) is proved similarly. Consider the R-bilinear map
S': H x HI -> HomR(Hl,Hl) R(4) given by setting Т,1Л2(х) = qixq2- The
resulting R-linear map T: Hl ®R Hl -► HomR(H,Hl) is an algebra homo-
morphism between algebras of the same dimension. The injectivity of Ф
can be checked on a natural basis for Hl ® H. 
We now come to the first main result of the section. Before stating the
result, we make the observation that for any (r,s), the complexification of
the algebra Clr s is just the Clifford algebra (over C) corresponding to the
complexified quadratic form, i.e„ C£r s ®R C C£(Cr+s, q ® C). (This fol-
lows easily from Proposition 1.1.) However, all non-degenerate quadratic
forms on C" are equivalent over C£„(C). Hence, setting
^c(z)= E 2j
J=1
and defining
C£n Ct(C",qc),	(4.9)
we have that
C£n s C£n,0 ®R С С£Я_1Д ®R C £ • • • = C£On ®R C (4.10)
Theorem 4.3. For all n 0, there are “periodicity” isomorphisms
C^n + 8,o = о ® C£8 о	(4.11)
С^о,л+8 = С10,л ® C%,8	(4.12)
C£„+2^C£n®cC£2	(4.13)
28	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
where
C£8,o = C£o,8 = R(16)	(4.14)
C£2 = C(2).	(4.15)
Therefore, by using the identities (4.4) and (4.5), all the algebras Ctn Q, CtQn
and C£n can be easily deduced from the following table.
Table I
	i	2	3	4	5	6	7	8
	с	н	н ® н	Н(2)	С(4)	ад	R(8) ф R(8)	R(16)
С£Ол		ад	С(2)	Н(2)	Н(2) ф Н(2)	Н(4)	С(8)	R(16)
ее.	СФС	С(2)	С(2) ф С(2)	С(4)	С(4) ф С(4)	С(8)	С(8) ф С(8)	С(16)
Proof. From (4.1) and (4.2) we see that for any n, we have C£n+8 0
С£и0 ® C£02 ® C£2,o ® C£o>2 ® C£2i0. Using (4.0) and Proposition
4.2 we see that С£и+8>0 С£и>0 ® И ®	® R(2) ® R(2) С£и 0 ®
R(4) ® R(4) Ctn 0 ® R(16). This establishes (4.11). The periodicity (4.12)
is proved similarly. To prove (4.13), note from (4.10) that Ctn+2
С£и+2,о ® С = C£n>0 ® C£o>2 ® C s C£n ®c C£2.
Using the isomorphisms (4.1) and (4.2), and the facts (4.4) to (4.8), one
can work out the first two rows of the table in “criss-cross” fashion (start-
ing with the initial data (4.0)). The third row of the table now follows by
taking the tensor product of corresponding terms in either of the first two
rows with C. 
Combining Table I with the fundamental periodicity isomorphism (4.3)
and the fact that C£t t s R(2), we achieve the complete classification in
Table II.
By now the reader has probably noticed some of the intrinsic beauty
of this constellation of algebras and its interrelationships. There are some
observations one can make from the table that are interesting exercises
to prove. For example,
C£r s C£r_4 s+4	(4.16)
C£r s+1 s C£s r+1 (symmetry about the axis у = x 4- 1). (4.17)
Remark. The above classification reduces the Clifford algebras to fa-
miliar matrix algebras over К — R, C or HL Of course it is also useful to
think of this result as introducing hidden and unexpected structure in the
algebras K(2W). This information can be quite interesting as we shall see
when we discuss vector fields on spheres in §8.
Table II. CCr s in the box (r,s)
8	R(16)	0(16)	H(16)	Н(16) ф H(16)	0-0(32)	C(64)	R(128)	R(128) © R(128)	R(256)
7	C(8)	0-0(8)	H(8) ф H(8)	H(16)	C(32)	R(64)	R(64) © R(64)	R(128)	C(128)
6	0-0(4)	0-0(4) © 0-0(4)	0-0(8)	C(16)	R(32)	R(32) © R(32)	R(64)	<C(64)	0-0(64)
5	0-0(2) © 0-0(2)	0-0(4)	C(8)	R(16)	R(i6) ® R(i6)	R(32)	C(32)	H(32)	0-0(32) © 0-0(32)
4	0-0(2)	C(4)	R(8)	R(8) ф R(8)	R(16)	C(16)	0-0(16)	0-0(16) © 0-0(16)	0-0(32)
3	C(2)	R(4)	R(4) © R(4)	R(8)	Q8)	H(8)	H(8) ф H(8)	0-0(16)	C(32)
2	R(2)	R(2) ф R(2)	R(4)	C(4)	H(4)	Н(4) ф H(4)	0-0(8)	C(16)	R(32)
1	r © r	R(2)	C(2)	H(2)	H(2) © H(2)	H(4)	C(8)	R(16)	R(16)© R(16)
0	R	c	H	н © и	H(2)	C(4)	R(8)	R(8) © R(8)	R(16)
	0	1	2	3	4	5	6	7	8
§4. THE CLASSIFICATION
30	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
§5. Representations
Most of the important applications of Clifford algebras come through a
detailed understanding of their representations. This understanding fol-
lows rather easily from the classification given in §4.
We begin with a general definition. Let К be a vector space over a
field к and let q be a quadratic form on V.
Definition 5.1. Let К 2 к be a field containing k. Then a X-representa-
tion of the Clifford algebra Ct(V9q) is a k-algebra homomorphism
p:Ct(V9q)-----► HomK(FT,FT)
into the algebra of linear transformations of a finite dimensional vector
space W over K. The space W is called a C£(K,g)-module over K. We
shall often simplify notation by writing
р(ф)(и>) = <p • w	(5.1)
for (p g Ct(V9q) and w e W9 when no confusion is likely to occur. The
product (p • w in (5.1) is often referred to as Clifford multiplication.
Note. By a k-algebra homomorphism we mean a к-linear map p which
satisfies the property р((рф) = p((p) ° р(ф) for all (р9ф e CZ(V9q).
We shall be interested in K-representations of C£r s where К = R, C or
HI. Note that a complex vector space is just a real vector space W to-
gether with a real linear map J: W -► W such that J2 = — Id. A complex
representation of C£r>s is just a real representation p: C£r>s -► HomK(iy,IF)
such that
p(<p) ° J = J ° p(<p)	(5.2)
for all (p g Clrs. Thus the image of p commutes with the subalgebra
span{Id,J} C. (This algebra is called a “commuting subalgebra” for p.)
Strictly analogous remarks apply to quaternionic representations of
C£r s. Here the real vector space W carries three real linear transfor-
mations I9 J and К such that
I2=J2 = K2 = -Id
IJ=-JI=K9 JK=-KJ = I9 K1=~IK = J.
This makes W into an HI-module. A representation p:C£rs->
Homin',W) is quaternionic if
р(ф) О I = I О р(ф), р(ф) о J = J О р(ф), р(ф) О К = К О р(ф)
(5.3)
for all <р e C£r s. That is, p has a commuting subalgebra spanR{Id,/,J,K}
isomorphic to H.
§5. REPRESENTATIONS
31
Remark 5.2. Any complex representation of C£r>s automatically ex-
tends to a representation of Ctr s C s C£r+s. Any quaternionic repre-
sentation of C£r>s is automatically complex (by restricting to С с Hl). Of
course the complex dimension of any HI-module is even.
The above remarks will prove useful when we carry these constructions
over to vector bundles in Chapter II.
We now come to the notion of irreducibility.
Definition 5.3. Let V,q,k K, be as in definition 5.1. A K-represen-
tation p:Cl(V,q) -► HomK(W,jy) will be said to be reducible if the vector
space W can be written as a non-trivial direct sum (over K).
W = Wx ф W2
such that р(<р)(Иу) £ for j = 1,2 and for all (p e Ct(V,q). Note that in
this case we can write
P = Pi ® Pi
where = p((p)\Wj for j = 1,2. A representation is called irreducible if
it is not reducible.
It is more conventional to call a representation “irreducible” if it has
the property that there are no proper invariant subspaces. However, since
C£n is the algebra of a finite group (see the discussion following Proposi-
tion 5.15), the two concepts are easily seen to agree in this case.
Proposition 5.4. Every К-representation p of a Clifford algebra C£(K,q)
can be decomposed into a direct sum p = pv ф • • • ф pm of irreducible rep-
resentations.
Proof. If p is reducible, it can be decomposed as a direct sum p = рх ф p2.
If either pr or p2 is reducible, p can be further decomposed. This process
must stop because of the finite dimensionality of the module. 
We shall be interested here, of course, only in equivalence classes of
representations.
Definition 5.5. Two representations pj:Ct(V,q) -► Нотк(И<,И5) for
j = 1,2 are said to be equivalent if there exists a К-linear isomorphism
F: Wr -► W2 such that F ° Pi((p) ° F~l = p2((p) for all (p e C£(K,q).
From §4 we know that every algebra C£r s is of the form K(2m) or
K(2m) ф K(2m) for К = R, C or HI. The representation theory of such
algebras is particularly simple.
Theorem 5.6. Let К = R, C or HI, and consider the ring K(n) of n x n
К-matrices as an algebra over R. Then the natural representation p of K(n)
on the vector space Kn is, up to equivalence, the only irreducible real repre-
sentation of K(n).
32	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
The algebra K(n) ф K(n) has exactly two equivalence classes of irre-
ducible real representations. They are given by
Pi(<Pi,<p2) = P(<Pi) and p2(<Pi,<p2) = P(<Pz)
acting on Kn.
Proof. This follows from the classical fact that the algebras K(n) are sim-
ple and that simple algebras have only one irreducible representation up
to equivalence. See Lang [1]. 
From the classification of §4 (see Table II) we immediately conclude
the following:
Theorem 5.7. Let vr s denote the number of inequivalent irreducible real rep-
resentations of C£r>s, and let v„ denote the number of inequivalent irre-
ducible complex representatons of C£„. Then
_ (2 if r 4- 1 — s = 0 (mod 4)
r’s (1 otherwise
and
r [2 if n is odd
= <
"	[1 if n is even.
This is a good time to recall (cf. Theorem 3.7) that there are isomorphisms
ar,^ar°+liI	(5.4)
for all z\s, and consequently
Ct„^C£„°+1	(5.5)
for all n. Since
Spinr>s c C£r°s c C£r°+s	(5.6)
we see that it is the irreducible representations of C£r_ls and C£r+s-1
that are relevant to constructing irreducible real and complex representa-
tions of Spinr s.
From this point on we shall restrict our attention to the algebras
Cln = C£„ 0 (and С£и = Cln ®K C) in order to simplify the exposition.
Corresponding facts for the general case Clr s are easy to deduce if the
reader is interested. We shall begin with a summary of information easily
deduced from the classification theorem 4.3.
We begin with some definitions. For each n, let dn = dimK(IF) where
W is an irreducible R-module for C£„. Similarly, let d„ = dimJW") where
W is an irreducible complex module for C£„ (and therefore for C£„ =
§5. REPRESENTATIONS
33
C£„ ®R C). Let Kn = R, C or IH1 denote the maximal commuting subal-
gebra for an irreducible real representation of C£„. Thus if Kn — C, this
representation is automatically complex. If Kn = H, it is automatically
quaternionic. (Note that in the cases where C£n has two distinct irreduc-
ible representations, dn9 d„ and Kn are the same for both.)
An object which will be of interest later on is the following. Let (or
denote the Grothendieck group of equivalence classes of irreducible
real (respectively, complex) representations of C£n. This is merely the free
abelian group generated by the distinct irreducible representations over
R (or C). Since any representation can be decomposed into irreducibles,
it naturally corresponds to an element in this group (with positive
coefficients).
Theorem 5.8. For 1 < n <, 8, the elements vn = v„ 0, v£,	d„> Knt
and (defined above) are as given in Table III.
Table III
n	ce.	V*	dK	K,	SR,	Ct,		£	
1	c	1	2	c	Z	С ф c	2	1	z ® z
2	H	1	4	H	z	C(2)	1	2	z
3	H ф H	2	4	H	z ® z	C(2) ф C(2)	2	2	z ® z
4	H(2)	1	8	H	z	C(4)	1	4	z
5	C(4)	1	8	c	z	C(4) ф C(4)	2	4	z ® z
6	R(8)	1	8	R	z	ОД	1	8	z
7	R(8) ф (R(8)	2	8	R	z ® z	C(8) ф C(8)	2	8	z ® z
8	(R(16)	1	16	R	z	C(16)	1	16	z
For n > 8 these elements can be computed from the following facts, which
hold for all m9k 1.
+ 8k	Vm + 2k =	(5.7)
da + ak = 24kdm	£ + 2* = 24m	(5.8)
Wm + fik=Wm		(5.9)
^m + 8k		(5.10)
Proof. This is a direct consequence of Theorem 4.3. 
We shall now consider the key role played by the volume element in
determining irreducible representations. Recall from §3 that the volume
element in C£n is defined as
cw =	• • • еи
(5.П)
34	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
where el9... ,en is an orthonormal basis of R". It is well defined up to
sign and is fixed after a choice of orientation on R". In the complex case
we have a corresponding element cdc g C£„ given by
coc = fl- 2 co,	(5.12)
called the complex volume element Note that when n = 2m, we have
сус = ime, • • • e2m
(Note also that coc — cd only in dimensions seven and eight modulo 8.
Other conventions for a complex volume element are possible. This one
is particularly useful in studying elliptic operators.)
Recall from Proposition 3.3 that if n is odd, then co and coc are central.
Furthermore, by (3.7)' we have that
co2 = 1	if n = 3 or 4 (mod 4),	(5.13)
(coc)2 = 1 for all n-	(5.14)
Thus there are algebra decompositions
C£„ = С£и+ ф C£;	for n = 3 (mod 4)	(5.15)
C£n = С£л+ ф C£'	for n odd	(5.16)
where
C£* = (1 ± co)C£n and C£* = (1 ± сус)С£и. (5.17)
(see Proposition 3.5). These decompositions correspond to the ones given
in Table III.
Proposition 5.9. Let p : Cln -► HomK(IT, W) be any irreducible real repre-
sentation where n = 4m 4- 3. Then either
p(co) = Id or p(co) = — Id.
Both possibilities can occur, and the corresponding representations are in-
equivalent. (They represent the two generators о/ЭИи.)
The analogous statements are true in the complex case for C£„, n odd.
Proof. Since p(co)2 = p(co2) = Id, we can decompose W into W =
ф W~ where W* and W~ are the +1 and — 1 eigenspaces for p(co)
respectively. Since cd is central, the spaces W+ and W~ are C£n-invariant.
By irreducibility either W* = W or W~ — W. This proves the first
statement.
The inequivalence of representations p + and p_ with p±(co) = ±Id
is evident, since if F: W -► W is an isomorphism and if p(co): W W is
a scalar multiple of Id, then F ° p(co)« F~l is the same scalar multiple of
Id.
§5. REPRESENTATIONS
35
To see that both possibilities exist we take irreducible factors of C£„
acting on Ct* and on C£~ by multiplication from the left.
The complex case is proved in the analogous manner by using coc. 
Proposition 5.10. Let p:Cln -► Нотк(РУ,1У) be an irreducible real repre-
sentation where n = 4m, and consider the splitting
w=w+ ©w~
where W± = (1 ± p(co)) • W (as in Proposition 3.6). Then each of the sub-
spaces W* and W~ is invariant under the even subalgebra C£°. Under
the isomorphism (3.18) C£° = these spaces correspond to the two
distinct irreducible real representations of
The analogous statements are true in the complex case for n even.
Proof. The invariance of W+ and W~ under C£° is evident from the fact
that co commutes with everything in C£° (see (3.9)). Under the isomor-
phism Cln_t Д C£° given in (3.18), we see that the volume element co' =
ex • • • en,1 of С1п-! goes to the volume element co e C£°. (To see this
note that (e}e„) • • • (e„_ 1S„) = (—1)2<"“	1 =j4 • • • en
since n = 4m.) It follows that co' = Id on and co’ — Id on W . Hence,
by Proposition 5.9 these representations of C£n_r are inequivalent.
The complex case is proved in the same manner using the volume form
coc for C£„. 
The representations of the algebras C£n give rise to important represen-
tations of certain groups.
Consider the spin group
Spin,, с C£? c Cln.	(5.18)
Definition 5.11. The real spinor representation of Spin,, is the homo-
morphism
Ди: Spin„ —► GL(S)
given by restricting an irreducible real representation C£„ -► HomK(S,S)
to Spin,, c Cf° c= C£„.
Proposition 5.12. When n = 3 (mod 4) this definition of Д„ is independent
of which irreducible representation of C£n is used. For n ф 0 (mod 4) the
representation Д„ is either irreducible or a direct sum of two equivalent irre-
ducible representations. (The second possibility occurs exactly when n = 1
or 2 (mod 8).) In the other cases there is a decomposition
Д4т = Д4т ® Д4т	(5.19)
where Д4т and Д4т are inequivalent irreducible representations o/Spin4m.
36	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proof. Recall that if n = 3 (mod 4), then the automorphism a: C£„ -♦ Ct„
interchanges the factors Ct* and Ctn- (since a(co) = -co). Consequently,
Ct° sits diagonally in the decomposition Ctn = Ct; ф Ct;, i.e.,
Ct® = {(<p, a(<p)) e Ctn+ ф Ct; : <p e Ct;}	(5.20)
The two irreducible representations of C£„ differ by the automorphism
a, and are clearly equivalent when restricted to C£°. This proves the first
statement of the proposition.
It is evident from Table III that the restriction of an irreducible real
representation of C£n to C£° = Cln_r is still irreducible if n = 3, 5, 6, or
7 (mod 8), and must be two copies of an irreducible representation when
n = 1 or 2 (mod 8). When n = 0 (mod 4), we know from Proposition 5.10
that the restriction to Cl® splits into two inequivalent irreducible repre-
sentations. To complete the proof we observe that any irreducible repre-
sentation of Cl® restricts to an irreducible representation of Spin„ because
Spin„ contains an additive basis for Cl®. 
Remark 5.13. Note that the spin representations are complex for n = 2
or 6 (mod 8), and are quaternionic for n = 3,4 or 5 (mod 8). (The maximal
commuting algebra is determined by С1п_г Cf°.)
The analysis above carries over to the complex case.
Definition 5.14. The complex spin representation of Spin„ is the homo-
morphism
A„c:Spinn—>GLC(S)
given by restricting an irreducible complex representation C£„ -►
Homc(S,S) to Spin„ c Cl® c Cln.
Proposition 5.15. When n is odd, this definition of Af is independent of
which irreducible representation of C£„ is used. Furthermore, when n is
odd, the representation A^ is irreducible. When n is even, there is a de-
composition
Ф	(5.21)
into a direct sum of two inequivalent irreducible complex representations of
Spinn.
Proof. The proof is entirely analogous to that of Proposition 5.12. 
It should be pointed out that the spin representations defined above
do not descend to the group SO„ = Spinn/Z2 since A„(— 1) = —Id.
It is worthwhile noting that representations of Cln also give rise to re-
presentations of the Clifford group. This is the finite group Fn c= Cl*
§5. REPRESENTATIONS
37
generated by an orthonormal basis e19... ,e„ of R". It can be presented by
the abstract elements el9... ,en, — l subject to the relation that — 1 is
central and that (-1)2 = 1, (ej2 = — 1 and = (- l)eje£ for all i j.
The Clifford algebra is nearly the group algebra RFn of Fn. More explicitly
C^RFJR-K-n+l}
It is clear that representations of C£„ correspond exactly to linear repre-
sentations of Fn such that ( — 1) acts by —Id.
This group enables us to draw an important conclusion:
Proposition 5.16. Let C£n -► Нотк(1У,1У) be a real representation of C£„.
Then there exists an inner product <•/> on Wsuch that Clifford multipli-
cation by unit vectors e e R" is orthogonal, i.e., such that
<e • w, e • w'> = <w, w'>	(5.22)
for all w,w' g Wand for all e g R" with ||e|| = 1. If Kn (= R, C or H) is a
commuting subalgebra for the representation, then the inner product can be
chosen to be Kn-invariant, that is, so that J is orthogonal when Kn = C and
so that I, J and К are each orthogonal if = H.
In particular, the spin representations Ди are unitary if n = 2 or 6 (mod
8) and symplectic if n = 3, 4 or 5 (mod 8).
Proof. Choose a K„-invariant inner product and average it over the finite
group Fn. Note that if e = £ afi where £ aj = 1, then
(ew,ew) = E aj<ejw, ejw) + E
since (е^,е^) — (w, w) and for i^j, (e^ejW) =	=
(e^w, w) = — (ejW, CjW) = 0. For the last statement, recall that An comes
from a representation of C£° s Cin_t. 
Corollary 5.17. Let <•,•> be the metric discussed in Proposition 5.16. Then
for any v c R",
<t? • w, w') = — v • w'y	(5.23)
for all w,w' g W. That is, Clifford multiplication by any vector v e R" is a
skew-symmetric transformation of W.
Proof. Assume v 0. Then <t> • w,w'y = <(d/||d||) • v • w, (r/||r||) • w") =
(l/||t?| |2) < t>2 • w, v • wf) = — <W, V • w'). 
It is worth noting that the irreducible representions of С£2и have a par-
ticularly nice description. Introduce on C" the standard hermitian metric
(z,0 S E z£.	(5.24)
38
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
and use this inner product to define a complex linear contraction map
(v t):ApcCn----► Л^С"
for v g C" by formula (3.21). We then define	-* Л£С" by setting
fv(<p) = VA<P-VL<P.	(5.25)
Then since v л v = 0, (u L)(r L) = 0, and v L (г л <p) = ||r||2<p — v л (v L <p),
we see that
Л°Л(ф)=-М2ф.	(5.26)
Note that since the inner product is C-antilinear in the second variable,
the map v -► fv is only R-linear. Nevertheless, writing R2"	C", we see
from universality (Proposition 1.1) that property (5.26) determines a unique
extension of f to a representation
f : Cl2n----► Нотс(Л*С", Л*С").	(5.27)
Since the complex dimension of this representation is 2", we see that it
must be the irreducible one.
We now make some remarks concerning tensor products. Suppose W
is a К-module for C£„ (where К = R or C) and let V be any vector space
over K. Then W V is also a К-module for C£n where by definition
Ф • (w ® v) = (<pw) ® v.	(5.28)
Therefore, if and W2 are К-modules for C£„, then W2 is a
К-module for C£n in two distinct ways. We set
Ар(нч ® W2) = (cpwr) ® w2,
P<p( ® w2) =	® (<pw2).
Then A and p are commuting representations of C£„. Furthermore, the
product
® W2) = (<p!wt) ® (cp2W2)
is a representation of the (ungraded) tensor product Cln ®
Proposition 5Л8. Let Ct2n -► Homc(S,S) be an irreducible complex repre-
sentation of C€2„. Then the tensor product representation of С€2и ®c C€2w
on S ®c S is equivalent to the representation Ф on Ct2„ itself given by
setting
ФФ1,Ф2(Ф) = Ф1 • Ф • <l>2
Proof. Since Ct2n ®c C£2„ = Cf4n (see Theorem 4.3), we see that
S ®c S must be an irreducible module for reasons of dimension. Since
dimc(S ® S) == 22" = dimc(Ct2„), the representations must be equivalent.
§5. REPRESENTATIONS
39
Corollary 5.19. Let pn: Spin„ -» SO(R") denote the standard n-dimensional
representation o/Spin„. Then in the complex representation ring o/Spin2m
(cf. Adams [1]) we have the equation
(Д^+Д^)-(Д^+Д^)
= 2(1 + p$m +	+ ... + A"-lpU +
where p2m denotes the complexification of p2m.
Proof. The tensor product A2m (x) A2m is obtained by embedding Spin2m
into C£2m ® C£2m diagonally (g i-> g (x) g) and restricting the tensor
product representation. By Proposition 5.18 this is equivalent to the
adjoint representation Фд(<р) = д<рд* = gtpg"' = Adg(<p). Under the
correspondence С£2и A£C2w, this representation is equivalent to
(1 + Pim + A2p2« + • • • + A2wp2m) ® C. However, by Hodge duality
App2m^ A2m~pp2m. 
Results analogous to Proposition 5.18 and Corollary 5.19 hold for the
algebras C£8m and for the real spin representation A8m = A8m + A8m.
Note that the tensor product of irreducible real representations of C£„
and C£8 gives an irreducible real representation of C£n+8 = C£„ ® C£8.
Similarly the complex tensor product of irreducible complex representa-
tions of C£„ and C£2 gives an irreducible complex representation of
С£и+2	® C£2. In general, however, С£я ® C£m is not a Clifford
algebra. Thus, to find a multiplicative structure in the representations of
Clifford algebras it is natural to consider the category of Z2-graded mod-
ules. A Z2-graded module for C£„ is a module W with a decomposition
W=W° ®Wl such that
C£* • Wj £ p0» + J)(mod 2)
for 0 i, j 1.
Proposition 5.20. There is an equivalence between the category of Z2-
graded modules over Ctn and the category of ungraded modules over C£n_ P
It is defined by passing from the graded module W° © W1 over C£n to the
module W° over £ Cfn_v
Proof. The inverse procedure is given by assigning to a C£°-module
IV0, the Z2-graded module
W = C£„ ®c£o W 0
(Left multiplication by C£„ on Cfn makes W into a Z2-graded module.)
The remainder of the proof is straightforward. 
There is a natural definition of the Z2-graded tensor product of Z2-
graded modules W = IV0 © IV1 and V = И0 © И1 over C£„ and Ctm
40	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
respectively. We set
(W ® K)° = W° ® V° + IV1 ® V1
(PV ® И)1 = W° ® V1 4-	® V°.
The action of C£n ® C£m on W ® V is given by
(Ф ® Ф) • (w ® v) = (— \)P4((pw) ® (0)
where deg(^) = p and deg(w) = q. Under the isomorphism Ctn+m =
C£n ® C£m, induced from mapping R" © Rw -► C£n+m
(e I—► e ® 1	if e e R" c C£n
[e I--► 1 ® ё	if ё e Rw c C£m,
V ® W becomes a Z2-graded module over C£n+m. This construction
holds for either real or complex modules.
In analogy with the above we define Яйп (Яй„) to be the Grothendieck
group of real (complex) Z2-graded modules over C£n. Note that by Prop-
osition 5.20 there are natural isomorphisms
<=9Wn-i and ^Яй^.	(5.29)
The arguments just given have established the following:
Proposition 5.21. There are natural pairings
(5.30)
(5.31)
induced by the graded tensor product. These pairings are associative
and give $1* =	and 9Й* =	the structure of graded
rings.
These pairings are important in the relation of Clifford algebras to real
and complex К-theory (see §9).
§6. Lie Algebra Structures
This section shall be concerned with the Lie algebra of Spinn. Recall that
the group of units C£* is a Lie group with Lie algebra cl* = (C£„,[•»•])
where [<p,	= q> -ф — ф • <p. There is an exponential mapping exp: cl * ->
C£* given by the standard series (see Remark 2.1). The group Spinn is an
explicitly defined, compact subgroup of C£*. We shall now investigate its
associated Lie subalgebra $ptnn in C£n.
§6. LIE ALGEBRA STRUCTURES	41
Recall that there are canonical embeddings c C£n for all p.
Proposition 6.1. The Lie subalgebra of (C£n,[•,•]) corresponding to the
subgroup Spin„ g C£* is
$pinn = A2R".	(6.1)
In particular, A2R" is closed under the bracket operation.
Proof. The Lie subalgebra $pin„ is the vector subspace of C£n spanned
by the tangent vectors to the submanifold Spinn at 1. Fix an orthonormal
basis ei9... ,en of R" and consider for each pair i < j, the curve y(t) =
(ef cos t + ej sin t) • ( — cos t + sin t) = (cos2t — sin2 0 + 2^- sin t cos t =
cos(20 + sin^Oe^. This curve lies in Spinn by definition of Spinn, and its
tangent vector at y(0) = 1 is 2^. Hence, $pinn contains the vector sub-
space spanR^eJ^j = A2R". Since dimR($pinn) = n(n — l)/2, we conclude
they are equal. 
We now'recall that the Lie algebra of the orthogonal group SOn is ex-
actly the space
$оп = {Л: R"----► R": A is linear and skew-symmetric} (6.2)
There is a natural isomorphism A2R" Л induced by associating to a
pair of vectors v,w e R", the skew-symmetric endomorphism “v л w” de-
fined by
(и л w)(x) = <t?,x>w — <w,x>d,	(6.3)
and then extending by universality. Note that л ер for i < j, corresponds
to the elementary skew-symmetric (i,j) matrix:
This is a standard basis of $on.
Recall now that the adjoint representation gives a surjective homomor-
phism
Spin, SO„.
(Since Spin„ c C£°, we have Ad|Spinn = Ad|Spifln.) This induces an asso-
ciated Lie algebra isomorphism
sptn„-^ so»	(6.4)
42	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proposition 6.2. The Lie algebra isomorphism (6.4) induced by the adjoint
representation is given explicitly on basis elements {eiej}i<j by
S0(efe;) = 2Ci л	(6.5)
Consequently for r,w e R",
SJ 1(v л w) = |>,w]	(6.6)
Note. This factor of | plays a delicate role in geometric applications.
Proof. Consider the curve y(t) = cos(t) + 8т(г)е^ in Spin„ with y(0) = 1
and /(0) = e^j. Then
So(^j) = 4 £o(X0)
dt t=o
and to compute this we apply it to a vector x e R". Since
£o(y(0)W = y(0*y(0" \
and since (y-1)'(0) = — ?'(0) = — we have that
E0(efey)(x) = e^x - xe^
= e^jX 4- (efx 4- 2<ei,x»ej
= e^x — e^jX — 2<e/,x>ei 4- 2<еьх>е;.
= 2(ef л e7)(x)
To prove (6.6), note that on basis elements, i[eheJ = Ke^j — е}еД = jetej.

This Proposition has the following immediate corollary:
Corollary 63. Let A:Spinn -► SO(W9 be a representation obtained by re-
striction of a representation C£n -* Hom(JV,py) of the Clifford algebra
C£n => Spinn. Let A* :$on $o(JV) be the associated representation of the
Lie algebra (obtained by first pulling back $on to the double covering via
Eq x). Then on the elementary transformations v л w e $on,
A*(t> л w) =	•	(6.7)
where the dot indicates Clifford module multiplication on W.
In terms of the standard basis {e, л e^^j
л ej) =	(6.8)
Suppose now that Ctn -» Нот(И/,Ил) is a complex representation of
C£n> and fix an element weW. The subgroup
Gw={ge Pin„ :gw = w}	(6.9)
§6. LIE ALGEBRA STRUCTURES
43
is called the isotropy group of w. Its Lie algebra is the subalgebra
Qw = {(pe spin,,: <p  w = 0}	(6.10)
Two elements w,w' g W are considered to be different as spinors (or more
precisely, to have distinct orbit types) if their isotropy groups Gw and Gw,
are not conjugate in Pinn. One crude measure of this difference is the
following:
Definition 6.4. The rank of the (generalized) spinor w is the rank of the
Lie group Gw. This is the dimension of a maximal torus in Gw or, equiv-
alently, of a maximal abelian subalgebra of
In a compact Lie group, every abelian subgroup is contained in a maxi-
mal torus, and all maximal tori are conjugate (cf. Adams [1]). Hence any
maximal torus Tw of Gw is contained in a maximal torus T of Pin„ which
we can assume to be the following standard one associated to a fixed
orthonormal basis {el9... ,en} of R":
T = < {j (cos вк 4- sin вке2к- te2k): 0 0k < 2n for each к >
The Lie algebra of T is given by
t = < 21 Лке2к-ie2fc • Ak e R for each к >
[k= 1	J
We now use our distinguished orthonormal basis to decompose the
module W. For each k, 1 2k n, define
= ~ i^2k- le2k
and note that
cuj,... ,си[и/2] pairwise commute,	(6.11)
a)k = 1 for each k,	(6.12)
= — eka)k	for each k.	(6.13)
Suppose now that V g W is a linear subspace which is e2k~ x-invariant
and e2k-invariant for some fixed k. Then by (6.12) we know that V =
ф V- where V± = (1 ± G>k)K are the ± 1 eigenspaces of on V. Fur-
thermore, by (6.13) we see that ekK+ = K_ and ekV_ — V+. In particular,
dim V± = | dim К
We shall now use this process to decompose the module IV. We begin
with the decomposition W = ф ИС by cdv Since commutes with
e3^4,... ,e„, we see that each of the subspaces W+ and ИС is e3>... ,en-
invariant. Hence we can similarly decompose each subspace W+ and W_
by w2 Ю get = W++ ф ИС- and WC = WC+ ф Each sub-
space W±± is e5,... ^„-invariant. Continuing inductively, we produce a
44	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
decomposition
W = ®Fr±...± = ©и;	(6.14)
a.
where dim W* = (dim W)/2{nl2} for each a and where a = (ab ... ,a[n/2])
ranges over the 2[n/2]-possibilities having ak = + or — for each k. (Note
that if W is an irreducible module, then dimc Wa = 1 for all a.)
The maximal torus T preserves each subspace Wa. In fact, given g —
(cos 0k + i sin 0ka>k) e T, we see that
0|ж. = е1Таквк = е‘<л-в>
Similarly for	= i £ e t we have
ф|ж« = ' E aA »<M>	(6.15)
The set of vectors e t* are called the weights of the representation.
The I occurs because in general theory the weights are normalized by
relating them to the weights of the adjoint representation.
We now return to our given spinor w e W. With respect to the decom-
position (6.14) we write
Л
From (6.10) and (6.15) we conclude the following:
Proposition 65. The maximal abelian subalgebra of Gw is
= {i E : <M> = 0 for all a such that wa / 0}
Corollary 6.6. rank w = [n/2] — dim spanR{a : wa 0}
If w = wa for some a, i.e., if all but one component vanish, then w is
clearly of maximal rank. Those elements that take the simple form w = wa
for some choice of orthonormal basis in R", are called pure. Pure spinors
are related to complex structures, twistors spaces and calibrations. They
will be discussed in detail in Chapter IV.
§7. Some Direct Applications to Geometry
In this section we shall use the classification of Clifford modules given
above to construct families of pointwise linearly independent vector fields
on spheres, projective spaces and other elliptic space forms. We shall also
apply the methods to study the “hyperplane” bundle over complex and
quaternionic projective space. This allows us to estimate the geometric
dimension of TIPn(C). In almost all cases the families constructed in this
manner are maximal.
§7. APPLICATIONS TO GEOMETRY
45
We begin with the following observation:
Proposition 7.1. Suppose RN+1 is a module for the algebra Cln. Then there
exist n pointwise linearly independent tangent vector fields on the sphere SN
and also on the projective space IPN(R) = SN/Z2.
Proof. Choose an inner product in RN+1 so that Clifford multiplica-
tion by unit vectors in R" is orthogonal (see Proposition 5.16). Let SN =
{x g RN+1 :||x||2 = 1}. Choose a basis . ,vn for R", and to each Vj as-
sociate the vector field Vj on RN + 1 defined by
F/x)s tyx j= l,...,n
(where the dot denotes Clifford multiplication). Since the linear transfor-
mation xi->p x is skew-symmetric (see Corollary 5.17), we have that
<VJ(x),x> = <VjX,x> = 0. Hence, the vector fields Vj are tangent to SN. It
remains to show that . ,КЯ are pointwise linearly independent. Fix
x g SN and consider the linear map ix: R" TXSN g RN+1 given by
Ц0 = V ‘ X
The image of ix is the linear span of KJx),... ,K„(x), so it suffices to prove
that ix is injective. However if ixv = v • x = 0, then v • v • x = — ||d||2x = 0
and so v = 0.
Since Vj(—x) = — these vector fields descend to (pointwise linearly
independent) vector fields on PN(R). 
The question now is: given an integer N, what is the largest number of
independent vector fields on SN that can be constructed in this manner?
That is, what is the largest integer n such that RN+1 is a C£„-module? We
recall that the dimension of an irreducible C£n-module is always a power
of 2. Hence, we want to find the largest power of 2 which divides N + 1.
That is, we write N + 1 = p2m where p is odd, and then we consult Table
III to find the largest n such that dn = 2W. The result is the following clas-
sical result of Radon and Hurwitz.
Theorem 7.2. On the sphere SN (and on the projective space PN(R)) there
exist n pointwise linearly independent vector fields where n is computed as
follows. Write N + 1 = 24a+b(2t + 1), 0 b < 3. Then
n = 8a + 2Ь - 1.	(7.1)
Proof. One need only check this when a = 0, and then note that for each
increase of n by 8 the dimension of the vector space for an irreducible
representation of increases by 24. Note that when N is even, the num-
ber of such vector fields is zero as it must be since the Euler characteristic
is non-zero in this case. Note also that this construction gives three vector
fields on S3, seven on S7 and eight on S15. 
46
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
One of the deep results of algebraic topology is the following:
Theorem 7.3 (J. F. Adams [2]). The number of vector fields constructed on
SN above is the largest possible number of pointwise linearly independent
vector fields that can exist on SN.
It is worth noting that this construction also gives rise to vector fields
on many elliptic space forms. If the representation of Ctn on RN+1 is com-
plex or quaternionic, then Clifford multiplication by v e R" commutes with
complex scalar multiplication. Therefore, if Д = e2nl,p is a pth root of unity,
then the vector field V(x) s v • x on has the equivariance property
К(Дх) = Д • V(x)	(7.2)
for all x g SN. This means precisely that the vector field V(x) is invariant
under the diffeomorphism SN SN given by scalar multiplication by Д.
The Zp-action on SN generated by fi is free. From (7.2) we conclude that
V descends to a vector field on the quotient S*fZp which is a lens space
of simple type LN(p) = LN(p;l,... ,1).
Analogous remarks hold when the representation of C£n is quaternionic.
Here we may replace Zp by any finite multiplicative subgroup of H. Such
subgroups are constructed as follows. The unit sphere S3 c (HI is a Lie
group isomorphic to Spin3 (see the paragraph below). Let £0:S3 -► SO3
be the 2-fold covering homomorphism. Then for any finite subgroup
Го<= SO3, the inverse image Г = £0 ^Го) is a finite subgroup of S3 c H.
Of course, the symmetry groups of the regular polygons, the so-called di-
hedral groups, and the symmetry groups of the Platonic solids give many
examples of finite subgroups of SO3. The (^-images of dihedral groups
are called binary dihedral groups. There are also the binary tetrahedral
group, the binary octahedral group, and the binary icosahedral group, cor-
responding to the -images of the symmetry groups of the tetrahedron,
octahedron and icosahedron, respectively.
From our remarks above we conclude the following two theorems.
Theorem 7.4. On each simple lens space LN(p) = SN/1LP, for p > 1, there
exist к pointwise linearly independent vector fields where, if N + 1 ~
2w(2t + 1), then
к = 2m — 1
Moreover, this is the maximal number of pointwise linearly independent vec-
torfields possible on LN(p) if m = 1 or 2 modulo 4.
Theorem 73. There exist q pointwise linearly independent vector fields
on SN/T where Г is any finite subgroup of S3 c H and where if N + \ =
§7. APPLICATIONS TO GEOMETRY
47
2m(2t 4- 1) and m = 4a 4- b, 2 < b < 5, then
_ (8a 4- b 4- 1	if b 5
q=(ia + b + 2	if b = S'
Moreover, this is the maximal number of pointwise linearly independent vec-
torfields possible on the elliptic space form SN/T if m is congruent to 1 or
2 modulo 4.
Similar constructions can be made for complex and quaternionic pro-
jective space. Of course in these cases the Euler characteristic of the mani-
fold is not zero, and so every tangent vector field must vanish somewhere.
However, we can pass to the stabilized bundle TX @ Rm where ROT de-
notes the trivial bundle of dimension m. Clearly there exists some bundle
E over X so that TX © Rw E © Rw+"~* where n = dim X, к = dimR E
and к is as small as possible (for any choice of m). The dimension of E is
called the geometric dimension of TX.
Let (Pn(K) denote the n-dimensional projective space over the field K.
For К = R, C or H there is a tautological К-line bundle rj -► P"(K) whose
fibre above a point [if] g Ря(К) is the one-dimensional linear subspace
f c Kn+i corresponding to [if]. For К = R or C we have the following
fact:
T(P"(K)) ® K= n* ®	<	(7.3)
(n 4- 1)-times
where a/* is the dual bundle
f]* = HomA(^K).
To see this, we first show that
Т[ЛР"(К) Hom^1).	(7.4)
Each one-dimensional К-linear subspace I g Kn+l can be canonically
identified with the (К-linear) orthogonal projection map 7t,:Kn+1 ->l.
Note that nj = я,, я,я,± —	= 0 and п, 4- я^ = Id. Let ift, |t| < e, be
a smooth family of К-lines with if0 = t, and set я = (d/dt^Jf=0. Then by
deriving the identities above we have that я,я, 4- я,я, = я,, я^я^ 4- я^я^± =
я,±я, 4- я^я, = 0, and я, 4- я,± = 0. It follows that я,я,х = 0 and я,я, = 0.
Непсе я, g Hom^(if, if1). On the other hand it is easy to construct an ele-
mentary basis of HomK« if1) as tangent vectors я, for curves <f(t) with
^(0) = if.
From the exact sequence
0----► £----► K"+1----► if1---► 0
48
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
we obtain the exact sequence
0----► Hom^, <0-----► Hom^, Kn+i)-------► Hom^, f1)------► 0. (7.5)
From (7.4) this gives an exact sequence of bundles over Pn(K):
0----> HomKfr,rf)---► g* ®® g*----------► TPn(K)----► 0 (7.5)
(n 4- 1) times
which holds for К = R, C or H. When К = R or C, we have Hom^, a/) s
® = К ( = the trivial К-line bundle), and this establishes (7.3).
Suppose now that L:Kn+1 -> Km is a К-linear map (for any m 1),
Then L defines a section of ® • • • ® (m times) as follows. Fix a
К-line t c Kn + 1. Then L\e:f Km is exactly m К-linear functions on Л
Thus, we have an embedding
Нотк(Ки+ \ Km) а Г(ц» ®- • ® q*\	(7.7)
m times
into the space of sections of the bundle ® • • • ® ^*. A section corre-
sponding to a К-linear map L is nowhere zero if and only if L(x) 0 for
any x 0. Similarly, L1?... JLP give pointwise linearly independent sec-
tions of ® • • • ® if and only if Lx(x),... ,Lp(x) are linearly inde-
pendent at each non-zero xe Kn+i. The К-representations of C£p give us
precisely p К-linear maps ... JLP:KN -► KN which satisfy this condi-
tion of point wise independence. Thus, by consulting Table III, we have
the following result analogous to that of Theorem 7.4 (cf. Lawson-
Michelsohn [1]).
Theorem 7.6. Let (n 4- 1)^* = ^* ® • • • ® denote (n 4- lycopies of the
“hyperplane” bundle over complex projective n-space R"(C). If n 4- 1 =
2w(2t 4-1), then there exist к sections of (nA- 1)^* where
к = 2m — 1.
Therefore
ydn < 2n — 2m 4- 3
where ydn denotes the geometric dimension of the tangent bundle TR"(C).
We also have the following result which is analogous to that of The-
orem 7.5.
Theorem 7.7. Let (n 4- 1И* = £* ® • • • ® £* denote (n 4- \)-copies of the
“hyp er plane” bundle over quaternionic projective n-space IP”([H1). Ifn 4- 1 =
49
§8. APPLICATIONS TO LIE GROUPS
2w(2t 4- 1) then there exist к sections of (n 4- 1)£* where
2m — 3
к = 2m — 2
2m — 1
if m = 0 (mod 4)
if m = 3 (mod 4)
otherwise.
Therefore
An — 2m 4- 7
gdn < <4n - 2m 4- 6
An — 2m 4- 5
if m = 0 (mod 4)
if m = 3 (mod 4)
otherwise
where gdn denotes the geometric dimension of the bundle ТРЛ(1Н1) ф
HomH(«).
Note that we are unable to make a conclusion about the geometric
dimension of the tangent bundle itself because HomH(^Л) is not trivial.
§8. Some Further Applications to the Theory of Lie Groups
It is interesting to note that the classification of Clifford modules gives
an immediate proof of certain well-known isomorphisms between low-
dimensional Lie groups. It also leads to the fact that S1 = Spin7/G2 and
to the principal of triality for Spin8. We would like to thank Reese Harvey
for pointing this out to us.
We begin with some classical definitions. Let C" and NT carry the stan-
dard “hermitian” inner products
(x,y) = £ Xjyj	(8.1)
where for a quaternion x == x0 4- iXi 4- jx2 4- kx3, the conjugate is defined
by x = x0 — ixx — jx2 — kx3. Then we have the following definitions of
the classical groups:
U„ = {g g Homc(C",C"): (gx.gy) = (x,y) for all x9y g C"}.
SUn = {g e U„: detc(0) = 1}
Spn = {в e	: (gx,gy) = (x,y) for all x,y e H"}
Under the natural isometries R4" s C2" s H" (and R2" = C") one finds
that
Sp„ <= SU2n and U„ <= SO2„
(8-2)
50
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Elementary linear algebra proves that
fdim(SO„) = |n(n — 1)
Jdim(SU„) = n2- 1	(8.3)
[ dim(Sp„) = n(2n + 1),
and it is not difficult to see that each of these groups is connected.
When the integers p,q,r are sufficiently large (say 7), the groups Spinp,
SUe, Spr are all distinct. However, in low dimensions there are certain
exceptional isomorphisms between these groups. These isomorphisms are
easily deduced from the Dynkin diagrams. However, the approach re-
quires the non-trivial classification of compact, simply-connected Lie
groups. The same isomorphisms can also be deduced from Table II, which
was comparatively easy to establish.
Theorem 8.1	. There exist the following isomorphisms between low-dimen-
sional Lie groups:
Spin3 SU2 Spi
Spin4 Spin3 x Spin3
Spin5 s Sp2
Spin6 SU4
Furthermore, Spin7 has a faithful 8-dimensional real representation, and
Spin8 has three inequivalent 8-dimensional real representations.
Proof. Recall that Spinn c C£° ~ C£n_p Furthermore any representa-
tion of С£и_! on CN or HN can be assumed to have the property that,
when restricted to the group Spin„, it preserves the hermitian inner prod-
uct (8.1). Consequently, since Cl2 s H has a faithful one-dimensional
quaternionic representation, we get an injection Spin3 SpP The first
isomorphism follows easily. Since C£3 = H ® H, we get an injective map
Spin4 Spi x SpP Since dim(Spin4) - dim(Spi x SpJ and Spin4 is con-
nected, we get the second isomorphism. Since C£4 = H(2), we get an in-
jection Spin5 Sp2 and the third isomorphism follows. Since Cl5 = C(4),
we get an injection Spin6 U4. By the simplicity of (the Lie algebra of)
Spin6, or by Lemma 8.5 below, we see that Spin6 must lie in the kernel
of the homomorphism detc:U4 ->Up Thus, we have Spin6 SU4 and
the fourth isomorphism follows.
The existence of a representation of Spin7 on R8 is obvious since
C£6 = R(8). Furthermore, since C£7 = R(8) ® R(8), we see that the two
spin representations Д8 and Д8 of Spin8 are on R8. There is also the
§8. APPLICATIONS TO LIE GROUPS	51
adjoint representation Ad of Spin8 on R8. To see that these representations
are all distinct, it suffices to consider the central elements. Set
= {1,-1,co,-co}	Z2 © Z2
where co denotes the oriented volume element of Cl7 s Cf8. This group
lies in the center of Spin8. (In fact it is the center.) From Propositions 5.9
and 5.10 we know that Д8(co) = Id and Д8(co) = —Id. Since Д8 come
from representations of Cf7 we have
Д+(со) = Id,	Д8+(-со) = -Id,	Д8+(-1) = -И	(8.4)
Д8(со) = -Id,	Д8(-со) = Id,	Д8(-1) = -Id.	(8.5)
From the definition it is clear that if Ad denotes the adjoint representation
of Cf8 on R8, restricted to Spin8 c= C£8 = C£7, then
Ad(—l) = Id.
Recall now that equivalent irreducible representations must agree on
central elements. (Note, for example, that if there exists an isomorphism
F:R8->R8 with the property that F о Д8(#) ° F"1 = Д8 (g) for all
g e Spin8, then in particular, Д8 (g) = Д8 (g) for all g e 3f.) Consequently
the three representations Д8 , Д8 and Ad are inequivalent. 
It is an interesting and often useful fact that two 8-dimensional rep-
resentations of C£7 can be explicitly generated using the Cayley numbers.
Recall that the Cayley numbers О can be defined as pairs of quaternions
with multiplication given by
(a,b) • (c,d) = (ac - db, da + be).	(8.6)
The multiplication so defined is neither commutative nor associative.
However, every non-zero element has a multiplicative inverse. Further-
more, given a Cayley number x = (a,b), we write x = (a, — b) and define
real and imaginary parts of x by setting
Re(x) = |(x -I- x); Im(x) = |(x — x)
An inner product on О is defined by <x,y> = Re(xy). It has the property
that |xy| = |x| |y| for all x,y g О (where |x|2 = <x,x> as usual). An impor-
tant fact concerning the Cayley numbers is that any subalgebra of О gen-
erated by two elements is associative.
We now consider R7 = Im(O) and R8 = О with the above inner prod-
uct. For any v g Im(O) we define a linear endomorphism Av of R8 by
setting
ЯДх)= v•x
(8.7)
52	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
for x e О = R8. From the associativity of the algebra generated by x and
u, or by a direct computation from (8.6), we see that
Я2 = -H2Id	(8.8)
for all v e Im(O) = R7. Consequently, from the universal property (Propo-
sition 1.1), we know that Я extends to a representation
A: C£7-----► HomR(R8,R8)	(8.9)
of Cl7. For dimensional reasons this representation must be irreducible.
The other irreducible 8-dimensional representation
p = A ° a
is generated by the mapping pv(x) = - v • x. Observe that under the con-
jugation map c(x) = x, pv becomes equivalent to right multiplication by
v, i.e., pv = c ° pv с c is given by
pv(x) = (— г • x) = — xv = xv
(since for v e Im О we have v = —r).
These two representations are equivalent when restricted to Spin7, but
they are inequivalent on Spin8 c= s C£7.
We now consider the action of Pin7 on R8 given by the representation
A above. From (8.7) it is clear that the orbit of 1 contains all elements
eelm(O) with |e| = 1. That is, this orbit contains the “equator”
S6 = S7 n Im(O). It also contains e • S6, which is a great sphere passing
through the “north pole” 1, for each e g S6.
Since the orbit Pin7 • 1 is a compact embedded submanifold of S7, we
conclude that it is S7. It now follows that the orbit of Spin7 must be
7-dimensional and, hence, also equal to S7. We have proved the result of
A. Borel.
Theorem 8.2	. The 8-dimensional spin representation of Spin7 is transitive
on the unit sphere S7.
§8. APPLICATIONS TO LIE GROUPS	53
With a little more work it is possible to prove that the isotropy sub-
group {g g Spin7:^(l) = 1} of 1 g S1 is exactly the group G2 = Aut(O)
(see Harvey-Lawson [3] for example). Hence, we have the diffeomorphism
S1 = Spin7/G2.
We pass on now to the group Spin8. Recall from Theorem 8.1 that this
group has three distinct homomorphisms Д8 , Д8 , Ad:Spin8 -► SO8 with
kernels isomorphic to Z2. Writing the center S£ = {1, — l,co, — co} as before,
we have the following table (cf. (8.4) and (8.5)).
9	co	—co	-1
Л,+(я)	Id	-Id	-Id
A.'(9)	-Id	Id	-Id
Ad(p)	-Id	-Id	Id
We now consider the pair of homomorphisms Д8 and Д8 . From general
covering space theory there must be a lifting of Д8 over Д8 which takes
the identity to itself, i.e., a map a: Spin8 Spin8 such that the diagram
Spin8
j/* a8+	(8.11)
Spin8 ——► SO8
Д8
commutes and <r(l) = 1. The map a satisfies the condition
= <r(0iM02)	(8.12)
for all gt,g2 in a small neighborhood of 1, since both Д8 and Д8 are group
homomorphisms. In fact, the relation (8.12) holds for all дъд2 g Spin8. To
see this note that both sides of (8.12) are well defined on Spin8 x Spin8
and that the set where (8.12) holds is both open and closed. (Alternatively,
one could use the fact that (8.12) is an equation between real analytic
maps.) Thus, a is a group homomorphism. Since a is a covering map
between simply-connected spaces, it must be injective, and so a is a group
automorphism.
It is now clear from (8.11) that a carries the kernel of Д8 onto the
kernel of Д8 . Since кег(Д8) = {1, —co} and кег(Д8 ) = {l,co}, and since
ff(l) = 1, we conclude that
a( — co) = co
Since co and —co are central, a must be an outer automorphism.
54	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Lifting Ad over the homomorphisms A8 and A8 and applying the
arguments above we construct outer automorphisms r + and т of Spin8
with the property that
t*(—l) = co and t'(—1)=—w.	(8.13)
By definition we have that A8 от1 = Ad and A8 ° a = A8 . Furthermore,
the associated Lie algebra maps (A8)* and Ad* are isomorphisms, and
we have that
= (A*)-1 - Ad* and r* = (A8)* 1 о (Д’)*.	(8.14)
At this point the automorphisms a and are only defined modulo
inner automorphisms. To get concrete representatives we must choose
concrete representatives for the maps A8 and Ad.
We shall give such an explicit construction for a. Choose an ortho-
normal basis eu... ,e8 of R8 and recall that the Lie algebra $pin8 = A2R8
has an orthonormal basis • ej}i<J. The map C£7 Л C£8 is induced by
the assignment
ej--->e/?8 j = l,...,7.	(8.15)
Consequently, the preimage of spin8 under this map is just
spin8 = R7 © A2R7 cz Cf7	(8.16)
where i—► eze8 for 1 < i < 7 and where efij i-> eie8ejes = for 1 s i < j 7.
We now consider the two representations Я and p of C£7 on R8 = О
that were constructed above. Since Ay(x) = v • x and pv(x) = —vx for
v g R7, we see that
f^=~Pv	for«peR7
\^ = PV	for<peA2R7.	1610
Now restricted to spin8 SR7 ® A2R7 we have that A = (A8)♦ and
p = (A8 )*. Consequently, ст* = (Д8)“1 ° (Д8 )ж has the property that
J—if e R7
'4 <p	|81’>
In particular, (a*)2 = Id and so by exponentiation we have that
ст2 = Id.	(8.19)
We now let O8 and I8 denote respectively the groups of outer and inner
automorphisms of Spin8. There is a natural homomorphism
O8//8----> Aut(^f).	(8.20)
It is easy to see that Aut(^f) = Aut(Z2 ф Z2) s S3. In fact, we have natu-
rally that Aut(^f) S Perm(— l,co,—co). Since ct(—co) = co and cr(^f) = S£,
§8. APPLICATIONS TO LIE GROUPS	55
we have that either a( — 1) = — 1 or a( —1) = —co. Since a2 = Id, we
conclude that
a( — 1) = — 1 and a(co) = — co.
This information, together with (8.13), easily proves the following:
Theorem 83. The homomorphism O8/I8 -► Aut(<2f) S3 is surjective. In
particular, since O8/I8 is a finite group (cf. Helgason [1]) there exists an
element in O8/I8 of order three.
Those readers similar with the classification of Lie groups know that
this map O8/I8 -► S3 is reflected in the representation of O8/l8 as the
group of symmetries of the Dynkin diagram of Spin8. This representation
is faithful, and so O8/I8 S3.
It can be shown that in fact there exists a non-trivial element A e O8
such that
A3 = Id.
This element is called the triality automorphism. Continuing with calcula-
tions as above, this automorphism can be constructed explicitly.
It has most likely occurred to the reader that the methods of Theorem
8.1 can be applied more generally to the groups Spinr>s for any r and s.
Indeed, this does produce further exceptional isomorphisms between
low-dimensional Lie groups. We recall the following classical groups.
SL„(R)= {g g НопМЙЭД : detR(^) = 1}
SL„(C) = {g g Нотс(Сл,Сл): detc(^) = 1}
SL„(H) = {g g Нотн(Нл,Нл): detc(a) = 1}
For the last definition we have fixed a presentation D-0” = (C2",J) where
J:C2n -► C2" is C-antilinear and J2 = —Id. We observe that the complex
determinant is in fact real-valued on Нотн(1Н)л,1Н1л) = {g e Нотс(С2л,С2л):
go J = J о g}. To see this, let с:С2л->С2л denote complex conjuga-
tion and set c(g) = c ° g ° c. Then since c2 = 1 and since g and J com-
mute, we have detc(c(^) ° с о J) = detc(c ° g ° J) = detc(c о J о g). Hence,
detc(cto)) = detc(0) and since detc(c(^)) = detc(0), the determinant is real.
56	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Now, each of the groups SL„(K) is connected, and elementary linear
algebra shows that
dimR(SL„(R)) = n2 — 1
< dimR(SL„(C)) = 2(n2 - 1)	(8.21)
dinMSUH)) = 4n2 - 1.
Recall that for r,s > 0, the group Spinr,s has two connected components.
We let Spin®, denote the component containing the identity. It is obvious
that there is an isomorphism
Spinr,s Spinsr,
and an elementary calculation shows that
dim(Spinr,s) = i (r + s)(r + s — 1)	(8.22)
for all r,s. A careful look at Table II (p. 29) now proves the following.
Let SL„(R) -> SL„(R) denote the (2-sheeted) universal covering group for
n^3.
Theorem 8.4. There exist the following isomorphisms between low-dimen-
sional Lie groups.
Spin^ SL2(R)
Spin* x SL2(C)
Spin?4 SL2(H)
Spin2,2 = SL2(R) x SL2(R)
Spin*,3 = SL4(R)
Furthermore Spin4,4 has three inequivalent 8-dimensional real represen-
tations.
Proof. Recall from Theorem 3.7 that for r 1 we have
Spinr,s c C£*s Ctr_l s.
Consequently, from Table II, we have the following embeddings: Spin2, x c
GL2(R), Spin31 <= GL2(C), Spin51 <= GL2(H) x GL2(H), Spin2t2 g
GL2(R) x GL2(R) and Spin3 3 c GL4(R) x GL4(R), where GL„(K)
denotes the set of invertible elements in HomK(K",K"). We now observe
that in each of these embeddings the identity component Spin*s is actually
contained in the subgroup SL„(K) or in SL„(K) x SL„(K) for the latter
cases. This is an immediate consequence of the following lemma.
§8. APPLICATIONS TO LIE GROUPS
57
Lemma 8.5. Let p: C£r_ l s-► HomK(lzJz) be any K-representation for
К = R or C. If r + s 3, then
= ± i
for all g e Spinr s c C(r_ l s.
Proof. If r + s 3, then every element g e Spinr s can be written as a
product g =	• • • gm where e Spinr s satisfies
g]=±l	(8.23)
for all j. To see this, write g = • • • v2m where Vj g Rr+s and q(vj) = ± 1.
Write vtv2 =	where t)6Rr+s satisfies q(v) = ±1 and where
9(^1) = 4(1^2) = 0. Set gi = and g2 = +vv2. Then gj =	=
—v2v2 = ± 1, and similarly, g% = +1. Of course vtv2 = дгд2. Continuing
in this manner proves our claim.
Since dim^K) is even, we see that for g = gx • • - gm as above, we have
П
. i
[detK(p3)]2
2
= Il det«(p3j)
= П det^ + Id) = 1. 
J
The first, second and fourth isomorphisms of Theorem 8.4 now follow
from dimension considerations (cf. (8.21) and (8.22)).
For the fifth isomorphism, consider the image of Spin3>3 under the two
projections SL4(R) x SL4(R) SL4(R). At least one of these must be
non-trivial and, therefore, locally injective by the simplicity of the Lie
algebra. Since dim(Spin3>3) = dim(SL4(R)) = 15, this projection is a cov-
ering map Spin3>3 -> SL4(R). Since Spin3 3 is simply connected, it lifts
to an isomorphism Spin3t3 Л SL4(R). The third isomorphism is proved
similarly.
Looking more closely we can see that the two projections Spin3 3
SL4(R) are both non-trivial and inequivalent. To prove this we consider
the volume element co = ег • e6 where e2 = e2 = e2 = — e4 — —e2 =
= — 1 (see Proposition 3.3). This element is central in Spin3 3 and
satisfies co2 = 1. It clearly lies in Spin3 x Spin3 c= Spin3 3 and is therefore
connected to the identity. The module V R8 for C£3 3 decomposes as
V = V+ © V~ where V1 = (1 ± co)V are invariant subspaces for Spin3 3.
Since co = +Id on K1, we see that the representations are inequivalent.
They are each non-trivial, since otherwise we would have the identity
co = 1 (or co = — 1) in Spin3 3, which is clearly false.
To prove the final statement we again consider the volume element
co = eY • • • e8 in Spin4>4. As above we see that co is central in Spin4>4 and
is connected to the identity. Since co2 = 1, the module W R16 for C£4>4
58	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
decomposes as W = W+ © W~ where W1 = (1 ± co)W are each invari-
ant under Spin4>4. Let Д1 denote these two 8-dimensional representations,
and consider the central subgroup
S£ = {1,—co} =
in Spin4>4. Then Д+, Д' and Ad have precisely the values given in the
table (8.10) derived above for the case of Spin8. It follows that these three
8-dimensional representations of Spin4 4 are distinct. 
The above arguments actually prove slightly more. Let
SL*(K) = {g e Hom^K",*"): detK,(a) = ± 1}
where К' = К if К = R or C and К' = C if К = H. There is a short exact
sequence
1 —+ SL„(K) —+ SL*(K) Z2--------------> 0.
From the proof of Theorem 8.4 we can actually conclude that
Spin2,i = SLJ(R)
Spin3,i = SL*(C)
Spin5>1 s SLJ(H)
Note. For further results of the type given in Theorem 8.4, the reader is
referred to the beautiful book of Reese Harvey [1].
§9. К-Theory and the Atiyah-Bott-Shapiro Construction
In this section we shall present the essentials of К-theory in a fashion
which will be useful later in our discussion of the Atiyah-Singer Index
Theorem (Chap. III). Following these basics we shall present the con-
struction of Atiyah, Bott and Shapiro which relates the Grothendieck
groups of real Clifford modules to the KO-theory of spheres and that of
complex Clifford modules to the К-theory of spheres. Their fundamental
isomorphisms explicitly identify Bott periodicity with the periodicity
phenomena in the theory of Clifford algebras. For an elaboration of the
details presented here, the reader is referred to Atiyah-Bott-Shapiro [1]
and Karoubi [2].
Throughout this section all spaces will be assumed to be compact. If
X is any such space, we denote by V(X) the set of all isomorphism classes
of complex vector bundles over X. The set V(X) is an abelian semigroup
if we define addition by direct sum. We let F(X) be the free abelian group
generated by the elements of V(X) and let E(X) be the subgroup of F(X)
§9. K-THEORY AND THE ABS CONSTRUCTION
59
generated by elements of the form [K] + [РУ] — ([К] ф [И']) where +
denotes addition in F(X) and ф denotes addition in V(X).
Definition 9.1. The К-group of X is defined to be the quotient
K(X) = F(X)/E(XY
Note that K(X) is an abelian group. The elements of K(X) are called
virtual bundles.
If V and W are bundles over X and Y respectively then V ® W is a
bundle over X x Y. When X — Y the diagonal map A: X -► X x X can
be used to define an interior tensor product in K(X) by
[u] • [v] = A*[u ® v].	(9.1)
This gives us
Proposition 9.2. The group K(X) has a ring structure with multiplication
given by (9.1).
Let a: V(X} -► K(X) be the composition V(X) F(X) - F(X)/E(X).
Proposition 93. If G is any abelian group and f: V(X) ->G any semi-group
homomorphism, then there is a unique homomorphism f: K(X) —► G such
that fa = f.
Proposition 9.4. K(X) is the universal group with respect to maps of the
type described in Proposition 9.3
Suppose that /: X -► Y is a continuous map and consider the map
/*:У(У) V(X) given by the induced bundle construction. Since this
is a semi-group homomorphism, it descends to a homomorphism /♦:
Х(У) -► K(X). One easily checks that К thereby becomes a contravariant
functor from the category of compact spaces to the category of abelian
groups.
Suppose now that S is any abelian semi-group with unit and let
A:S -> S x S be the diagonal map. This is a semi-group homomorphism.
If we let JT(S) be the set of cosets of A(S) in S x S, then JT(S) is a quotient
semi-group. Since the interchange of factors in S x S induces an inverse
in JT(S), Jt(S) is actually a group.
Definition 9.5. Jf'(X) is defined to be Ж(К(Х)).
Proposition 9.6. JT(X) is isomorphic to K(X).
Proof. For an abelian semi-group S with 0 we define ps: S -> JT(S) to
be s (s,0) followed by the natural projection S x S -> JT(S). If g: S T
60
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
is a semi-group homomorphism, then there is induced a map	: Ж(5)—►
JT(T) so that ° fls = f}T о g. Now let S be V(X) and Tbe any abe-
lian group. Then is an isomorphism. This shows that is uni-
versal with respect to semi-group homomorphisms from K(X) to abelian
groups. 
Corollary 9.7. Every element of K(X) can be represented in the form [K] —
[IV] where [K],[PV] g V(X).
Lemma 9.8. Let n:V -► X be a vector bundle over a compact Hausdorff
space X. Then for some N there is a continuous map f:V^>CN which
is injective and linear on each fibre.
Proof. Cover X with a finite number of open sets U19... ,Ur over which
there exist trivializations a;-: ti~ HUj) Л Uj x Ck and set = pr^ ° a;- where
pr/ Uj x C* -► C* is projection. Let {^j}rj=i be a partition of unity sub-
ordinate to pThen/ = Ф • • • Ф GMr): К->Скф,,*фСк
is the desired map. 
Corollary 9.9. To each bundle V over X there exists a “complementary”
bundle V1 such that V ф V1 is trivial. Hence, each element of K(X) can
be represented in the form [К] — [t*], where tn is the trivial bundle over
X of dimension N.
Proof. Let f:V^>CN be the map in Lemma 9.8 and define K1 =
Uxex/CK)1- Clearly V ® V1 X x C\ Hence, [K] = [<] - [K1] in
K(X) and an arbitrary element [IV] — [K] in K(X) can be rewritten as
[Ж® K1] - [<|. 
Definition 9.10. We define the real X-ring for X, KO(X), just as we
defined K(X) by replacing V(X) by KR(X), the set of isomorphism classes
of real vector bundles. The same construction and considerations* apply.
Analogues of the following definitions can also be made for XO-theory.
We will assume them without specifically stating them.
We would now like to use the X-groups to define a generalized coho-
mology theory. For this reason we now consider X to be a space with a
distinguished basepoint, pt g X.
Definition 9.11. The reduced X-ring, K(X), is defined to be the kernel
of the natural projection K(X) -► X(pt) s Z, so that K(X) is an ideal of
K(X).
In fact the exact sequence
0----> K(X)----► K(X)-----> X(pt)---> 0	(9.2)
splits in an obvious fashion.
§9. K-THEORY AND THE ABS CONSTRUCTION	61
Definition 9.12. Suppose У is a non-empty closed subset of X. Then
we define the relative К-groups as follows:
K(X,Y) = K(X/Y)
where X/Tis taken to have Y as its basepoint.
If Y is empty, X/Y is defined to be the space
X+ = Xu{pt}	(9.3)
where pt is a disjoint point which will play the role of basepoint.
Observation 9.13. On a non-basepointed space X we have the identi-
fication
K(X) « K(X+) = K(X, 0).
Definition 9.14. We define the wedge X v Y and the smash product
X л Y of two spaces Х,У with basepoints ptx and pty by
X v Y = (X x pty) u (ptx x У) cz X x Y
X aY = X x У/X v У.
We also define the (reduced) suspension S(X) of X by
E(X) = S1 л X.
Iterating this i times gives us the i-fold suspension 5?(X). For all i there is
a homeomorphism 5?(X) » S* л X.
Definition 9.15. When X is a compact basepointed space, or when
(Х,У) is a compact pair, we define
K~'(X) = K(£'(X))
К-1(Х,У) = K~i(X/Y) = K'^X/Y)).
For spaces X which are not necessarily basepointed we define, in the spirit
of 9.13,
K-\X) = К \Х9 0) = K(2?(X+)).
Since the functor К is representable (see Chap. Ill, Theorem 8.6) there
is an exact sequence for basepointed pairs (Х,У)
К(Х,У)-----► K(X)----► К(У)
which we may now extend to a Barratt-Puppe sequence (Barratt [1]):
------►	K~‘(X,Y)-----► K~\X)-----►	(9.4)
К~\У)------►------► K°(X)-----► K°(Y).
We write K“* for the graded functor K~\ i > 0.
62	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Remark 9.16. If У is a retract of X, then for all i
0----► K~\X,Y)------► K~\X)------► K~\Y)------► 0
is a split short exact sequence and K~l(X) К"‘(Х,У) ф K~l(Y). This
follows from the Barratt-Puppe sequence (9.4).
Now if X and Y are spaces with basepoints then
К~‘(Х x У) K~\X л У) ф К~\Х) ф К~\У)
since X is a retract of X x У and У is a retract of X x У/ X.
We would like now to use the ring structure on K(X) to enable us to
define a ring structure on K~*(X).
Proposition 9.17. Given X and Y and i,j > 0 there is a pairing K~'(X) ®
K~j(Y) -► K~'~j(X л У) which is given by tensor product.
Proof. For a bundle E on S1 л X and a bundle F on S' л У we have
the tensor product bundle E ® F on (Sl л X) x (SJ л У). This induces a
pairing
K(Sl л X) ® K(Sj л У)-----► K((S'1 л X) x (Sj а У)).
But K((Sl л X) л (SJ а У)) is the kernel of X((S1’ л X) x (S' л У)) -
X(Sf л X) ф K(Sj л У). So we have a pairing K(Sl л X) ® K(Sj л У) ->
К((& л X) л (S' л У)) = K((Si+j) л X л У) as desired. 
Replacing X by X+ and У by У+ in Proposition 9.17 gives us a pairing
K~l(X) ® K~j(Y) -► K-i“'(X x У). We easily conclude the following.
Corollary 9.18. The pairing of Proposition 9.17 makes K“*(pt) a graded
ring. Furthermore, for any basepointed space (X,pt), this pairing makes
K~*(X) a graded module over K'*(pt).
Thus far everything we have said for К-theory holds equally true for
KO-theory. We will now describe periodicity and at this point the descrip-
tions must diverge. We will discuss the basic complex case first and then
proceed to the real case. We will simply state the results. A proof based
on the theory of Fredholm operators is given in §10 of Chapter III. For
alternative proofs the reader is referred to Bott [1], [4].
Bott Periodicity Theorem 9.19 (the complex case). The ring K“*(pt) is a
polynomial algebra generated by an element £ e K~2(pt) s K(S2), i.e., there
is a ring isomorphism
s ад.
(9.5)
§9. K-THEORY AND THE ABS CONSTRUCTION	63
Note.~The element £ is in fact represented by the virtual bundle £ = [H] —
t1 g K(S2) where H denotes the “tautologically defined” complex line bun-
dle over S2 = P^C) and t1 denotes the trivial line bundle.
The isomorphism (9.5) says in particular that the map /z{:K-,(pt) -►
K-,"2(pt) induced by multiplication by is an isomorphism for all i. In
this form, the theorem extends to arbitrary compact Hausdorff spaces.
Recall from 9.18 that for any pointed space (X,pt) the ring K~*(X) is a
module over K“*(pt).
General Bott Periodicity Theorem 9.20 (the complex case). Let X be any
compact Hausdorff space. Then the map
given by module multiplication by £, is an isomorphism for all i 0.
Note. Replacing X by X/Y, we get corresponding isomorphisms
^:К~\Х,У)---------------------------► K~'~2(X9Y)
for any pair (X, Y) of compact Hausdorff spaces.
The situation in KO-theory is slightly more complicated.
Bott Periodicity Theorem 9.21 (the real case). The ring KO“*(pt) is gene-
rated by elements
у g KO-1(pt), у g K0"4(pt), x g KO"8(pt)
subject only to the relations
= 0,	r]3 = 0, rjy = 0, y2 — 4x
i.e., there is a ring isomorphism
KO~*(pt) s Z[ti,y,xy(2fi,fi3,tiy,y2 - 4x>	(9.6)
As before we have that for any space X, KO ~ *(X) is a KO ~ *(pt)-module.
Theorem 9.22. Let X be a compact Hausdorff space. Then the map
px-.K0~\X)--------------------------► K0’*’8(X)
given by module multiplication by x e KO-8(pt), is an isomorphism for all
i£0.
As before there are also isomorphisms K0~\X,Y) ^>K0~'~8(X,Y)
for any pair (Х,У) of compact Hausdorff spaces.
64	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
The best explicit representatives for the elements tj9 x and у in KO'*(pt)
are given via the Atiyah-Bott-Shapiro isomorphism. To present this iso-
morphism, and also to adapt К-theory easily to the theory of elliptic
operators, we present now an alternative definition of the groups K(X9Y)
and KO(X9Y). The discussion is completely parallel in the real and com-
plex cases. We shall just use the generic term “vector bundle.”
We begin with the following definition. We assume throughout that Y
is a closed subspace of X,
Definition 9.23. For each integer n 1, consider the set &n(X9Y) of
elements V = (Ko, V19...	... ,<ти) where Ko,... 9Vn are vector bundles
on X and where
°—* K»|rк|гK|r—>0
is an exact sequence of bundle maps for the restriction of these bundles to
Y. Two such elements V = (Ko,... ,Ки;аь... ,a„) and V' = (K'o,... 9V'n'9
ap... 9a'„) are said to be isomorphic if there are bundle isomorphisms
(pi•: Vt Vi over X so that the diagram
K-l|r К у
(Pi-i	<PI
и;|,
commutes for each i.
An element V = (Fo,... 9Vn9 o19... ,a„) is said to be elementary if there
is an i such that
(a)	Vi = Vi^i and af = Id
(b)	Vj = {0} for j i or i — 1.
There is an operation of direct sum ф defined on the set £fn(X9Y) in
the obvious way. Two elements V,V' g £fn(X9Y) are defined to be equiv-
alent if there exist elementary elements En ... ,Efc,Fb ... ,F, g £fn(X9Y)
and an isomorphism
V ® Ej ® • • • ® V' ® Fj ®	® F,.
The equivalence class of an element (Ko,... ,Vn;vu... ,a„) will be denoted
by [Ko, • • • i, • • • The set of all equivalence classes in £fn(X9Y)
will be denoted by Ln(X9Y).
The set Ln(X9Y) is an abelian group under the operation ф.
Our first main proposition is the following, whose proof is left to the
reader. Consider the natural map <£n(X9Y) -► &n + 1(X9Y) which asso-
ciates to each element (Ko,... 9Vn-9 a1?... the trivially extended element
(Ko,...,^;^,...^).
§9. K-THEORY AND THE ABS CONSTRUCTION
65
Proposition 9.24. For each n 1, the induced map Ln(X,Y) -► L„+1(X,Y)
is an isomorphism.
We set L(X,Y) = limL„(X,Y). Each inclusion Ln(X,Y) -► L(X,Y) is an
isomorphism, and it would have sufficed for many purposes to consider
only the case n = 1.
Our second main proposition is the following.
Proposition 9.25. There exists a unique equivalence of functors %: L(X, Y) ->
K(X,Y) with the property that
X([K» • • • Л]) = t (- 1)*TO when Y = 0.	(9.7)
k = 0
Proof. This equivalence will be essentially determined by defining it on
L^X.Y). Given an element V = [K0,Vi;a] g L^X.Y) we associate to it an
element /(V) g K(X,У) by the following “difference bundle construction”.
Set Xk = X x {к} for к = 0,1 and consider the space Z = Xo Xr ob-
tained from the disjoint union Xo U Xt by identifying у x {0} with
у x {1} for all у g Y. The natural sequence
0----► K(Z,X1) K(Z) K(XJ----------------► 0
is split exact since there is an obvious retraction
p:Z-----
Furthermore, there is an isomorphism
gr.KtZ'XJ K(X,Y)
induced by the map of pairs (X,Y) -> (Z,XJ which identifies X with Xo.
From our element V = [K0,Kij 67 ] we define a vector bundle W over Z
(well defined up to isomorphism) by setting = Vk and identifying
over У via the isomorphism a. Setting = p*(Vi) we have [PY] —
[WJ g ker(i*). Hence, there is a unique element /(V)gK(X,Y) with
= [PY] — [W\]. This defines the homomorphism %:LfX,Y) -►
K(X,Y). It clearly has property (9.7).
It is now straightforward to verify that any homomorphism L^X.Y) -►
K(X,Y) with property (9.7) is an isomorphism, and furthermore, any two
such homomorphisms agree. The reader is referred to Atiyah-Bott-Shapiro
[1] for details. 
As a result of this proposition we shall henceforth drop the notation
L(A’,Y). We will however discuss elements [Ко/и^а] g K(X,Y) whose
meaning is now obvious.
For our later discussion of elliptic operators it will be useful to note
that the multiplication in K(X,Y) can be realized explicitly in J$f\(X,Y)
66
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
as follows. Choose V = (К0,Кх;а), W = (РУ0,И^;т) g S\(X,Y) and for con-
venience introduce metrics in each of the bundles. We then define the
tensor product U = V ® We££\(X9Y) to be the element U =
where
UQ = (Ko ® PK0) Ф ® W.) U. = (Kt ® %) ф (Ko ® WJ
and
/а ® 1 — 1 ® t*\
p = I	)•
\1 ® т a* ® 1 /
Under the construction above, this tensor product on <fZ\(X,Y) carries
over to the standard product on K(X,Y).
This is a convenient time to introduce К-theory for locally compact
spaces. К-theory in this setting will be important for certain geometric
constructions we shall make in Chapter III.
Definition 9.26. For any locally compact space X we define
Kcpt(X) = K(X+)
where X + = X u {pt} denotes the one point compactification of X. The
higher groups are defined by setting
№) = Kcpt(X x Rl)
for i 0.
The groups are functors on the category of locally compact
spaces and proper maps. Collectively they comprise the X-theory of X with
compact supports. They enjoy the multiplicativity properties that we pre-
sented above in the compact case. Note incidentally that if X is compact,
then K-J(X) = K~\X).
Any element in Kcpt(X) can be represented as the formal difference of
two bundles E and F on X, each of which is trivialized at infinity (i.e.,
trivialized outside some compact set in X). In fact, if О с X is an open
subset of the locally compact space X, then there is a natural extension
homomorphism Xcpt(0) -> Kcpt(X) induced by the map X+ -► X+/
(X+ —	= ®+. Taking products with R1 gives extension homomorphisms
--->K^(X)
for all i 0. Of course for any closed subset Y a X we have the functorial
restriction homomorphism
k-{(X)—>Kc;j(Y).
The kernel of this map is defined to be the relative group K~^(X,Y\ For
any pair (X,Y) where X is a locally compact space and Y is a closed sub-
§9. K-THEORY AND THE ABS CONSTRUCTION	67
space, there is a long exact sequence for the -groups, analogous to
(9.4) above. We leave as an exercise for the reader the verification of the
following isomorphism:
К-;(Х,У) Kcpt((X - Y) x R‘).
There exist definitions of “L-type” for the groups Xcpt(X). In analogy
with the above we can define L„(X)cpt to be the equivalence classes
where Ko,... ,Vn are vecctor bundles on X, and
where 0 -► Vo	Vn -> 0 is an exact sequence of bundle maps
defined on the complement of some compact subset in X. As above there
are natural isomorphisms Lx(X)zpt L2(X)cpt Л ... Kcpt(X). Thus, in
particular, any element of Kcpt(X) can be represented by a triple [Ko, Vi; a]
where o:V0 Vx is a bundle isomorphism defined in a neighborhood of
infinity.
All of the discussion above applies equally well to real bundles and
yields groups KO~p\(X) for any locally compact space X.
The Bott Periodicity theorems carry over to locally compact spaces in
the following elegant form:
Kcpt(X) Kcpt(X x C)
(9.o)
KOcpt(X) s KOcpt(X x R8).
These isomorphisms are induced by multiplication by fundamental ele-
ments £ g Kcpt(C) and x g KOcpt(R8) respectively. In the last part of this
chapter we shall produce explicit representatives for these elements using
Clifford modules.
We are now in a position to discuss the isomorphism of Atiyah, Bott
and Shapiro. Let W — W° ф W1 be a Z2-graded module over the Clifford
algebra C£„ = C£(R"). Let Dn = {x g R": ||x|| < 1} be the unit disk and set
S"-1 = dDn. We now associate to the graded module W, the element
<p(W) = [E0A;m] e K(D",S"-1)	(9.9)
where Ek = Dn x Wk is the trivial product bundle, and where p: Eo Л Er
is the isomorphism over S"'1 given by Clifford multiplication:
/x(x,w) = (x, x • w).
One easily checks that the element cp(W) depends only on the isomor-
phism class of the graded module W and, furthermore, that the map
W i—► cp(W) is an additive homomorphism. Hence, (9.9) gives us a homo-
morphism
---->Х(РЛ,5Л"1)	(9.10)
where is the Grothendieck group of complex graded C£„-modules
defined in §5.
68
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Consider for a moment the natural inclusion i:R" R" + 1 given by
setting i(xx, ... ,хи) = (xb ... ,x„,0). This induces an algebra homomor-
phism i*: С£и C£„+ p Restricting the action from С£и+ x to С£и thereby
induces a homomorphism i* :2R£+1 -►
Suppose now that IT is a graded C£„-module which can be obtained
from а С£и + 1-module in the above fashion. This means that the Clifford
multiplication of R" on W extends to all of R" + 1. Hence we may extend
the isomorphism д, defined on S"-1 = 5D", to all of Dn by setting
= (x,(x + Vl - IW|2e„+1) • w)
where еи+1 g R" + 1 is a unit vector orthogonal to R". Since this extended
map is an isomorphism over all of D", the associated element cp(W) must
be zero. It now follows that the homomorphism (9.10) descends to a homo-
morphism
Фи:<Л*<+1---------> K(Dn,Sn~l)	(9.11)
where i*: x -► is the restriction map defined above.
Exactly the same construction applied to the real case, gives us a homo-
morphism
---► KO(Dn,Sn-').	(9.12)
The isomorphisms (5.29), defined by taking the even part, determine
isomorphisms
1 s t /i*< and </i*< +, =
Set Qj =	1 and Qn —	+1. The periodicity pheno-
mena (4.11), (4.13) and (5.9) determine periodicity isomorphisms
Й+2 = Й and Q„+ft = Q„-
Elementary algebraic arguments show that
if n is even
if n is odd
if n = 0 or 4 (mod 8)
if n = 1 or 2 (mod 8)
otherwise.
As an example consider Q2 =	considered as the quotient of the
groups of ungraded modules. Since Ctj = C, the Ct j-modules are just
complex vector spaces, and the isomorphism A Z is generated by
taking the complex dimension. Similarly, since CC2 = H, the Cf2-modules
are quaternionic vector spaces and 5R2 Д Z is generated by taking the
quaternionic dimension. The map t*:9W2-+ JRj is generated by consider-
ing a quaternionic vector space to be a complex vector space under
restriction of scalars. Clearly this is just the map Z -> Z given by mul-
§9. K-THEORY AND THE ABS CONSTRUCTION	69
tiplication by 2, since the complex dimension is twice the quaternionic
dimension. Thus we have Q2 = Z2.
The reader may find the verification of these isomorphisms in general
to be an interesting exercise. Full details are contained in Atiyah-Bott-
Shapiro [1].
We recall now from Proposition 5.21 that the graded tensor product
of modules gives a multiplication ®	®l^+m for all m,n. This
makes (ЭД*/**ЯИ* + 1) =	/l’*W+i) into a graded ring.
On the other hand, by definition we know that
K(D\Sn~l) = K(Sn) = К ""(pt)
and therefore the direct sum @n^0K(Dn,Sn~r) = K"*(pt) is also naturally
a graded ring (cf. Corollary 9.18). (The analogous comments apply in the
real case.) One of the main results of Atiyah-Bott-Shapiro [1] is the
following.
Theorem 9.27 (the Atiyah-Bott-Shapiro Isomorphisms). The maps (9.11)
and (9.12) defined above induce graded ring isomorphisms:
<p#:(SW*/i*SW* + i) —K”*(pt)
Ф#:(ТО#Л*ОТ# + 1)^КО-*(рО.
Since the periodicity of the quotients	+ i is an elementary al-
gebraic fact, this theorem appears to give a new algebraic proof of the
Bott Periodicity theorems. However, the argument given in Atiyah-Bott-
Shapiro [1] to establish the isomorphisms of 9.27 actually invokes the
Bott result. Nevertheless the existence of these isomorphisms is a profound
and important fact. It goes a long way towards explaining the fundamental
role played by Clifford algebras in the index theory for elliptic operators.
Remark 9.28. Theorem 9.27 gives us explicit generators for K“*(pt)
and KO"*(pt) defined via representations of Clifford algebras. For ex-
ample, let Sc = Sq © Sq be the fundamental Z2-graded representation
space for Ct2„ where = (1 ± cuc)Sc. (The sign of the complex volume
element cuc depends on a choice of orientation in R2".) There is an iso-
morphism ®l2„ = Z © Z with distinguished generators given by Sc and
its “flip” Sc, the same graded module with the factors interchanged. (This
corresponds to a reversal of orientation in R2".) The generator of i*9JI2w +1
is [Sc] + [Sc]. Hence, the group K"2"(pt) Kcpt(R2") Z is generated
by the element
^с,и — [^c ,SC; p]
where px: S£ -> denotes Clifford multiplication by x e R2".
70
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
The real case is entirely analogous. Let S = S+ ® S be the fundamen-
tal graded module for Ct4„ where S1 = (1 ± m)S. Then
^4и = [S+,S ;д]
is a generator of the group K0“4"(pt) K0cpt(R4") Z.
Using the structure of Clifford modules we can easily compute that
ffc,B =	ff8n = (ff8)" and 4<r8 = (<r4)2.
§10. KR-Theory and the (l,l)-Periodicity Theorem
In this section we present a theory which, in a sense, contains both K-
theory and KO-theory. It was invented by Atiyah in the middle 1960s
and was motivated in part by the study of indices for families of real
operators. (There is a detailed discussion of this at the beginning of §16
in Chapter III.)
The higher groups in this theory carry a natural double-indexing KRr,\
r ^0, s 0. Interestingly, there is an isomorphism of the Atiyah-Bott-
Shapiro type given in §9, which relates Kr,s(pt) with real modules over the
Clifford algebra Cfr 5. In this sense, KR-theory is the analogue of K-
theory and KO-theory suggested by passing from and Cln to Ctr s.
Basic references for this material are Atiyah [2] and Karoubi [2].
We consider here the category of Real spaces, i.e., spaces with involu-
tion. This is the category of pairs (X,cx) where X is compact and cx:
X -► X is a map with = Idx. The map cx may itself be the identity.
Natural examples of such spaces are provided by complexifications of real
algebraic varieties where c is given by complex conjugation. Such an
example is (P"(C),c) where in homogeneous coordinates c([z]) = [f]. The
fixed point set here is IP"(IR) <= P"(C).
Definition 10.1. By a Real vector bundle over a Real space (X,cx) we
mean a pair (F,cK) where n: V X is a complex vector bundle over X
and where cv: V V is an involution such that the diagram
V	V
n	n
X	X
commutes and cv is C-antilinear on the fibres of V. We denote by VR(X,cx)
the abelian semigroup of isomorphism classes of Real bundles over (X,cx).
Proceeding as in §9 we then define the associated Grothendieck group
KR(X,cx). It is called the Real-K-group of X. We shall generally drop the
explicit mention of the involution cx and simply write KR(X).
§10. KR-THEORY AND (1, ^PERIODICITY	71
Remark 10.2. Note that if cx = Idx then we have a natural identifica-
tion K0(X) s KR(X). This identification associates to any real bundle FR
on X, the pair (FR ® C,c) where c denotes complex conjugation in the
fibres. Fr is recovered as the fixed-point set of c.
The groups KR(X) = ker(KR(X) -► KR(pt)) and KR(X,Y) = KR(X/Y)
are defined exactly as in §9 and give functors on the obvious categories
ofjspaces. The higher groups: KR'^X] = KR^X) and KR-\X9Y) =
KRf^X/Y)), are also defined as in §9. We have the exact sequence for
compact pairs (А",У):
...----► KR~l(X,Y)-----► KR-‘(X)-----► KR~l(Y) (10.1)
----► KR~i+1(X9Y)------► ...
An interesting facet of KR-theory is that it carries naturally a doubly-
indexed family of higher groups. Let Rr,s = IRr ®	= “IRr © Ш5” be
the Real linear space with involution c(x,y) = (x, — y). A basic case is
R1,1 £ C with complex conjugation. Consider now the compact Real
subspaces
D” = {(x,y)e R": ||x||2 + ||y||2	1}
Sr’s = {(x,y) 6 Rr,s: ||x||2 + ||y||2 = 1},
and for any compact Real pair (Х,У) define
Kr,s(X,Y) = KR(X x Drs, X x Sr’s и У x Drs). (10.2)
(The order of (r,s) here is the opposite of that in Atiyah [2].) Note that
it is immediate from the definitions that
KRi0 = KR~‘ for all i 0.	(10.3)
The sequence (10.1) generalizes to give exact sequences
...----► KRrs(X,Y)-----► KRr'\X)-----► KRr,s(Y)
----► KR'~1,S(X,Y)-----► ...	(10.4)
for all s 0. To prove this one establishes the exact sequence for compact
triples in KR-theory above, and then applies it to the triple (X x D°'s, X x
S° s и У x DO s, X x S0’»).
One of the basic facts about KR*'* is that the exterior tensor product
induces a bigraded multiplication
KRr s(X,Y) ® KRrs(X',Y’)-------► KRr+r',s+s'(X",Y")	(10.5)
where X" = X x X' and У" = X x У' u X' x У. In particular, as in §9,
the “coefficients” KR*’*(pt) are a bigraded ring, and for any compact
space X, KR*’*(X) is a graded KR*’*(pt)-module.
72	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
^Consider for a moment the group KR14(pt) =	=
KRfP^C)). One can show that this group is isomorphic to Z with genera-
tor £ = [H] — tx where H -► PX(C) is the tautological (or “Hopf”) com-
plex line bundle over P\C) with its natural Real structure. (The fibre
for / g P*(C) is the line / itself.)
One of the fundamental results in this theory is the following (see Atiyah
[2]):
The (l,l)-Periodicity Theorem 10.3. Let X be any compact Hausdorff space.
Then the map
p^:KRr,s(X) KRr+1’s + 1(X)	(10.6)
given by module multiplication by is an isomorphism for all r 0, s 0.
Consequently, for any compact Real pair (X,Y) we have the isomorphism
KRr+1’s+1(X,r)	(10.7)
Corollary 10.4. There exist natural isomorphisms
KRrs^KRsr	(10.8)
for all r^.s^.0 generalizing (10.3).
Using (10.8) we can extend the definition of KR~' to all integers i. The
exact sequence (10.1) then extends to infinity in both directions.
Remark 10.5. There are many further internal symmetries in this the-
ory. Using the multiplication in the fields R, C and H, Atiyah [2] shows that
for any compact space X there are isomorphisms
KR(X x S°’p) KR~2p(X x S°’p)	(10.9)
for p = 1, 2, and 4. (Recall that dim S0,p = p - 1.) This isomorphism for
p = 1 gives the complex Bott Periodicity Theorem. From the case p = 4
one can deduce the real periodicity theorem.
Remark 10.6. All of the discussion in §9 concerning the functors L„ and
their equivalence with К carries over to the Real situation. Therefore, in
particular, elements in KR(X,Y) can be represented by classes of the
form [F0,*b<] where Ko and are Real bundles on X and where
a: Vo -> Vt is a Real bundle isomorphism defined over У.
Remark 10.7. If X is a locally compact space, then the groups KRcpt(X)
are defined exactly as in 9.26. This group is generated by Real bundles on
X which are trivialized at oo. Using the L-construction, it can also be
generated by triples [Kb^jcr] where Vo and are Real bundles on X and
where a: Vo -► Vr is a Real bundle isomorphism defined outside of some
compact subset of X.
§10. KR-THEORY AND (1, 1)-PERIODICITY	73
In this context the higher KR-groups can be written in a particularly
nice form:
KRr’°t(X) = KRcpt(X x ВТ’*).	(10.10)
Of course if X is compact, then KRcpt(^) = KRr,s(X)- If we identify C =
R1’1 as before, then the (1,1)-Periodicity Theorem has the particularly nice
form:
KRcpt(X) KRcpt(X x C)	(10.11)
for any locally compact space X,
In light of the Remark 10.6 above it is natural to examine the Atiyah-
Bott-Shapiro construction in this theory. To do this we define the Real
Clifford algebra Ct(IRr’s) to be the Clifford algebra generated by Rr,s and
the positive definite quadratic form q(x9y) = ||x||2 4- ||y||2, taken together
with the algebra involution c: C£(Rr’s) -► Ct(Rr,s) generated by the involu-
tion (x,y) i-> (x, — y) of Rr,s.
Definition 10.8. By a Real module over the algebra Ct(Rr,s) we mean
a finite-dimensional complex representation space W for Ctr>s together
with a C-antilinear involution c: W -> W so that
c((p • w) = c((p) • c(w) for all (p g Ct(Rr,s) and all w g W. (10.12)
If in addition W — W° ® W1 is a Z2-graded module with the property
that c(JV') = W' for i = 0,1, then PF is called a Real Z2-graded module for
C£(Rr’5).
We denote by 9WRr,s and ®lRr s the Grothendieck groups of equivalence
classes of Real modules (and Real Z2-graded respectively) for Ct(IRr’s).
There is an important basic relationship between these Real algebras
and the algebras Ctr>s defined in §3. To begin, consider the complexifi-
cation Ct(Rr’s) of C£(IRr’s) and note that the involution c has a unique
extension to a C-antilinear involution on Ct(Rr,s). Any Real Ct(Rr,s)-
module is naturally a Ct(Rr,s)-module by extension, and the condition
(10.12) continues to hold. Considering these algebras and their modules
is equivalent to considering those above.
Now the main claim is that Ct(IRr’s) is the appropriate complexification
of Cfr 5. Under the ordinary complexification, all the algebras Ctr s with
the same value of r 4- s become isomorphic. However, if we also introduce
the involution, this is not so.
Recall that Ctr s is the Clifford algebra generated by IRr ® IRS with the
quadratic form qftS(x9y) = ||x||2 — ||y||2. Let s and 2Rr s denote the Gro-
thendieck groups of R-modules (and Z2-graded IR-modules respectively)
for the Clifford algebra Ctr>s.
74	I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proposition 10.9. There is a natural equivalence
Wr>s	(10.13)
for each (r,s) defined by assigning to a real C£rs-module W, the complex
module W ®R C with involution given by complex conjugation and with the
Cl(Rr,s)-multiplication engendered by setting
(x,y) • w = xw 4- iyw for (x,y) g Rr ©	(10.14)
Furthermore, under the natural inclusion i: Rr © Rs Rr+1 ® Rs given
by i(x19... ,xr,y19... ,ys) = (xn . .. ^„O,^,. .. ,ys) the diagram
®®r+l,s	> ®lRr+M
9Wr,s ---9WKr,$
commutes. Hence, there are natural graded isomorphisms:
+	+	(10.15)
All the analogous statements hold in the ^2-graded case.
Proof. The multiplication given in (10.14) has the property: (x,y) • (x,y) •
w = — (||x||2 4- ||y||2)w and therefore extends to Ct(Rr’s). Furthermore, we
see that c[(x,y)  w] = (x 4 iy)w = (x — iy)w = c(x,y) • c(w) where both ( )
and c denote complex conjugation. Hence, the map (10.13) is well defined.
The inverse is given by taking the real module to be the fixed-point set
of c and replacing multiplication by (x,y) e Rr © Rs with multiplication by
x — iy. The remaining details are left as an exercise for the reader. 
Remark 10.10^ As in previous cases the graded tensor product of mo-
dules makes and into graded rings. The multiplication is
preserved by the equivalence (10.13) in the graded case. In particular, the
map
+	+ 1 #	(10.16)
is a ring isomorphism.
The advantage of the groups ®lRr>s is that they are natural for extend-
ing the Atiyah-Bott-Shapiro construction given in §9. On the other hand,
the groups 50lrs and the quotients 50lr s/i*50lr+ M are particularly easy to
compute by using the results of §§4 and 5.
Suppose that W = W° ® Wl is a Real Z2-graded module for Ct(Rr,s),
and consider the associated element <p(W) g KRcpt(Rr,s) KRr,s(pt) given
by
<p(W) = [Eo.Ep/1]
§10. KR-THEORY AND (1, 1>PERIODICITY	75
where Ek = W’s x Wk for к = 0,1, are the trivial product bundles and
where д:Е0 -► Er is defined by
/z(z,w) = (z, z • w)
for z = (x,y) g Rr,s and for w g W°. Since z • z • w = — ||z||2w, the map д
is a bundle isomorphism outside the origin and so [Ео,£\; д] is an ele-
ment of KKcpt(Rr’s) (cf. Remark 10.7). This gives a graded ring homomor-
phism
(p:№R*'*------► KR*’*(pt).	(10.17)
Arguing as in §9 one can show that (p = 0 on + and so (p des-
cends to the quotient. The arguments of Atiyah-Bott-Shapiro [1] then
carry through to prove the following (see Atiyah [2]).
Theorem 10.11. The map defined above,
+	---► KK**(pt),
is a graded ring isomorphism.
Via the isomorphism (10.16) this relates the groups KRr,s(pt) to the alge-
bras C£r s. In particular, the (l,l)-Periodicity Theorem is reflected in the
(l,l)-periodicity for these algebras (cf. (4.3) of Theorem 4.1). Furthermore,
since KRs,0(pt) = KR~s(pt) = KO"s(pt) and C£s>0 C£s we recapture
here the real ABS-isomorphism of Theorem 9.27.
From (l,l)-periodicity we see that
KR°’°(pt) s KRr i(pt) KR2’2(pt) Z.
For our later discussion of the Ct„-linear Atiyah-Singer operator (in III. 16)
it will be useful to have certain explicit generators for these groups, which
we shall now present Recall that as a Real space IR"’" is just C" with
involution given by complex conjugation, and therefore we can write
KR"’"(pt) = KRcpt(C"). We will give an explicit generator for KRcpt(C")
by using the Clifford algebra Ct„ = Ct(C") and its natural /2“8га^п8
CtH = Ct? ® Ct,1. For (p g Ct„ let R^.Ct, -► Ct, and L^:Ct„ -► Ct,
denote right and left multiplication by (p respectively. For x -I- iy g C",
we consider the map
(Rx + iLy): Ct?----► Ct,1	(10.18)
which, since (Rx + iLy)(Rx — iLy) = — (||x||2 + l|y||2)Id, is invertible when
x + iy#0. This map (10.18) is clearly a Real endomorphism, i.e.,
(Rx + iLy)(p = (Rx -I- iL_y)<p.
Proposition 10.12. The element
£n = [се»,cel; Rx + Щ 6 krcp.(C")
is a generator of the group KRcpfCn} = KR"’"(pt) s Z for all n 1.
76
I. CLIFFORD ALGEBRAS AND SPIN GROUPS
Proof. Because of the isomorphism of graded algebras =
Cti ® • • • ® C£j (which follows from Proposition 3.2) it suffices to con-
sider the case n = 1. The element can be shown to coincide with the
generator of the image of (p in Theorem 10.11 as follows. To begin note
that since Ct? = C • 1, right and left multiplication coincide on Ct?.
Consequently, £ [C,C;m] where m(xy}iC -► C is given by scalar multi-
plication: m(xy)(w) = (x + iy)w.
On the other hand we have = Z ® Z with generators given as
follows. For (x,y) g R1,1, define a C-linear map С © C -► С © C by the
matrix
At (x,y), the square is — (||x||2 4- ||y||2)Id. Hence, this action extends to
make С ф C (with involution given by complex conjugation) into a com-
plex Z2-graded Ct(R14)-module. This is the first generator. The second
generator is obtained by interchanging factors. That these are indepen-
dent generators is easily verified by working through the natural iso-
morphisms
= 9Л01
(cf. Proposition 10.9 and Theorem 3.7). These isomorphisms commute
with i*, and working out the details for the simple case 5R0>1/i*5Rlfl
shows that either of ourdistinguished generators becomes a generator of
the quotient: SOlRj4/i*5RR24 KK1J(pt). Clearly under the isomorphism
cp of Theorem 10.11 our module above with multiplication 10.18 becomes
the element [C,C; m] as claimed.
CHAPTER II
Spin Geometry and the Dirac Operators
In this chapter one finds the soul of the book. It is here that we examine
the structures and concepts that are central to differential geometry—
manifolds, vector bundles, connections, curvature, etc. If one also intro-
duces metrics (in the sense of Riemann), then it is unavoidable that Clifford
algebras and spin groups will enter the discussion. This is for the following
reason.
There is a principle by which the natural operations on vector spaces
such as direct sum, tensor product, exterior power, etc., carry over canoni-
cally to vector bundles. In the same fashion, the natural operations for
vector spaces with quadratic forms carry over to vector bundles with
metrics. In particular, suppose that n: E -► X is a riemannian vector
bundle. Then in each fibre, Ex = я-1(х), the quadratic form ||u||2 = <u,u>
can be used to construct a Clifford algebra Ct(Ex). The result is a bundle
Ct(E) X of algebras over X called the Clifford bundle of E. It carries
all the natural properties of Clifford algebras such as the Z2-grading, the
transpose endomorphism and the L-operator. This rich structure is basic
for the study of E itself.
In the light of Chapter I it is natural to ask whether one can also find
a vector bundle $(E) -► X with the property that each fibre $(E^ is an
irreducible module over Ct(Ex). The answer is in general no. We shall
discuss the obstruction involved. By taking E = TX this will lead to the
notion of a spin manifold and spin cobordism. The bundle $(TX), when it
exists, plays a central role in the study of the global geometry and to-
pology of X.
In general, given any bundle of modules S over Ct(TX), furnished with
a suitable metric and connection, one can associate to S a self-adjoint,
first-order elliptic operator D: T(S) -► T(S) called the Dirac operator for
5. (This is done in §5.) From the algebraic operations of Chapter I one
can often construct natural splittings S = S+ ® S” with respect to which
the Dirac operator has the form
where D+: T(S+) -► T(S ) and D :T(S ) -► T(S+) are formal adjoints of
one another. This gives a unified procedure for constructing all the “clas-
78
II. SPIN GEOMETRY AND DIRAC OPERATORS
sical” elliptic operators on a manifold, namely: the Euler characteristic
operator, the signature operator, and the Atiyah-Singer operator. Details
of this are given in §6.
In §7 we introduce the notion of a Ctk-linear operator and discuss in
detail some of the basic examples. Over a compact manifold any Ctk-
linear Dirac operator D has an analytic index indk(D) g KO"k(pt) defined
by applying the Atiyah-Bott-Shapiro isomorphism to the residue class of
the Clifford module [ker D] (see 1.9).
Roughly speaking, every Dirac operator represents the square root of
a Laplace operator. In euclidean space this statement is unambiguously
true. Over general manifolds, however, the difference D2 — V*V between
the square of the Dirac operator and the standard “connection laplacian”
is a certain universal expression involving curvature and Clifford multi-
plication. Deriving such formulas and using them to draw global con-
clusions about curvature and topology is referred to as “Bochner’s
method.” We shall show how our universal formula specializes to give
the classical Bochner formulas on exterior differential forms, as well as
the Lichnerowicz formula for spinors. Using curvature identities derived
in §5, we shall then give a succinct proof of the theorem of Gallot and
Meyer concerning curvature and homology spheres. We shall also derive
formulas for the Atiyah-Singer operator with coefficients in a bundle.
This will be quite useful later in studying manifolds of positive scalar
curvature.
§1. Spin Structures on Vector Bundles
Let n: E -► X be a real n-dimensional vector bundle over a manifold X.
We assume this bundle is equipped with a riemannian structure, that is,
a positive definite inner product continuously defined in the fibres. Such
a structure always exists.
We assume also that the bundle is oriented, i.e., that there is an orienta-
tion continuously defined on the fibres. This structure does not always
exist. To analyze the situation, we consider the bundle Pq(E) of ortho-
normal frames in E. This is the principal O„-bundle whose fibre at a point
x g X is the set of orthonormal bases of Ex = n~ \x). The bundle of orien-
tations in E is then just the quotient Or(E) = Po(E)/SO„, where two bases
of Ex are identified if the orthogonal matrix transforming one to the other
has determinant +1. Note that Or(E) is a 2-sheeted covering space of X
and that E is orientable if and only if this covering space is the trivial
one. We recall now the following elementary fact:
Lemma 1.1. Let Cov2(X) be the set of equivalence classes of 2-sheeted
covering spaces of X. Then there is a natural isomorphism
Cov2(X)^H\X;Z2).	(1.1)
§1. SPIN STRUCTURES ON VECTOR BUNDLES
79
Note. This is a special case of the isomorphism:	« {equivalence
classes of principal G-bundles on X}, which is proved in Appendix A.
Proof. Assuming X is connected, we can decompose the isomorphism
(1.1) as follows: H\X; Z2) Д HomfH^X), Z2) Л Hom(7q(X), Z2) Л
Cov2(X). The second isomorphism follows from the fact that Ht(X) is
the abelianization of n^X). The third isomorphism is a restatement of
the elementary fact that 2-sheeted coverings of X are in one-to-one
correspondence with subgroups of index 2 in n^X). The case where X
is not connected now follows immediately. 
From (1.1) we now see that, for each vector bundle E over X, the 2-
sheeted covering space Or(£) determines an element w^E) e Hx(X;Z2),
called the first Stiefel-Whitney class of E. Directly from the definitions
we have the following fact.
Theorem 1.2. A vector bundle E over X is orientable if and only if w^E) = 0.
Furthermore if w^E) = 0, then the distinct orientations on E are in one-
to-one correspondence with elements of H°(X;Z2).
The second statement simply says that there are two possible orienta-
tions of £ over each connected component of X.
The definition of w^E) given above is in accord with the one given
via classifying spaces (see Арр. B). To prove this it suffices to show the
following:
(i) This definition of vvx is natural, i.e., w^f+E) = /♦w^E) for any
bundle £ over X and any continuous map f :X' -► X.
(ii) This definition of wt(E) gives the non-zero element in
H1 (ii)(BO„;Z2) Z2 when E is the universal n-plane bundle over the
classifying space BO„.
Fact (i) is obvious since Po(f*E) = /*PO(£) and, therefore, Or(/*£) =
/*Or(£). Fact (ii) is true since otherwise every n-plane bundle would be
orientable.
By establishing properties (i) and (ii) it is possible to prove the equiv-
alence of a number of quite different definitions of w^E). For example,
suppose X is connected. Then from the fibration O„ -► PO(E) -► X, there
is an exact sequence.
0	H°(X;Z2)-------► H°(Fo(£);Z2)-----► H°(O„;Z2)	H\X;Z2).
(1.2)
We can define wr(E) — wE(gi) where gt is the generator of H°(O„;Z2).
This definition of wx(£) has property (i) since the sequence (1.2) is natural
and property (ii) since the sequence (1.2) is exact and for the universal
80
II. SPIN GEOMETRY AND DIRAC OPERATORS
bundle E, PO(E) = EO„ is contractible. From the exactness of (1.2) it is
clear that w^E) = 0 iff PJiE) is disconnected, i.e., iff E is orientable.
We shall examine other equivalent definitions of later in this section.
At the moment we pass on to the next possible simplification of the
structure group of a bundle E. Note that if E is orientable, then choosing
an orientation is equivalent to choosing a principal SO„-bundle P$o(E) c
Po(£). This embedding is, of course, compatible with the action of SO„ c
O„. Having thereby made the structure group of E О-connected, one
might ask whether it is possible to make the structure group 1-connected.
This leads us to the concept of a spin structure.
Let £ be an oriented n-dimensional riemannian vector bundle over a
manifold X, and let Pso(£) be its bundle of oriented orthonormal frames.
Recall that for n 3 we have the universal covering homomorphism
£0:Spin„ -► SO„ with kernel {1,-1}^Z2.
Definition 1.3. Suppose n 3. Then a spin structure on E is a principal
Spin„-bundle PSpin(£) together with a 2-sheeted covering
^:PSpin(E)-----► Pso(£)
such that £(pg) = £(pK0(g) for all p e PSpin(£) and all g e Spin„.
When n = 2, a spin structure on £ is defined analogously, with Spin„
replaced by SO2 and £0: SO2 -► SO2 the connected 2-fold covering. When
n = 1, Pso(£) = X and a spin structure is simply defined to be a 2-fold
covering of X.
Note that the diagram
Pspin(£)	Pso(E)
(1-3)
where n and n' are the bundle projections, is commutative. Note also that
restricted to the fibres corresponds to the covering £0. The diagram of
fibrations is
Spin, - SO,
z2
PSpin„(£)Pso(£)
n'
X
On the other hand, suppose : PSpin(E) -> Pso(^) >s a 2-sheeted covering
§1. SPIN STRUCTURES ON VECTOR BUNDLES	81
which is non-trivial on the fibres of X, i.e., so that the diagram
Spin„ —-—► SO„
ME) — Лю(Е)
commutes. Then setting я' = я ° £ makes PSpin(E) a fibre bundle over X.
To make this a principal Spinn-bundle we must lift the action of SOW on
Pso(£) to a compatible action of Spin„ on PSpin(E). The proof that this
lifting exists is a straightforward application of elementary covering space
theory. We conclude the following (even for n = 1 and 2).
Theorem 1.4. The spin structures on E are in natural one-to-one corre-
spondence with 2-sheeted coverings of Pso(E) which are non-trivial on the
fibres of л.
This can be reinterpreted via Lemma 1.1 as follows:
Corollary 1.5. Suppose X is connected. Then the spin structures on E are
in natural one-to-one correspondence with elements of Hl(P3O(E); Z4) whose
restriction to the fibre of P3O(E) is non-zero.
We are now in a position to discuss the question of existence and
uniqueness of spin structures. Associated to the fibration SO„ 4 P^E)
А X there is an exact sequence
0----> H\X; Z2)	H^P^E); Z2) H^SO,; Z2)	H2(X; Z2)
(1.4)
which can be deduced from the Serre spectral sequence. In analogy with
the definition of via the sequence (1.2) we make the following definition:
Definition 1.6. The image w2(E) = wE(g2) e H2(X; Z2) of the generator
g2 of H1(SO„;Z2) s Z2 is the second Stiefel-Whitney class of the oriented
bundle E.
To prove that this (or any other) definition agrees with the one given
via classifying spaces, it again suffices to show (cf. (i) and (ii) above):
(i') This definition of w2 is natural.
(ii') This definition of w2(E) gives the non-zero element in
H2(BSO„;Z2) Z2 when E is the universal oriented n-plane bundle
over the classifying space BSO„.
Property (i') follows from the naturality of the sequence (1.4). Property
(ii') follows from the exactness of (1.4) and the fact that Pso(E) = ESO„
is contractible.
82
IL SPIN GEOMETRY AND DIRAC OPERATORS
From Corollary 1.5 and the exactness of the sequence (1.4) we im-
mediately conclude the following.
Theorem 1.7. Let E be an oriented vector bundle over a manifold X. Then
there exists a spin structure on E if and only if the second Stiefel-Whitney
class of E is zero.
Furthermore, if w2(E) = 0, then the distinct spin structures on E are in
one-to-one correspondence with the elements of H1(X;Z2).
Note that this result holds for all n. When n = 2, the class w2 is just the
mod 2 reduction of the Euler class. More generally, the following is true.
Remark 1.8 (cf. Milnor-Stasheff [1]). If E is the real 2n-dimensional
bundle underlying a complex n-dimensional vector bundle E, then
w2(E) = cJEMmod 2)
where cJE) is the first Chern class of E, (To see this, it suffices to observe
that the map H2(BSO2„;Z2) -► H2(BU„;Z2), induced by the inclusion
U„ c SO2„, is an isomorphism.)
Remark 1.9. Note that the spin structure on a bundle E is independent
of the bundle metric on E in the following sense. A spin structure on E
uniquely determines a spin structure for any other metric. This follows
from Theorem 1.4 and the observation that the inclusion P^E) c
PGL+(E), where PGL+(E) is the bundle of all oriented bases in E, is a
homotopy equivalence.
Remark 1.10. To choose an orientation and a spin structure for E is
in particular to find structure groups for E which are respectively 0 and
1-connected. Conversely, suppose E is equivalent to a vector bundle with
a Lie structure group G. If G is connected, then E is orientable; and if G
is simply-connected, then E is spin.
It should be noted that the process of finding successively more highly
connected structure groups for E terminates at the spin level. This is
because for any simply-connected Lie group G, n2(G) = 0, and if tt3(G) =
0, then G is contractible.
The conditions vvx = 0 and w2 = 0 can be interpreted geometrically as
follows:
Proposition 1.11. Let E be a vector bundle over a manifold X, Then E is
orientable if and only if the restriction of E to any circle embedded in X is
trivial.
The proof is obvious. There is an analogue for w2.
§1. SPIN STRUCTURES ON VECTOR BUNDLES
83
Proposition 1.12. Let E be an oriented vector bundle of dimension 3
over X. Then E is spin if and only if for any compact surface E and any
continuous map f : E -> X, the bundle f*E is trivial.
Suppose, furthermore, that X is simply-connected and of dimension > 4.
Then E is spin if and only if the restriction of E to any 2-sphere embedded
in X is trivial.
Proof. The group H2(X;Z2) is generated by maps f :E -► X of compact
surfaces. Hence, w2(E) — 0 о /*w2(E) = w2(/*E) = 0 for all such f о
f*E is trivial for all such f (since an oriented bundle of dimension 3
over a surface is trivial if and only if w2 = 0). This proves the first state-
ment. For the second statement it suffices to note that when л^Х = 0 and
dim(X) > 4, the group H2(X;Z2) is generated by embedded 2-spheres. 
Remark 1.13. This second statement can be refined somewhat. If л^Х =
0, then H2(X;Z2) is generated by immersions of S2, and if dim X > 4, the
immersions can be deformed to embeddings.
We now consider some alternative definitions of the classes and w2.
Recall that (cf. App. A) for any topological group G, we can view the
equivalence classes of principal G-bundles on X as elements in a “Cech
cohomology space” Hr(X;G). As noted in Milnor [7] the short exact
sequence of topological groups
1----► SO„ O„ Z2----------------► 0	(1.5)
gives an exact sequence
Hl(X; SO„) Hl(X; O„) Hl(X; Z2).	(1.6)
Given an n-dimensional bundle E on X, we define the first Stiefel-Whitney
class by setting wJE) = P*([PO(£)])- This definition is easily seen to have
properties (i) and (ii) and it therefore agrees with our previous definitions.
In a similar way the short exact sequence
0----► Z2-----► Spin„ -U SO„------► 1	(1.7)
gives an exact sequence
H°(X; SO„)	Z2)----► Hl(X; Spin,)	(1.8)
№(X;SO„) -U H2(X;Z2)
(cf. Hirzebruch [1]). Thus, for an oriented bundle E we define the second
Stiefel-Whitney class by setting w2(E) = <5([Pso(£)])- This definition has
properties (i') and (ii') and therefore agrees with our previous ones.
With this definition it is transparent that w2(£) = 0 if and only if Pso(^)
is equivalent to the Z2-quotient of a principal Spin„-bundle on X.
84
II. SPIN GEOMETRY AND DIRAC OPERATORS
Remark 1.14. This is a convenient time to inject a minor word of cau-
tion. It was pointed out by Milnor that the Spinn-bundles associated to
distinct spin structures on E may be equivalent as abstract principal bundles.
Recall that spin-structures on E are in one-to-one correspondence with
elements of H1(X;Z2). However, by (1.8) we see that the equivalence classes
of principal Spin„-bundles with Z2-quotient equal to [PSo(E)] are in one-
to-one correspondence with elements of H1(X;Z2)/^°(H°(X;SOW)). Now by
definition we have that H°(X;SO„) = C(X,SO„), the space of continuous
maps from X to SO„. The map
: C(X,SO„)----► H1(X;Z2)	(1.9)
is given as follows. For a map f:X -► SO„, set =/*(1) where 1 is
the generator of H1(SO„;Z2).
The map (1.9) is often surjective. For example, any class which is the
mod 2 reduction of an integral class lies in the image. To see this let
/0:X -► S1 represent the integral class and define f = i о f0;X -► SO„
where i:Slc->SO„ is not homotopic to zero. Thus, if H1(X;Z2) =
® Z2, then is surjective. Similarly if dim X < n, then is also
surjective. To see this note that every class in H1(X;Z2) can be induced by
a map to iP"([R) c SO„. Thus, in either of these cases, all the principal
Spin„-bundles associated to distinct spin structures on E are abstractly
equivalent.
As an example of this phenomenon consider an oriented bundle E over
the circle S1.
Since H1(S1;Z2) Z2, there are two spin structures on E. However, any
principal Spin„-bundle over S1 is equivalent to the trivial one since it
admits a cross-section (as does any fibre bundle over S1 with connected
fibre).
Finally we mention a direct definition of and w2 via homotopy
theory. From the sequence (1.4) there is a fibration
BSO„------► BO„ BZ2 = K(Z2,1).
A map fE: X -► BO„ (classifying a bundle E) has a lifting to BSO„ iff w о fE
is homotopic to zero. Recall that [X,K(Z2,1)] £ H1(X;Z2). The class
w о fE is the first Stiefel-Whitney class w^E).
From the sequence (1.6) we have a fibration K(Z2,1) -► BSpin„—
BSO„. It follows that the cofiber of BE, is K(Z2,2), i.e., there is a fibration
BSpin, BSO„ K(Z2,2).
We can now define w2(E) as we defined wJE) above.
We complete this section with an observation concerning Whitney sums.
Proposition 1.15. Given three vector bundles E', E” and E = E' ® E" over
a manifold X, a choice of orientation on any two of them uniquely determines
§2. SPIN MANIFOLDS AND SPIN COBORDISM	85
an orientation on the third. Similarly, a choice of spin-structure on any two
of them uniquely determines a spin structure on the third.
Proof. The statement concerning orientations is obvious since for finite
dimensional vector spaces V = V' ф V”, an orientation on any two canon-
ically determines an orientation on the third.
Suppose now that E, E' and E" are orientable. Then w2(E) = w2(E') +
w2(E"). Hence, if anv two are spin, so is the third.
Suppose under the correspondence of Corollary 1.5 a! e H^P^E');^)
and a” e Hx{Pso(E,y,Z2) represent spin structures on E' and E” respec-
tively. Now consider the cartesian product bundle E' x E” -► X x X. We
let
Д: Pso(E' © E")----► PSO(E' x E")
denote the diagonal map. The class in Hi(Pso(Ef ф E");Z2) which will
represent the spin structure determined by a' and a” will be A*b where
ЪеНуР^Е’ x E”);Z2) will be the unique class which extends a’ x 1 +
1 x a" e H\P3O(E') x Р^Е");^) under the inclusion Р^{ЕГ) x Pso(E") cz
PSO(E' x E"). That there is a unique b can be seen from the following
diagram:
0 -> H\X X X;Z2)---► H\P^Ef X E");Z2)----> Hl(SOn+n;Z2)--> H2(X x X:Z3)
b b b b
О - H\X X X;Z2) - Н'(Р^Е') x PSO(E");Z2) -> //‘(SO, x SO„.;Z2) - H2(X x X;Z2)
That any two of the classes a', a" and A*b determines the third is now easy
to see. Let 7tv: Pso(H -► X be the projection map where V = E, E' or E”.
Then any spin structure y' (under the correspondence of Corollary 1.5)
on E' may be written a' -I- t$ u' and any spin structure y” on E” may be
written a” -I- tc^u” for u',u" e H1(X;Z2). Then following the above pre-
scription / and y" determine A*b -I- 71^ФЕ»(и' -I- u"). Clearly any two of
these determine the third. 
§2. Spin Manifolds and Spin Cobordism
We are now in a position to discuss the notion of spin structures on mani-
folds. For convenience all manifolds are assumed to be of class C00.
Definition. A spin manifold is an oriented riemannian manifold with
a spin structure on its tangent bundle.
The Stiefel-Whitney classes wJX) of a manifold X are defined to be the
Stiefel-Whitney classes of its tangent bundle TX. Hence, by Theorem 1.7
we have the following.
86
II. SPIN GEOMETRY AND DIRAC OPERATORS
Theorem 2.1. An oriented riemannian manifold X admits a spin structure if
and only if its second Stiefel-Whitney class is zero. Furthermore, if w2(X) = 0,
then the spin structures on X are in one-to-one correspondence with elements
ofH\X',Z2\
Recall (cf. Remark 1.9) that the choice of spin structure for one rieman-
nian metric on X canonically determines a spin structure for any other
riemannian metric on X, Here we are using the fact that, for any metric,
the inclusion PSO(X) cz PGL+(X) of the bundle of oriented orthonormal
tangent frames into the bundle of all oriented tangent frames, is a homo-
topy equivalence.
Consider now a diffeomorphism f:X Д X of a spin manifold X. If
f is orientation preserving, then there is an induced diffeomorphism
df:PGL+(X) -* PGL+(X) of the bundle of oriented tangent frames. This
map carries fibres to fibres and therefore induces a permutation of the
possible spin structures on X (considered as 2-fold coverings of FGL+(X).)
If the given spin structure remains fixed, then f is called a spin structure-
preserving diffeomorphism.
It is a classical result of W.-T. Wu [1] that the Stiefel-Whitney classes
of a compact manifold X depend only of the homotopy type of X. There-
fore, the property of having or not having a spin structure is a homotopy
invariant, and in particular, it remains the same under changes of the dif-
ferentiable structure on X. Wu proves his result by giving a homotopy-
theoretic formula for the total Stiefel-Whitney class w=l+vv1+w2 + ...
of X. Define v=l+v1+v2 + ...e H*(X;Z2) by the requirement that
(v u <X] = Sq(a)[X]
for all a e H*(X;Z2) where Sq = 1 + Sq1 + Sq2 4-... is the Steenrod
square automorphism, and where [X] e H„(X;Z2), n = dim(X), denotes
the fundamental class. Wu’s formula states that
w = Sq(t>)
(see Milnor-Stasheff [1] for details). Since Sq'(a) = 0 if i > deg(a), we see
that vk = 0 for к > [dim(X)/2]. Thus, for example, if dim(X) = 3, we see
that v = 1 4- Vi = 1 4- Wj, and it follows that w2 = w2. In particular, if
X is orientable, then it is automatically spin. Suppose now that dim(X) = 4
and = 0. Then v = 1 4- v2 = 1 4- w2, and we see that w2 is characterized
by the fact that (w2 u a)[X] = (a u a)[X] for all a e H2(X;Z2),
We will now examine some examples of spin manifolds. For convenience
we shall use the expression X is spin to mean that wt(X) = w2(X) = 0,
(i.e., that X carries at least one spin structure for any riemannian metric
on X). Recall that any complex manifold is canonically oriented. Further-
more, by Remark 1.8 we know the following:
§2. SPIN MANIFOLDS AND SPIN COBORDISM	87
Remark 2.2. A complex manifold X is spin if and only if its first Chern
class satisfies ct(X) = 0 (mod 2).
Example 2.3 (the trivial examples). It follows immediately from Theo-
rem 2.1 that any 2-connected manifold carries a unique spin structure.
The obvious examples of this type include homotopy spheres, Stiefel mani-
folds, and simply-connected Lie groups. Of course, any manifold whose
tangent bundle is stably parallelizable is spin. This includes, for example,
the inverse image of any regular value of a smooth map f:Un+p-> R₽. It
also includes any Lie group and any orientable manifold of dimension 3.
Example 2.4	. Let P"(k) denote the n-dimensional projective space over
the (skew) field k. Then
P"(R) is spin iff n = 3 (mod 4)
P"(C) is spin iff n is odd
P"(H) is spin for all n.
To see this recall that the total Stiefel-Whitney class of P"(K) isw=l +
Wi + vv2 + ... = (1 + #)"+1 where g, the generator of the cohomology
ring, has dimension 1, 2 and 4 for К = R, C and H respectively. When
К = R, the conditions = 0, w2 = 0 are equivalent to (n + 1) = n(n + 1)/
2 = 0 (mod 2). The cases К = C and К = H are obvious.
Example 2.5	. The manifold SOn has two distinct spin structures given
as follows: P(SOJ = SO„ x SOjy where N = n(n — l)/2. The two coverings
are:
PJSO,) = SO, x Spinw
^(SO„) = (Spin, x SpinN)/Z2
where Z2 acts on Spin, x SpinN by the map (g, h) i-+(—g, — h).
Example 2.6	. Let Eg be a compact Riemann surface of genus g. As
observed in 2.4 above, this surface is spin. (The Euler characteristic Ct(Eg)
is even.) There are exactly 22* distinct spin structures on X which can be
constructed as follows. Let №(Ев;(Р*) denote the equivalence classes of
holomorphic complex line bundles on Eg. This is an abelian group under
the operation of tensor product. There is a short exact sequence
0----> J-----►	H2(Eg;Z)-----> 0
where J s Н*фв№)/Н1&,в;2) T2". Let r0 e Н‘(Е9;$*) denote the tan-
gent bundle of Е0 and note that there exist exactly 22e elements т e
H^EgjtP*) such that r2 = r0. Recall that for any complex line bundle т
the natural map т -*t2 is of the form z -> z2 in the fibres. Hence, each
88	П. SPIN GEOMETRY AND DIRAC OPERATORS
bundle т with т2 = t0 determines a 2-fold covering of PSo(2^) c t0 which
is non-trivial on the fibres. These coverings realize all the distinct spin
structures on
This construction is a special case of the general fact (cf. Hitchin [1]
and App. D) that for any compact Kahler manifold X with cr(X) = 0
(mod 2), the spin structures on X are in one-to-one correspondence with
holomorphic square roots of the canonical bundle of X.
Example 2.7	. Let K"(d) denote the non-singular complex hypersurface
of degree d in 1Рл + ^C). That is, Vn(d) is given in homogeneous coordinates
[Zo,... ,Zn+1] for Рл + 1(С) as the zeros of a homogeneous polynomial
p(Z0,... ,Zn+J of degree d which satisfies the condition (Vp)(Z) 0 when
Z / 0 and p(Z) = 0. The diffeomorphism class of Vn(d) is uniquely deter-
mined by the integers n and d.
The first Chern class of Vn(d) for n > 1 is
ct =(n + 2 - d) g
where g is the canonical generator of H2(K"(d); Z) (the Kahler form induced
from 1Рл + 1(С)). It follows that
Vn(d) is spin	n -I- d is even.
Example 2.8	. Let Vn(dl9... ,dk) be the transverse intersection of hyper-
surfaces Vn+k = 1(dt)9... ,Kn+k + 1(dk) in Рл+к(С). Then for n > к
Ci = (n 4- к -I- 1 - di - ... - dk) • g
where g is the canonical generator of H2(Vn(di9... ,dk);Z) Hence,
к
Vn(d19... ,dk) is spin <=> n -I- к 4- 1 -I- £ df is even.
i=l
Example 2.9 (N. Hitchin [1]). Let f:M-> IP2(C) be a p-fold ramified
cover, branched over a non-singular curve of degree pq. Then M is spin
iff p is even and q is odd.
As we observed above, every oriented manifold of dimension 3 is spin.
In higher dimensions the spin condition has a nice geometric interpreta-
tion. We shall restrict our attention to manifolds X with the property that
the homomorphism H2(X;Z) -► H2(X;Z2), given by reduction mod 2, is
surjective. This holds whenever X is simply-connected.
Theorem 2.10. Let X be an oriented n-dimensional manifold as above. If
n 5, then X is spin if and only if every compact orientable surface embedded
in X has trivial normal bundle. If n = 4, then X is spin if and only if the
normal bundle to every compact orientable surface embedded in X has
even Euler class.
§2. SPIN MANIFOLDS AND SPIN COBORDISM
89
Proof. Since H2(X\T2) = H2(X\T) ® Z2, the group H2(Xf£2) is generated
by maps of compact orientable surfaces. By the classical theorems of
Whitney, every map of a surface into X is homotopic to an embedding
if n 5, and to a self-transversal immersion if n = 4. In this latter case,
we can remove small disks at each self-intesection point and attach an
embedded handle, thereby producing an embedded surface in the same
homology class. Consequently, H2(Ar;Z2) is generated by smooth embedd-
ings of compact orientable surfaces.
Let i:E^->X be such an embedding. Then i*w2(X) = i*w2(TX) =
w2(i*TX) = w2(TE © NE) = w2(TE) + w2(NE) = w2(NE). Evaluating
on the fundamental class, we have (w2(X),i+[E]) = (i*w2(X), [E]) =
(w2(NE), [E]). Consequently, if w2(X) = 0, then w2(NE) = 0. On the other
hand, Я2(Х;к2) is generated by such surfaces. We conclude that w2(X) = 0
if and only if w2(NE) = 0 for every compact orientable surface embedded
in X.
The normal bundle NE to E is orientable. Therefore, when dim(NE) 3
(i.e., when n 5), we have w2(NE) = 0 if and only if NE is trivial. When
dim(NE) = 2 (i.e., when n = 4), we have w2(NE) = /(NE) (mod 2). This
completes the proof. 
Corollary 2.11. Let X be a simply-connected manifold of dimension's.
Then X is spin if and only if every 2-sphere embedded in X has trivial
normal bundle.
Proof. Since H2(X;Z) = n2(X) by the Hurewicz Theorem, we have that
tf2(X;Z2) is generated by embedded 2-spheres. 
Corollary 2.12. Let X be a compact, simply-connected 4-manifold. Then X
is spin if and only if (y u y)[X] = 0 (mod 2) for each у e H2(X;Z) (i.e., the
intersection form is “even”).
Proof. Let у g H2(X;Z) [X,P°°(C)] be given by a smooth map f: X ->
P3(C). Make f transversal to a hyperplane P2(C) <= P3(C) and let E =
/-1(P2(C)). Then E represents the Poincare dual of y, and (y u y)[x] is
the self-intersection number of E, i.e., the Euler number of NE. 
From the classification of quadratic forms (e.g., Husemoller-Milnor [1])
we conclude that the signature of X must be a multiple of 8. This result
holds, in fact, for any topological 4-manifold with = w2 — 0. For
smooth manifolds there is the following deeper result (see Chap. IV).
Theorem 2.13 (Rochlin). The signature of a smooth compact spin 4-manifold
is a multiple of 16.
90
П. SPIN GEOMETRY AND DIRAC OPERATORS
Example 2.14. The complex hypersurface K2(4) = {(Zo,Z1,Z2,Z3):Zo +
Zf + Z2 + Z3 = 0} c: IP3(C) is a spin manifold with signature 16 (see
Example 2.7 above). This is the so-called Kummer (or КЗ) surface.
Remark. The statement of Corollary 2.12 fails when / 0. How-
ever, the following is true. For a compact orientable 4-manifold X, the
intersection form is even if and only if w2(X) is the mod 2 reduction of a
torsion class in Jf2(X;Z). This was proved by N. Habegger [1] who also
showed the following. Let X = K2(4)/Z2 be the quotient of the Kummer
surface by the involution (Zo, ZX,Z2,Z3) -> (Z19 — Z0,Z3,— Z2). Then X is
orientable and has even intersection form; however, sig(X) = 8. Of course,
X cannot be a spin manifold. In fact, w2(X) is the mod 2 reduction of the
unique torsion class in H2(X;Z).
The remainder of this section will be devoted to a discussion of spin
cobordism. We begin with two observations which follow immediately
from Proposition 1.15.
Proposition 2.15. The cartesian product of two spin manifolds is canonically
a spin manifold. Any submanifold of a spin manifold with a spin structure
on its normal bundle is canonically a spin manifold.
In particular, if Y is a compact manifold with boundary SY, then any
spin structure on Y induces a spin structure on 8 Y as follows. Let v be
the field of interior unit normal vectors along SY. Using v, one obtains
an embedding PSO(5Y) c P^Y), by completing each tangent frame to
S Y with the given normal vector. The spin structure, considered for example
as a 2-sheeted covering, can now be restricted to P^SY).
Definition 2.16. Two spin manifolds are said to be differentiably equiv-
alent if there is a diffeomorphism between them preserving orientations
and spin structures. A compact (not necessarily connected) spin manifold
is said to be spin cobordant to zero if it is differentiably equivalent to the
boundary of a compact spin manifold Y with (orientation and) spin struc-
ture induced from Y as above.
Let fXJpin denote the free abelian group generated by equivalence classes
of compact connected n-dimensional spin manifolds, modulo the subgroup
generated by elements [A\] + ... 4- [A\] where Xi II... II Xk is spin co-
bordant to zero. Q^pin is called the л-dimensional spin cobordism group.
From Proposition 2.15 we know that the product of two spin manifolds
has a uniquely determined spin structure. This multiplication makes
QSpiii = @f=onspin into a graded ring, called the spin cobordism ring.
Note that the equivalence class (and therefore also the cobordism class)
of a spin manifold X is independent of the choice of riemannian metric
on X.
§2. SPIN MANIFOLDS AND SPIN COBORDISM
91
Remark 2.17. Given two spin n-manifolds and X2, we can form
leir connected sum Xr# X2 and equip it with a spin structure so that
\ # X2 and XY II X2 are spin cobordant. Thus every spin cobordism
ass is represented by a connected manifold (see Milnor [7]).
The connected sum operation is a special case of the general procedure
f doing surgery on spin manifolds (see Milnor [5] and Kervaire-Milnor
1]). In particular one can show that for n 3 every spin cobordism class
represented by a simply-connected manifold. For n 5, every spin
□bordism class is represented by a 2-connected manifold (see Corollary
.11). Thus by Poincare duality and the h-cobordism theorem = 0.
For the special case n = 1, a spin structure is defined to be a 2-fold
overing of PsoCS1) = S1. Consider S1 as the boundary of the 2-disk with
s unique spin structure. Then the spin structure induced on S1 is the
onnected 2-fold covering. Interestingly, the disconnected 2-fold covering
> not cobordant to zero, although two copies of it clearly is (a good
xercise). Thus, Q^pin Z2.
It is also true that the square of the circle with the bad spin structure
5 not zero in Qlpin ~ Z2 (another good exercise).
We now briefly review the Thom construction. Let Xn be a compact
i-dimensional spin manifold and choose a smooth embedding Xn -► S"+k,
: > n. The spin structures on Xn and Sn+k determine a unique spin struc-
ure on the normal bundle N(Xn) of Xn (see Proposition 1.15). Hence, there
s a bundle map
Xn BSpint
classifying N(Xn\ where Ek is the universal к-plane bundle. The map f
descends to a map of Thom spaces /:т(ЛГ(Х")) -► r(Ek) = MSpink. If we
identify N(Xn) with a tubular neighborhood of Xn in S"+fc, then t(N(X"))
is the space obtained by collapsing the complement of this tubular neigh-
borhood to a point. Thus we get a map n: Sn+k -► t(N(X")), and the com-
position/- n:Sn+k -► MSpink determines an element
<X"> g 7iJ(MSpin) = lim 7r„+k(MSpink).
k-* oo
The classic result of Thom states that this map induces an isomorphism
QSpm ^(MSpin).	(2.1)
We observe now that there is a natural ring homomorphism
p,:Q*pin---► Q*°	(2.2)
92	П. SPIN GEOMETRY AND DIRAC OPERATORS
where is the oriented cobordism ring. In fact, in low dimensions we
have the following (cf. Milnor-Stasheff [1; §17] and Milnor [7]):
n		
0 1 2 3 4 5 6 7 8	z Z2 Z2 0 Z (generated by the Kummer surface surface V2(4) in Example 2.14) 0 0 0 Z ф Z (generated by P2(IHI) and a manifold L6 such that 4L8 is spin cobordant to V2(4) x V2(4))	z 0 0 0 Z (generated by P2(C)) z2 0 0 Z ® Z (generated by P4(C) and P2(C) x P2(C))
Of course the non-zero element x g Qipin given by the circle with the
“bad” spin structure, goes to zero in Q^°. Hence, we have x • Q^pin c ker p*.
In fact, it can be proved that x • O*pin = ker p* (see Anderson-Brown-
Peterson [1]).
The cokernel of p* is not so simple to describe. For example, the signa-
ture gives an isomorphism Q40 £ Z, and therefore using Theorem 2.13
and Example 2.14 we get a short exact sequence
0----> Q*pin	> Z16----> 0.
Similarly,
0----> Q|pin Q|°---------> Z27----> 0
is exact. The map p* tensored with the rational numbers is an isomorphism.
In fact, so is p* tensored with Z[|] (cf. Milnor [5]).
It is known that the oriented cobordism class of a manifold is determined
by its Pontryagin and Stiefel-Whitney numbers. Similarly, it has been
proved that the spin cobordism class of a manifold is completely deter-
mined by its Stiefel-Whitney and KO-characteristic numbers (Anderson-
Brown-Peterson [1]). A fundamental such KO-invariant is the ring
homomorphism
X,: Q*pin----> KO~ *(pt)	(2.3)
which we describe in §3 of this chapter. Recall that
л (mod 8)	0	1	2	3	4	5	6	7
ATO-(pt)	z	z2	z2	0	z	0	0	0
The homomorphism is an isomorphism for n 7.
§3. CLIFFORD AND SPINOR BUNDLES	93
One of the remarkable and very useful consequences of the Atiyah-
Singer Index Theory is that this invariant can be computed as the topolo-
gical index of an elliptic operator naturally defined on any spin manifold
in the cobordism class. This will be discussed in Chapter III.
One of the interesting uses of the invariant «я/ is the following. Let E"
be an и-dimensional homotopy sphere, i.e., a compact differentiable mani-
fold which is homotopy equivalent to the и-sphere Sn. Then E" is cobordant
to zero (cf. Kervaire-Milnor [1]), but it is not necessarily spin cobordant
to zero. In fact the homotopy n-spheres form a finite abelian group, ®„,
under the operation of connected sum, and : ®n KO ~ "(pt) is a homo-
morphism. The following is a consequence of deep results of Adams [3]
and Milnor [7].
Theorem 2.18. For n = 1 or 2 (mod 8) and n > 8, the homomorphism
-------------------------------------► Z2
is surjective.
§3. Clifford and Spinor Bundles
We begin this section by briefly describing the associated bundle construc-
tion. Let n:P -> X be a principal G-bundle over a space X, and let
Homeo(F) denote the group of homomorphisms of another space F. Gwe
Homeo(F) the compact-open topology. Then to each continuous homo-
morphism p.G -► Homeo(F), we construct a fibre bundle over X with
fibre F as follows. Consider the free left action of G on the product P x F
given by
<Рв(Р,Л = (Р9~1>Р(в)Л
for g g G and (p,/) g P x F. Define P xp F to be the quotient space (the
space of orbits) of this action. One easily sees that the projection P x F ->
P X descends to a mapping
np'.PxpF------>X
which is the fibre bundle over X with fibre F. It is called the bundle asso-
ciated to P by p.
If X and F are manifolds, G is a Lie group, and P is differentiable, one
can consider continuous homomorphisms p: G -► Diff(F), where Diff(F)
is the group of diffeomorphisms of F with the usual C00 topology. In this
case, the associated bundle np :P xp F -+ X is differentiable.
Note that if P is given by transition functions
dap • U a	Up > &
for Ua,Up g where is some open cover of X, then P xp F is given
94
И. SPIN GEOMETRY AND DIRAC OPERATORS
by the transition functions:
P 0 9ap • n Up-----► Diff(F).
Note also that if p: G -► GL(K) is a linear representation on a vector
space K, then P xp К is a vector bundle over X.
Example 3.0. Let X be a manifold and let p: X -> X be its universal
covering space. Then X is a principal TqQO-bundle. Choose p e
Hom(7r1XiZ2) H1(XjZ2), and consider Z2 as the group of permutations
of the set {0,1}. Then X xp {0,1} is a 2-sheeted covering of X (see Lemma
1.1).
Example 3.1	. Let X be a manifold and let P = PGL(X) be the principal
GL„(R)-bundle of tangent frames. Let pn: GL„(R) -► GL(R") denote the
standard representation, and let p* denote the dual representation (p*(g) =
Pnte-1)')- Then
TX = PGL(X) x Pn R" and T*X = PGL(X) x (R")*
where TX and T*X are the tangent and cotangent bundles of X respec-
tively. Similarly,
A‘TX = PGL(X)xAkp„A‘R"
\kT*X = PGL(X) xAkp. (A*Rn)*
(XXT^ = PGL(X)x0:pn((xXR")
where \kpn, Akp* and (^fspn are the induced exterior power and tensor
product representations.
Example 3.2	. Let X be an oriented riemannian manifold and let P =
РЮ(Х) be the SO„-bundle of positively oriented orthonormal frames. If
p„: SO„ -> SO(R") is the standard representation, then again
TX = РЮ(Х) хрч R"
ХкТХ = Рт(Х) xAkpn(A*R")
(ХУГХ = Р8О(Х) xerpn(0TR")
Note that in this case p* = pn. This corresponds to the canonical iso-
morphism TX s T*X given by the riemannian metric.
Example 3.3	. More generally, if E is any oriented riemannian vector
bundle over a space X, then
E = РЮ(Е) xpn R"
Ak(E) = Pso(E)xArpJAkR")
0r(E) = Pso(E)xe.Pn(0rR")
Again, since p„ = p*, E and E* are canonically isomorphic.
§3. CLIFFORD AND SPINOR BUNDLES
95
These examples suggest the following. Recall that each orthogonal
transformation of R" induces an orthogonal transformation of C£(R") =
Ctn. (It maps the tensor algebra to itself and preserves the ideal.) This
induced map on Ctn clearly preserves the multiplication. Hence we get a
representation:
c£(p„): SO„---► Aut(C£(R")).	(3.1)
Definition 3.4. The Clifford bundle of the oriented riemannian vector
bundle E is the bundle
C£(E) = Pso(E)xct(p„)C£(R")
associated to the representation (3.1).
E is a bundle of vector spaces with inner products, and C£(E) is just
the associated bundle of Clifford algebras. In fact C£(E) could be defined
as the quotient bundle:
ce(E) = ff <&e]Ii(E)
where 1(E) is the bundle of ideals, i.e., the bundle whose fibre at x e X is
the two-sided I(EX) in	generated by elements v ® v 4- ||d||2
for v g Ex.
It is evident that C£(E) is in fact a bundle of (Clifford) algebras over X.
The fibrewise multiplication in C£(E) gives an algebra structure to the
space of sections of C£(E).
It is also evident that each of the notions intrinsic to Clifford algebras
carries over to Clifford bundles. For example, there is a decomposition
C£(E) = C£°(E) ф Ct^E)	(3.2)
corresponding to the even-odd decomposition of the algebras. These are
the +1 and — 1 eigenbundles of the bundle automorphism
a:C£(E)-----► C£(E)	(3.3)
which extends the map E -► E sending v to — v. There is also an intrin-
sically defined bundle map
L:C£(E)-----► C£(E)	(3.4)
which in any fibre C£(EX) is given by
U.<p) = - E ek<pek	(3.5)
k = 1
where {ev ... ,en} is an orthonormal basis of Ex (see Chap. I).
The following is an elementary but important fact:
Proposition 3.5. There is a canonical vector bundle isometry
A:A*(E)-^C£(E)	(3.6)
96
II. SPIN GEOMETRY AND DIRAC OPERATORS
under which
A(AevenE) = C£°(E); A(AoddE) = Cf(E)	(3.7)
and
ДЛрЕ) = {(p e Ct(E): a □ Ц<р) = (и - 2р)<р}	(3.8)
for p = 0,... ,n.
Proof. The isomorphism л follows directly from the canonical isomor-
phism A:A*R" Cl(R") and the fact that Я ° A*p„ = c£(p„) ° A. Equa-
tions (3.7) and (3.8) follow from Proposition 3.8 in Chapter 1. 
As mentioned in the introduction, it is now natural to look for bundles
of irreducible modules over the bundle of Clifford algebras C£(E). Such
bundles can be constructed if w2(E) = 0.
Definition 3.6. Let E be an oriented riemannian vector bundle with a
spin structure £: PSpin(£) -► Pso(^)- A rea* spinor bundle of E is a bundle
of the form
S(E) = PSpin(£) x,
where M is a left module for Ct(R") and where p: Spin„ -► SO(Af) is the
representation given by left multiplication by elements of Spin„ c C£°(R").
Similarly, a complex spinor bundle of E is a bundle of the form
SC(E) = PSpin(E) Mc
where Mc is a complex left module for C£(R") ® C.
If the module M (or Mc) is Z2-graded, the corresponding bundle is said
to be 7L 2-graded.
Example 3.7. Consider Cl(R") as a module over itself by left multiplica-
tion The corresponding real spinor bundle
ClSpin(E) = PSpin(E) x,Cl(R")
is a “principal C£(R")-bundle”, i.e., it admits a free action of C£(R") on
the right. There is a natural embedding PSpin(E) <= ClSpin(E) which comes
from the embedding Spin„ c= Ct(R"). Hence, every real spinor bundle for
E can be captured from this one.
A similar remark holds for the complex case.
Of course, the bundle ClSpin(E) differs from the Clifford bundle C£(E).
They can be compared as follows. Consider the representation
Ad: Spin„-----► Aut(Cl(R"))	(3.9)
given by Ade(<p) = g(pg 1 for g e Spin„ c Cl(R"). Clearly Ad_ t = identity,
§3. CLIFFORD AND SPINOR BUNDLES	97
and so this representation descends to a representation Ad' of SO„. One
easily checks that Ad' is just the representation ct(p„) given in (3.1). It
follows that
C£(E) = PSpin(E) xAd C£(R").
This leads to the following.
Proposition 3.8. Let S(E) be a real spinor bundle of E. Then S(E) is a bundle
of modules over the bundle of algebras Ct(E). In particular the sections of
the spinor bundle are a module over the sections of the Clifford bundle.
The corresponding fact holds in the complex and Z2-graded cases.
Proof. The diagram
PsPi„(£) x C£(R") x	M	PSpin(E) x	M
Pg	Pg
PSpin(E) x C£(R") x	M	PSpin(£) x	M
given by
(р,Ф,т) -------------► (p,(pm)
-----► (pg~l,g<pm)
clearly commutes. Therefore, p descends to a mapping
p:Ct(E) ф S(E)-------> S(E),	(3.10)
which is easily seen to have the desired properties. The corresponding
argument goes through in the complex and Z2-graded cases. 
We say that two spinor bundles of E are equivalent iff they are equivalent
as bundles of C£(E)-modules. A bundle of (real or complex, graded or
ungraded) C£(E)-modules is called irreducible if at each x the fibre is
irreducible as a (real or complex, graded or ungraded) module over C£(EX).
Recall that every module for C£(R") can be written as a direct sum of
irreducible ones, and there are at most two equivalence classes of irreduc-
ible modules. Consulting §5 of Chapter I we obtain the following:
Proposition 3.9. Every spinor bundle of E (real or complex, graded or un-
graded) can be decomposed into a direct sum of irreducible ones. With the
assumption that X is connected, the number N of equivalence classes of ir-
reducible ones depends on the dimension n of E as follows.
98
II. SPIN GEOMETRY AND DIRAC OPERATORS
л (mod 8)	Real Ungraded	Complex Ungraded	Real Graded	Complex Graded
1	1	2	1	1
2	1	1	1	2
3	2	2	1	1
4	1	1	2	2
5	1	2	1	1
6	1	1	1	2
7	2	2	1	1
8	1	1	2	2
Thus, for n even, there is only one irreducible, ungraded spinor bundle
of E (over R or over C). If n = 6 or 8 (mod 8), the complex one is just the
complexification of the real one. If n = 2 or 4 (mod 8), the complexification
of the real one splits into two copies of the complex one. This, and the
corresponding information for n odd, can be easily deduced from the
tables in Chapter I.
We now observe that in certain dimensions any real spinor bundle auto-
matically carries a natural complex or quaternion structure. This is a con-
sequence of the fact that for certain n, each irreducible real C£(R")-module
carries a compatible C or H structure, i.e., the module multiplication is
C or H linear (and hence, the complex or quaternion scalar multiplication
in M descends to the quotient PSpin M). The appearance of such struc-
ture is periodic in n and can be deduced from Table III in Chapter I. We
conclude the following.
Proposition 3.10. Let Ebe a real n-dimensional bundle equipped with a spin
structure, and let S(E) be any real ungraded spinor bundle of E. If n= 1 or
5 (mod 8), then S(E) carries a complex structure such that Clifford multipli-
cation is complex linear in each fibre. If n = 2, 3 or 4 (mod 8), then S(E)
carries a quaternion structure so that Clifford multiplication is quaternion
linear in each fibre.
Let us now say a word about the Z2-graded case. There is a natural
one-to-one correspondence between classes of bundles of irreducible Z2-
graded modules over C£(E) = C£°(E) ф Ct^E) and classes of bundles of
irreducible modules over C£°(E). Given a bundle S(E) = S°(E) ф S1(E) of
the first kind, S°(E) is of the second. Given an S°(E) of the second kind,
the bundle
5(E) = C£(E) ®cto<£) S°(E)
is of the first.
Suppose now that n = 2m and SC(E) is the irreducible complex spinor
bundle of E. We shall show explicitly how to split SC(E) into a direct sum
SC(E) = S£(E) ф Sc(E)	(3.11)
§3. CLIFFORD AND SPINOR BUNDLES	99
of C£°(E)-modules. Interpreting Sc(E) as S°(E) and Sq(E) as Sq(E), or
the other way around, gives a Z2-graded module structure to SC(E). The
two possibilities are the two inequivalent graded modules appearing in
the table. The construction is as follows. Consider the global section coc
of C£(E) ® C which at x e X is given by
"c =	" e2m	(3.12)
for any positively oriented orthonormal basis {ev ... ,e2m} of Ex. Then
we have
a>l = 1	(3.13)
ea)c = — coce	(3.13')
for any e e СР(Е) ® C. We then define S£(E) and Sq(E) to be the +1
and — 1 eigenbundles for Clifford multiplication by coc. One easily sees
from (3.13') that these bundles have the same dimension and that they
form the Z2-graded modules as stated above.
These bundles can be written as associated bundles in the following
way. Let Д2„ and Д2т denote the two fundamental complex representa-
tions of Spin2m. Then
(3.14)
For n = 0 (mod 4) there is an analogous construction in the real case.
Let S(E) be the irreducible real spinor bundle of E and define a global
section co of C£(E) by setting
co =	(3.15)
at x 6 X where {e1;... ,en} is any positively oriented orthonormal basis
of Ex. Again, we have that
co2 = 1	(3.16)
coe = — eco for all e e C£°(E).	(3.17)
For equations (3.16) and (3.17) it is necessary that n be a multiple of 4.
This again determines a decomposition
S(E) = S+(£)® S"(E)
(3.18)
into the +1 and — 1 eigenbundles of the operator given by Clifford multi-
plication by co. They make S(E) a Z2-graded module in two distinct ways,
thereby accounting for the 2 in Table 3.1 above.
If n = 0 (mod 8), then S±(E) ® C s S£ (E). This corresponds to the
fact that in these dimensions, Д* are the complexifications of real
representations.
100	II. SPIN GEOMETRY AND DIRAC OPERATORS
If n = 4 (mod 8), then S*(E) ® C Sq(E) © Sc(E). In these dimen-
sions are quaternionic.
We now make a fundamental observation concerning these Z2-graded
bundles. Let
B(E) = {eeE:\\e\\^ 1}
be the unit disk bundle of E with boundary
0(E) = {e e E : \\e\\ = 1},
the unit sphere bundle. Let it: D(E) -> X be the bundle projection. Assume
n is even, so that Sq(E) are defined. Then the pull-backs of these bundles
over D(E) are canonically isomorphic on D(E) by the map
Me:(n*S^)e-----(3.19)
given at e g D(E) by
де(а) = e • a
that is, Clifford multiplication by e itself. Since e • e = — ||e||2 = — 1, each
map is an isomorphism. The pair of bundles 7i*Sq over D(E), together
with the isomorphism д: n*S£ h*Sq over D(E), given by (3.19), deter-
mine a “difference” element
Vc(E) =	e K(t(E))	(3.20)
where t(E) = D(E)/D(E) is the Thom space of E. Here К denotes reduced
complex К-theory (cf. 1.9).
If n = 0 (mod 4), the analogous construction clearly goes through in
the real case. Here we obtain an element
q(E) = [7i*S+,7i*S"; д] g KO(t(E)).	(3.21)
We are now in a position to define the map (2.3) discussed in the last
section. Let E8k be the universal 8k-plane bundle over BSpin8fc with its
unique spin structure. (E8fc is the pull-back of the universal 8k-plane bundle
over BSO8k by the map B£: BSpin8fc -> BSO8fc.) Let
6 KO(MSpin8Jk)
be the class (3.21) defined above. Now fix и and choose к sufficiently large
that we have the isomorphism Q^pin nB+8t(MSpin8J. Then a cobordism
class [X] e£lBpin determines a map fx:S"+Sk -> MSpin8lk. We define
: f^pin---► KO - "(pt)	(3.22)
by
4ЯЛ) =/MU) 6 W‘+") = KO(«") = KO-\pt).
§4. CONNECTIONS ON SPINOR BUNDLES	101
§4. Connections on Spinor Bundles
Suppose E is a smooth riemannian vector bundle over a manifold X and
that £: PSpin(E) -► PSO(E) is a spin structure on E. Then, of course, any
connection on Ps</E) can be lifted via £ to a connection on PSpin(E), and
this, in turn, defines a connection on the associated spinor bundles. In
this section we shall give an explicit computation of this spinor connection
and its associated curvature tensor in terms of the connection on E.
We begin by briefly recalling some facts from the theory of connections.
Let л: P -► X be a smooth principal bundle over a manifold X with group
G. Then G is a Lie group whose Lie algebra will be denoted by g. G acts
freely from the right on P. Each element Keg determines a vector field
V on P by setting
Vp = d/dt(p  exp(tK))|t=0.
The map V -► Vp gives an isomorphism
9 =	(4.1)
where is the tangent space to the orbit through p. The orbits are the
fibres of 7i, and the plane тГр can be thought of as the “vertical” space
through p. A connection is then a choice of an invariant field of comple-
mentary “horizontal” spaces.
Definition 4.1. A connection on P is a G-in variant field of tangent n-
planes т on P (n = dim(X)) such that the linear map я*: tp -► TKp(X) is
an isomorphism for all ре P.
At each peP, rp determines a linear projection Tp(P) -> У*р. The ca-
nonical isomorphism (4.1) then gives a linear map
%:Tp(P)------>g.	(4.2)
This defines a g-valued 1-form co on P, called the connection 1-form. It
has the following properties.
cd(V) = V for all Ve g.	(4.3)
= Ad„ - ico for all g e G acting on the manifold P. (4.4)
Note that given the connection 1-form co, one can recapture the connec-
tion by the relation
rp = ker(cop).
The curvature of the connection is the g-valued 2-form Q given by the
equation
Q = do + [co,co].
(4.5)
102	II. SPIN GEOMETRY AND DIRAC OPERATORS
(Note that by the skew-symmetry of the Lie bracket, [m,m](t;,w) =f
[сф),со(и>)] is a g-valued exterior 2-form.) This form has the following
properties.
Q(F, ) = 0 for all V e g	(4.6)
gm = Adg- i(Q) for all g e G	(4.7)
Example 4.2 (orthogonal connections). Let P = PSO(E) where E is a
smooth, oriented riemannian vector bundle. The Lie algebra of SO„ is the
space $o„ of real, skew-symmetric n x n-matrices. Hence, a connection 1-
form co can be considered as an n x n-matrix of 1-forms cd = ((c%)) where
co.. = — The corresponding curvature is a matrix of 2-forms Q = ((Qv))
where
Qy = d<oy + f aik л a)kJ.	(4.8)
fc = l
For an orthogonal matrix g, Adg(m) = gcog~r.
Let л ej denote the elementary skew-symmetric (i,j)-matrix. If
{eb... ,e„} denotes the canonical basis of R", this corresponds to the
transformation
(ef л e})(v) = <e(, v>e} - (cj, v)e{.	(4.9)
The connection and curvature forms can then be written as
« = - £ ".А л ei	(4.10)
(4.11)
Given a connection on the bundle PSO(E)as above, we can define a rule
for taking derivatives of sections of E. For any smooth vector bundle E'
over X, let Г(Е') denote the space of smooth cross-sections of E'.
Definition 4.3. A covariant derivative on E is a linear map
V.T(E)----->Г(Т*Х ® E)
such that
V(fe) = df® e+fVe	(4.12)
for all f e C°°(X) and all e e Г(Е).
Thus, given a smooth vector field V on X, we obtain a map : Г(Е)
Г(Е) called the covariant derivative with respect to V. At a given point
x g X, (Vve)x depends only on Vx and on the values of e in a neighborhood
of x.
§4. CONNECTIONS ON SPINOR BUNDLES	103
Proposition 4.4. Let cd be a connection 1-form on PSO(E) as above. Then a)
determines a unique covariant derivative on E by the rule
Vet = £ <5}i ® e}	(4.13)
J=1
where S = (e19... ,e„) is a local family of point wise orthonormal sections of
E9 i.e., a local section of PSO(E), and where co = <?*co.
This covariant derivative satisfies the rule
K<e,e'> = <VKe,e'> + <e, VKe'>	(4.14)
for all V g T(X) and e,e' g Г(Е), where <•, •> denotes the inner product in
E.
Conversely, any covariant derivative on E satisfying (4.14) determines a
unique connection 1-form by equations (4.13).
Note. A covariant derivative with property (4.14) will be called riemannian.
Proof. Let S = (el9... ,e„) be a frame defined on an open set U c X9
and let ((d>l7)) be any skew-symmetric n x и-matrix of 1-forms on U. Then
equation (4.13) together with property (4.12) defines a unique covariant
derivative on E\v. (To see this, note that any section e of E over U can be
written uniquely as e = ^ffij for fl9... 9fn g C°°(U). Then by (4.12) and
(4.13) we have Ve = £ dfj ® + £ f}<bkj ® ek = £ {< + £ <»„/,} ® ek.
This definition of V has properties (4.12) and (4.14) and is therefore a
riemannian covariant derivative on U.) Conversely, given & and a rieman-
nian covariant derivative V on I/, we have a skew-symmetric matrix of
1-forms ((a>l7)) on U uniquely defined by (4.13).
Consequently, to define a global covariant derivative on E it suffices
to assign to each local frame field S a matrix of local 1-forms ((<5i7)) sat-
isfying the following compatibility condition. Suppose S = (eb ... ,ея) and
= (e'19... 9e'n) are two orthonormal frame fields over an open set U9
and let <5 = ((c50)) and co' = ((c5-j)) be the associated matrices of 1-forms.
Then for each x e U9 there is a unique orthogonal n x и-matrix g(x) =
((fiij/x))) such that £(x) = &'(x)g(x)9 i.e.,
e,<x)= £ e'/xjg^x}.
Applying V and using (4.13) we find easily that
d)(x) = g " ^xJw'frMx) + g~ r(x)dg(x).	(4.15)
This transformation rule is the required compatibility condition.
Suppose now that PSO(E) is provided with a connection 1-form co. Then
given a local section S = (et,... 9en) of Pso(£) over an open set U, we get
104
11. SPIN GEOMETRY AND DIRAC OPERATORS
a skew-symmetric matrix of 1-forms <5 = ((d>0)) on U by setting <5 = <?*co.
Note that S determines a local trivialization
<p:U x SO„------► 7t"1(U)
of л: PSO(E) -► X by setting cp(x9g) = £(x)g. Conversely, cp determines £,
since &{x} = <p(x,e). Note that	is SO„-equi variant.
We now observe that in this local product determined by the connec-
tion 1-form can be written as
(<P*<g = Adg - i(c5) + g " ldg.	(4.16)
To see this we first write = coo 4- cdx where coo = Z	for local
coordinates x1 on X and where = £ Ь^х,д)йдц. Properties (4.3) and
(4.4) imply that = g~xdg. By definition we have coo = <?*co along
U x {e} c U x G. Property (4.4) then implies that coo = Adtf-i(c5) =
g~' ° <5 ° g at a general point (x9g).
If we choose a different cross section £' = (e\,... ,e'n) over U, we get a
new trivialization cp': U x SO„ n~ 1(U) by the formula cp'(x,g) = £'(x)g.
The change of trivializations Ф = (<?')”1 ° x SO„ U x SO„ is
given by
Ф(х,0) = (x9g(x)g)	(4.17)
where g:U -> SO„ is the change of frames as above, i.e., <?(x) = £'(x)g(x).
Clearly we have that
cp*cD = Ф*(ф'*С9).
Using (4.16) and (4.17) we can re-express this as
Ad9-1(<5) + g~'dg = Ad(g(x)g)- .(<£>') + (g(x)g)~ ld(g(x)g)	(4.18)
= Adg-i{Adg-.(x)(<5') + g~l(x)dg(x)} + g~ldg.
This equation immediately reduces to the compatibility condition (4.15).
Consequently, a connection 1-form on Pso(£) determines a riemannian
covariant derivative on E, as claimed. Conversely, given a riemannian
covariant derivative, we obtain local 1-forms <5 transforming according
to (4.15). This implies that the compatibility condition (4.18) for the ex-
istence of a global connection 1-form on PSo(E) is satisfied. This completes
the proof. 
Given a covariant derivative V on £, it is natural to ask whether the
second covariant derivatives commute in an appropriate sense. For this we
consider the composition
Г(Е) ЦТ* ® E) Г(Л2Т* ® E)
§4. CONNECTIONS ON SPINOR BUNDLES	105
where V is the natural prolongation of V defined on sections of the form
a ® e by V(a ® e) = da ® e — a л Ve, and we set R = V ° V
Proposition 4.5. Let cd, S and V be as in Proposition 4.4, and let Q be the
curvature 2-form of the connection. Then
Ret = £ Q>; ® e}	(4.19)
J=1
where Q = <?*Q.
Proof. From equation (4.8) we have
V(Ve() = v( £ а>л ® e})
\>=i	/
= £	® ej + Y a>kj л <5j,. ® ek
J=1	j,k=l
= £ 4. ® e}. 
J=1
Proposition 4.6. Let Q, and V be as in Proposition 4.5. Then for local
tangent vector fields V and W on X, we have
Ry,we =	— Wv ~	(4*20)
Proof. Note that
V^e^v/XeA^))
\ i	/
= Z e^fV^W) + £ e? • <5^)
M	j
and therefore
(V^ -	- V^je, = ej{ V  c^W) - W  d>j,(K) -
+ E e^V^W) - &jk(W)&kt(V)}
j*k
= E	+ E "д a
j	к
= ^e^V,W). 	(4.21)
j
Notice that from (4.14) we have the relation
У 4" <e, RVtWe У — 0.	(4.22)
Notice also that from the above it follows that the expression <RFJFe,e'>
is a tensor, that is, at any point x e X, it depends only on the quantities
Кс,И^,ех,е^ and not on the local fields V,W,e,ef extending them. Hence,
106
II. SPIN GEOMETRY AND DIRAC OPERATORS
given two tangent vectors K,PFat x e X, the curvature gives a well-defined,
skew-symmetric endomorphism
RVW:EX------> Ex	(4.23)
called the curvature transformation associated to V and W.
Suppose now that n: P -> X is a smooth principal G-bundle over X,
that p: G -► SO„ is a representation of G, and that
Ep = P xp R"
is the associated riemannian vector bundle (cf. II.3). Then given a connec-
tion on P there is a canonical connection induced on P(Ep) as follows.
Note that
P(Ep) =Pxf SO,
where an element g e G acts on SO„ via left multiplication by p(g). Given
a connection t on P, extend it trivially to P x SO„ and then push it
forward to P xp SO„. One can easily see that this gives a connection тр
on P(Ep).
There is a canonical, G-equivariant mapping
r:P----->P(Ep)	(4.24)
given by
p —► [(₽,«)]•
(where [(p,h)] denotes the class of (p,h) e P x SO, in the quotient P(Ep).)
If the representation p is faithful, the mapping (4.24) is an embedding.
For simplicity we assume p to be faithful.
Proposition 4.7. Suppose co and Q are the connection and curvature forms
on P respectively, and let cop and Clp denote the corresponding forms for the
induced connection on P(Ep). Then, considering P c P(Ep) as above, we
have that
o>p|p = P*a>
and
= РиД
where -► $on is the Lie algebra homomorphism associated to p:G
SOn.
Proof. This follows straightforwardly from the fact that the embedding
(4.24) is G-equivariant, i.e., that i(p • g) — i(p) • p(g). 
We are now in a position to discuss the connections on Clifford and
spinor bundles. Let E be an oriented riemannian vector bundle of dimen-
§4. CONNECTIONS ON SPINOR BUNDLES	107
sion и, and suppose E is furnished with a riemannian connection, i.e., a
connection t on PSO(E). The Gifford bundle C£(E) is associated to E by
the representation (3.1):
c£(pn):SOn-----► Aut(C£(R")).
Therefore, by the construction above, т induces a unique connection, t',
on Ct(E). Note that since ct(pj maps into the automorphisms of C£(Rn),
we have that the associated Lie algebra homomorphism is a map
сг(ри)*:*ои---->Der(C£(H)
where Der( ) is the Lie algebra of derivations; i.e., c£(pn)+ has the prop-
erty that for each element A e $on
{с£(рв)жЛ}(ф • ф) = ({с£(р„)фЛ}<р) -ф + <p • ({сЦрп)*А}ф) (4.25)
for all <р,ф g C£(R"). Recall, furthermore, that under the canonical iden-
tification Ct(R") = A*R", the representation c£(pn) becomes A*pn. To-
gether with Propositions 4.4 and 4.7, this gives the following.
Proposition 4.8. The covariant derivative V on Ct(E) acts as a derivation
on the algebra of sections, i.e.,
V(<p • ф) = (V<p) • ф + g, • (уф)	(4.26)
for any two sections cp and ф of Ct(E).
Furthermore, under the canonical identification СЦЕ) = A*(E), the co-
variant derivative V preserves the subbundles №(E) and agrees there with
the covariant derivative induced by the representation №pn (i.e., the usual
covariant derivative).
Corollary 4.9. The subbundles Cl°(E) and С£Х(Е) are preserved by V.
Furthermore, the “volume form" co = et • • • en is globally parallel; that is,
Vco = 0.
Therefore when n = 3 or 4 (mod 4), the eigenbundles CC^E) = {cp e Cl(E):
cocp — ± cp} are also preserved by V.
Proof. The first statement follows from the fact that Ct°(E) Acven(E)
and С£Х(Е) = Aodd(E). The second statement follows from the fact that co
corresponds to the unit section of A"(E). 
In the analogous way, Propositions 4.5 and 4.7 give the following.
Proposition 4.10. For any pair of tangent vectors V and W at x g X, the
curvature transformation
Rv^:Ct(Ex) -► C£(EX)
108
П. SPIN GEOMETRY AND DIRAC OPERATORS
is a derivation, i.e.,
’ Ф) =	’ Ф + <P ’ Ry.wW)	(4.27)
for all (р,ф g Ct(Ex). Furthermore, RVtW preserves the subspaces C£°(EX),
Ctl(EJ and Cl±(Ex) as above.
Suppose now that E carries a spin structure £:PSpin(E) -► PSO(P) and
let S(E) = PSpin(E) хд M be an associated real spinor bundle. Here M
is a left module over C£(R") and p: Spinn -► SO(M) is the resulting repre-
sentation (cf. §3). The connection t on PSq(E) lifts via the covering map
£ to a connection f on PSpin(E). This in turn induces a connection and
therefore a covariant derivative on S(E). Recall (Proposition 3.8), that the
sections of S(E) form a module over the sections of Ct(E).
Proposition 4.11. The covariant derivative V on S(E) acts as a derivative
with respect to the module structure over СЦЕ), i.e.,
V(<p • a) = (V<p) • a + (p • (Vff)	(4.28)
for any section (p of Ct(E) and any section о of S(E).
Corollary 4.12. If n = 3 or 4 (mod 4), the eigenbundles S±(E)={<pe
S(E):co(p = ±<p} are preserved by V.
Proof of Proposition 4.11. The representations ct(pn) (= Ad) and p pre-
serve the module multiplication, that is, p(g)((p • a) = {cl(pn)(g)(p} • {p(g)(r}
for all g g Spinn, cp g C€(R") and a g M (see the discussion surrounding
Proposition 3.8.) Differentiating at the identity, we get that for each element
Л g son = $pinn
 <0 = ({c«(p„).A}<p) • <7 + <p • ({m*A}<t).	(4.29)
The argument is now completed using Propositions 4.4 and 4.7 as before.

We also have the analogous statement for curvature:
Proposition 4.13. For any pair of tangent vectors V,W at x g X, the curva-
ture transformation
RVW:S(EX)------>S(EX)
is a module derivation, i.e.,
Ry,w(<P • <*) = Rv,w((p) -o + (p- Ry,w(p)	(4.30)
for all (p g Ct(Ex) and all в g S(Ex). Furthermore, Rv w preserves the sub-
spaces S±(EX) when they are defined.
§4. CONNECTIONS ON SPINOR BUNDLES
109
We shall now proceed to explicitly compute the connection and cur-
vature forms on S(E). To do this we need to examine the representation
Recall that son is 'generated by the elementary transformations x л у;
x,y g R", given by the formula
(x л y)(v) = <x,u>y - <y,u>x.	(4.31)
From Chapter I, §6, we have
л у) = |[х,у].	(4.32)
That is, л y)(<r) = |[x,y] • a for all а e M. Recall that on Spin„, we
have cl(p„) = Ad. Hence,
ct(pn)*x л у = Ad*x л у = adifx.y].	(4.33)
That is, Ad*(x л y)(<p) = | [[x,y],<p] for (p g Ct(R"). Note that equation
(4.29) follows immediately from (4.32) and (4.33).
Suppose now that g — (e19... ,en) is an n-tuple of pointwise orthonor-
mal sections of E defined over a contractible open set U c x. g is just
a section of PSO(E) over and it can be lifted to a section g of PSpin(£)
over U. There are two possible such liftings. They satisfy the relation:
£ о g = g.
The connection 1-form on PSpin(E) is just the lift £*co of the connection
1-form a> on Pso(£). To obtain a formula of the type given in Proposition
4.4 we want to pull £*co down to U by the local section g. This “pull-
down” is just
& = g*(£*O)) = ({ о g)*(co) = g*(O)).
Consequently, the scalar 1-forms c5fJ are just the 1-forms we obtained
earlier by pulling down the connection cu by the local frame field g.
Now for any spinor bundle S(E) we have a canonical embedding
PSpin(E) c Pso(S(£))- Thus, g can be considered as a section of Pso(S(E)).
Let a? denote the connection 1-form on Pso(S(E)). Then to apply Propo-
sition 4.4 we want to compute (5s = g*(a>s). However, by Proposition 4.7,
we know that co3 restricted to Pspin(£) ° Pso(^(P)) is just д*(£*су). Con-
sequently, we have
d? = д*с5.
Writing co = - £ (5^ л Cj (cf. (4.10)) and using (4.32), we can rewrite
this as	i<j
(5s = -i £ с5уе(е?	(4.34)
i<J
(Note that since e19... ,e„ are orthogonal, [ef9ej] =	= 2е^.)
110	II. SPIN GEOMETRY AND DIRAC OPERATORS
Finally, we note that the embedding PSpiri(£) c PSo(S(E)) can be inter-
preted to mean that every point of PSpin(£) determines an orthonormal
frame in S(E). In particular, the local section S determines a local section
= (o19... ,ctn) of Pso(S(E)). (The other choice of lifting S just deter-
mines the negative frame —Zf = { — oX9...9 — aN).)
Combining the remarks above, we have the following.
Theorem 4.14. Let co be the connection 1-form on PSO(E) an^ let S(E) be
any spinor bundle associated to E. Then the covariant derivative Vs on S(E)
is given locally by the formula
VSffa = 2 £ ®jl ® eiei ‘	(4-35)
i<j
where 8 = (en ... ,ert) is a local section of PSg(E)9 co = <£*(co), and where
Sf = (ffx,... ,crN) is a local section of PSo(S(E)) determined by 8.
Note. The frame field = (o19... ,aN) is, in fact, only determined up to a
constant orthogonal change of framing, that is, up to a choice of ortho-
normal basis at some point. (This corresponds to a choice of basis in
the module M9 i.e., on the matrix realization of the representation д.) This
family of framings is characterized by the following property. For any
I = (i19... 9ip)9 we have that eit • • * eipOj = Скц<тк where the coefficients
{Скц} are constants.
An analysis similar to the one above can be carried out for the curva-
ture 2-form. Since curvature is a tensor, we do not need to be concerned
in this case with the distinguished frame field on S(E). Hence, the cur-
vature of S(E) can be expressed in the following very pretty way:
Theorem 4.15. Let 12 be the curvature 2-form on PSo(E) and let S(E) be any
spinor bundle associated to E. Then the curvature Rs ofS(E) is given locally
by the formula
RSO = 2 E tyi ® eiej ’ G	(4.36)
i<J
where 8 = (el9... 9e„) is a local section of Pso(£\ £2 = <£*(12) and where a
is any section of S(E).
In particular, for any two tangent vectors V and Wat x e X9 the curvature
transformation RsVtW:S(Ex) -> S(EX) is given by the formula
Rv,w(o) = 1 X. (Rv.wtei),	(4.37)
where RVtW is the curvature transformation of Ex.
§4. CONNECTIONS ON SPINOR BUNDLES	111
Note that the expression
Wv.w = 1 X	e^ej	(4.38)
i<J
is independent of the choice of an orthonormal basis (ex,... ,ej for Ex
and is therefore invariantly defined. It is skew symmetric in V and W.
Consequently, 9F can be thought of as a 2-form on X with values in
Ct(E). The formula (4.37) can now be succinctly expressed as
Rsv^) = ^wo,	(4.39)
It is interesting to note that the arguments above can be carried through
also for the Clifford bundle C€(E) by using the equation (4.33). For tan-
gent vectors V and VK at x e X we define the invariant operator
^v\w = 1 X <Rytw^ e;>ade<ej	(4.40)
i<J
where, as before, (e15... ,en) is any orthonormal basis of Ex. We then ob-
tain the following expression for the curvature tensor on Cl(E) A*(E)
in terms of Clifford multiplication.
Theorem 4.16. For any two tangent vectors V and Wat xeX, the curva-
ture transformation RyW:Cl(Ex) -► C€(EX) is given by the formula
Rv.w((p) = *v.w(<p)	(4.41)
where w is the operator defined above in terms of the curvature trans-
formation Rv w of Ex.
It is an easy exercise to check that the restriction of the operator
®£x c C£(EX) agrees with KFtlF. One need only verify that =
(eiл f°r e G Ex-
Note that Theorem 4.16 does not depend on the existence of a spin
structure on E. In fact, it does not even depend on the existence of an
orientation on E.
We conclude this section with some remarks concerning the situation
where E = T(X), the tangent bundle of X. In this basic case we abbre-
viate notation by setting PSO(X) = ?so(T(X)), the (orthonormal) tangent
frame bundle of X, and Ct(X) = Ct(T(X)), the Clifford bundle of X.
Suppose now that PSO(X) is furnished with a connection, and let V
denote the corresponding covariant derivative. Then there is an invari-
antly defined tensor field associated to V as follows. Let V and W be
tangent vectors at a point x e X. Extend them to local vector fields (i.e.,
local sections of T(X)) and consider the expression
Tv,w = VyW-VwV-[V,W]	(4.42)
112	II. SPIN GEOMETRY AND DIRAC OPERATORS
where	is the Lie bracket. The value of Tv w at x can be easily
shown to be independent of the choice of vector fields extending Vx and
Wx (see Helgason [1, Chap. I]). Tv w is clearly bilinear and skew-sym-
metric. It therefore defines a global 2-form on X with values in T(X) called
the torsion tensor of the connection. The following result can be found
in any basic text in differential geometry.
Theorem 4.17. (The Fundamental Theorem of Riemannian Geometry).
Let PSO(X) denote the tangent frame bundle of a riemannian manifold X.
Then there exists a unique connection on PSO(X) with the property that its
torsion tensor vanishes identically.
This connection will be called the canonical riemannian connection on
X. It induces a canonical connection on Ct(X) = A*(X).
Note that if X admits a spin structure £: PSpin(X) -► PSO(X), then by
lifting, we obtain a canonical riemannian connection on PSpin(X). This,
in turn, induces a connection on any spinor bundle associated to PSpinP0.
This situation falls precisely into the general framework developed
above. Thus any spinor bundle for X is a bundle of left modules over
C£(X), and the canonical covariant derivative is a derivation of the mod-
ule multiplication (see Proposition 4.11).
For most of the basic facts of riemannian geometry, the reader is referred
to the general literature which is extensive and quite good. However, there
is one fact we use sufficiently often that it is worthwhile to cite it here.
Proposition 4.18. Let R denote the curvature tensor of a riemannian mani-
fold X, i.e., the curvature tensor of the canonical riemannian connection on
the tangent bundle TX. Then R satisfies the following identities:
RU VW + RVtWU + Rwu V = 0,	(4.43)
<Я^^,Г> = <Рж,г^О	(4.44)
for all tangent vectors U,V,W,Y g TxX, at all points x g X.
§5. The Dirac Operators
Let X be a riemannian manifold with Clifford bundle Ct(X), and let S
be any bundle of left modules over Ct(X) (i.e., a vector bundle over X
such that at each point x g X, the fibre Sx is a left module over the algebra
Ct(X)x.) Assume S is riemannian and is furnished with a remannian con-
nection. Then under these general hypotheses, we can define a canonical
first-order differential operator D: T(S) -► T(S) called the Dirac operator
§5. DIRAC OPERATORS
113
of S9 by setting
(5.0)
j=i
at x eX, where еъ... ,en is an orthonormal basis of Tx(X), where V
denotes the covariant derivative on S determined by the connection, and
where denotes the Clifford module multiplication. The operator D2 is
called the Dirac laplacian.
Recall that the principal symbol of a differential operator D: Г(Е) -► Г(Е)
is a map which associates to each point хе X and each cotangent vector
£g T£(X), a linear map o^D):Ex -► Ex defined as follows. If in local
coordinates we have
d|a|
D = E A/x) — and { = £ &хк,
|a|^m	^-X	fc
where m is the order of D, then
^D) = in E Aa(xK*.
|a| = m
The operator D is elliptic if is an isomorphism for all £ # 0 (see
III.l for further details).
Recall also that the riemannian metric induces a canonical isomor-
phism between T(X) and T*(X). Throughout this section we shall con-
sider them as so identified.
Lemma 5.1. Let D be the Dirac operator of the bundle S defined above.
Then for any £ e T*(X) s T(X) we have that
<№ = *	(5.1)
<^D2)= l^ll2	(5.2)
where the symbol on the right denotes Clifford multiplication by the vector
£ in (5.1) and the scalar ||^||2 in (5.2). In particular, both D and D2 are
elliptic operators.
Proof. Fix x g X and an orthonormal basis en ... ,en of Tx(X). Choose
local coordinates (xb ... ,x„) on X at x such that x corresponds to 0 and
corresponds to (8/dXj)Q for each j. Under the identification Tx(X)
Т$(Х) we have that e;- also corresponds to (dx;)0 for each j.
For any local trivialization of S near x, we have that Vej = (8/dXj)Q +
zero-order terms. Hence, at 0 we have that D = £ ej(8/8xj)Q 4- zero-order
terms. Consequently, for any cotangent vector £ = £ £j(dxj)0 at 0, we
have by definition of the symbol that a^D) = i £ efa = i£. This gives (5.1).
Then a5(D2) =	° tf^D) = — £•<!; = ||^||2, and the proof is complete.
114	II. SPIN GEOMETRY AND DIRAC OPERATORS
We now observe that in light of the basic examples it is natural to
require that the bundle S have certain additional properties. The first
property is that Clifford multiplication by unit vectors in T(X) be or-
thogonal, i.e., that at each x e X,
= <^19су2>	(5.3)
for all a 19a2 g Sx and all unit vectors e g Tx(X). Since e2 = — 1, this is
equivalent to the requirement that
4- <ffi,ea2> = 0	(5.3)'
for all such a2 and e.
Recall (from the end of §4) that the bundle Ct(X) carries a canonical
riemannian connection, whose associated covariant derivative will be
denoted by V. Our second requirement is that the covariant derivative
on S (which we also denote by V) be a module derivation, i.e., that
V(tp • a) = (V<p) • a 4- cp • (Va)	(5.4)
for all <p g Г(С£(Х)) and all о g T(S).
There is a surprisingly large and important collection of bundles with
the properties described above. Although these bundles can be quite
varied in nature, a substantial part of the theory concerning them can be
treated in a uniform way. For this reason we introduce the following
concept:
Definition 5.2. A Dirac bundle over a riemannian manifold X is a bun-
dle S of left modules over Ct(X) together with a riemannian metric and
connection on S having properties (5.3) and (5.4) above.
Before presenting examples of these bundles, we shall investigate some
of their elementary properties. Note that any Dirac bundle S has a canon-
ically associated Dirac operator. Furthermore there is an inner product
on T(S) induced from the pointwise inner product <•,•> by setting
=	(5.5)
Proposition 5.3. The Dirac operator of any Dirac bundle over a riemannian
manifold is formally self-adjoint, i.e.,
(Dcr1,ff2) = (cT1,DCT2)
for all compactly supported sections and a2.
Proof. Fix x g X and choose an orthonormal tangent frame field
(elf... ,en) in a neighborhood of x so that (7е.е;)х = 0 for all i,j. This can
be done for example, by extending a frame at x by parallel translation
§5. DIRAC OPERATORS	115
along geodesic rays emanating from x. Using properties (5.3)', (5.4) and
(4.14), we have that at x,
= £ <ejVejffi,<72>x = -£ <Vejffi,eJ-ff2>x
= -E {ei<ffueiff2> ~ <^u(Vejej)<72> -
j
= -E(eXffl’ejff2»x + <аъ°а2>х
= div(K)x + <ffi,Dtf2>x
where V is the vector field defined by the condition that
<Г,1У> =
for all tangent vectors 1У. The last line in (5.6) is established as follows
div(r)x=E<VeZ’e>>x
= E{*XW-<*W>>}x
= ZK<>4>}X
j
= -E{eJ<^eJ ^2>}x-
The first and last expressions in (5.6) are independent of the frame field
(ev... ,e„). Hence, we have established the equation
— div(K) 4-
on X. The proposition follows immediately. 
Note that if X is permitted to have a boundary dX, then the above
argument proves that
(Dffj, ff2) - (ff!, Dct2) = f <v • a!, ff2>	(5.7)
JvA
for compactly supported sections and rr2, where v denotes the outer
unit normal field to dX in X.
It is a general consequence of the ellipticity of D that any weak solution
to the equation Difr = 0 is of class C00 (cf. Theorem 5.2(i) of Chap. III).
Because of the formal self-adjointness this may be expressed as follows.
Let Tcpt(S) denote the Frechet space of C00 sections of S with compact
support. Then any continuous linear functional F on Tcpt(S) such that
F(D(p) = 0 for all (p g Tcpt(S), can be represented as F((p) = where
ф e r(S) and D$ = 0. In particular, any locally integrable section ф of S
which is orthogonal to DFcpt(S) (i.e., which satisfies the equation D$ =
0 weakly), is of class C00 and satisfies the equation D$ = 0 in the usual
sense.
116
II. SPIN GEOMETRY AND DIRAC OPERATORS
Similar comments apply to D2.
With this in mind, we define ker D = {cp e T(S): Dtp = 0} and ker D2 =
{cp g T(S): D2(p = 0}, and observe the following:
Theorem 5.4. Let D be the Dirac operator of any Dirac bundle over a com-
pact riemannian manifold X. Then
ker D = ker D2
and this space has finite dimension.
Proof. Finite dimensionality is a direct consequence of elementary elliptic
theory (cf. Theorem 5.2(ii) of Chap. III). For the rest, observe that
D2cp = 0 => ||Z><p||2 = (D(p9D(p) — (<p9D2<p) = 0 =*> D<p = 0. 
The results above have an important extension to bisections of a Dirac
bundle over any complete manifold. We begin with the following pre-
paratory lemma.
Lemma 5.5. Let D, S and X be as above. Then for any f g C°°(X) and any
(p g T(S), we have that
D(f(p) = (gradn-<P+W	(5.8)
Proof. D(fy>) = £ e, • Vejlftp) = E еД(ву/)Ф + Г?е/р} = (X (е/Ю • <P +
fD(p) = (grad f)  <p + fD<p. 
Remark 5.6. The relation (5.8) immediately extends to distributional
sections Ф of S. To see this, consider <p e Fcpt(S), and note that from (5.8)
and (5.3)', (/РФ, ф) = (Ф, D(fy>)) = (Ф,(grad /) • <p + fD<p) = (Р(/Ф) -
(grad /)• Ф, <p).
We shall now consider the space L2(S) of bisections of S. This is the
natural completion of Tcpt(S) in the Hilbert space norm introduced above.
We now consider D as a symmetric operator on Tcpt(S) and take its closure
(also denoted D), in L2(S). This gives us an unbounded operator in L2(S).
Its domain dom(D) consists of all cp g L2(S) for which there is a sequence
<<Pn>n=i <= rcpl(S) such that (pn -► <p and Dcpn -► ф = “Рф” in L2(S).
There is another extension of this operator to L2(S) which we shall
denote D*. The domain of D* consists of all (p g L2(S) such that the dis-
tributional image Dcp is also in L2(S). Of course, dom(D) <= dom(D*).
We now observe that D* is simply the adjoint of D. Recall that ф g
L2(S) is in the domain of the adjoint D* if the function (p <ф(р9фу is
continuous on dom(D). Since dom(D) is dense, there exists a unique ele-
ment D^ g L2(S) such that <D(p, фу = <ф, D^y for all (p g dom(D). Clearly
D^ is just the distributional image of ф under D> and so D‘ = D*.
§5. DIRAC OPERATORS
117
We now prove that on complete manifolds, any Dirac operator is
essentially self-adjoint (cf. Wolf [1]).
Theorem 5.7. Let X be a complete riemannian manifold and let D be the
Dirac operator of any Dirac bundle S over X. Then the closure of D in
L2(S) is a self adjoint operator. Furthermore,
ker(D) = ker(D2)
on L2(S).
Proof. Fix x0 g X and let d(x) be a regularization of the distance func-
tion from x0. Choose x e C°°(R) so that 0 x = 1, X(0 — 0 for t 2, /(t) =
1 for t g 1, and |/|	2. Then set
XnW = X| - d(x)).
We want to show that dom D = dom(D*). It suffices to prove that
dom(D*) g dom(D) since the reverse inclusion is obvious. Choose cp g
dom(D*) and define cpn = xn<P for each positive integer n. Then by Re-
mark 5.6 we know that Dcpn = (grad x„) • tp + xn&<P for each n. Clearly,
Xj)(p -► Dcp in L2(S). Furthermore, if we let Bp denote the metric ball of
radius p centered at x0 in X, then
||(grad %„) • <p||2 £ f 4 ||<р||2	0.
•/"In "n fl
Consequently, Dcpn -► Dcp in L2(S).
Thus, we are reduced to the case where cp g dom(D*) has compact
support. By a partition of unity over supp cp we can assume that cp has
compact support in a local coordinate system. Here, using Fourier trans-
form methods, we can construct a parametrix for D, i.e. a pseudo-dif-
ferential operator P such that
DP = 1 - and PD = 1 -
where and SF' are infinitely smoothing operators, and where P, $F and
all have Schwartzian kernels supported near the diagonal. Observe
now that since Dcp g L2(S), there exists a sequence i <= Fcpt(S),
with support uniformly bounded in this coordinate neighborhood, such
that фп -► Dcp in L2(S). We then set cpn = Рфп + SF'cp and observe that
since the Schwartzian kernels of P and SP' are supported near the diagonal,
each (smooth) cpn has compact support. That is, we have <tpn>®=i <=
rcpt(S). From the relations above, we see that cpn -► PDcp -I- SP'cp — cp,
and that Dcpn = DPij/n + D£P'cp = фп —	+ DSf'cp Dtp — SFDcp +
D£Pfcp = Dcp (since, clearly, we have D&” = £FD). This completes the
proof of the essential self-adjointness of D.
118	II. SPIN GEOMETRY AND DIRAC OPERATORS
We now prove that ker(D2) c ker(D). That is, we shall show that any
(necessarily smooth) bisection ф of S which satisfies the differential
equation D2cp = 0, also satisfies the equation D(p = 0. To see this, let
be the sequence of functions above and note that
о =	= (D(p,D^q>))
= (^<P,2z„grad(z„) • <p + riDtp)
= IlZn^vll2 + grad(x„) • <p).
Consequently, by the Schwartz inequality we have
||z„P<p||2 2||zBD<p|| llgrad (z„) • <p||.
Therefore,
||х„Оф|| 2||grad(x„) • ф|| t|M|,
and we conclude that \\D<p\\ = lim ||x„D(p|| = 0. This completes the proof.
Having discussed Dirac bundles in general terms, it is now time to look
hard at some important examples. We begin with the basic ones.
Example (an historical case). Let X = R", euclidean n-space, and let
S = R" x V where V is some finite dimensional module for C£„. In this
case the Dirac operator is a constant coefficient operator (on F-valued
functions) of the form
n d
o = I y*/-
k=i Sxk
where each yk is a linear map yk: V V and where
У ilk + УкУ) = ~2<5Л
for all j,k. If we choose a basis for F, these yk’s will be represented by
matrices. The relations above imply that
(Д
л
0
0
Д/
where Д = — £ d2/8xl is the positive laplacian in R".
This particular operator has historical roots in physics. In the 1920s, the
physicist P.A.M. Dirac was searching for a Lorentz-invariant first-order
differential operator whose square would be the Klein-Gordon operator.
Thus he was essentially led to search for a first order operator D of the
§5. DIRAC OPERATORS
119
form above which satisfied the equation D2 = Д. Realizing that the yfc’s
must be matrices, he was led immediately by this equation to the above
relations, which we recognize now as the generating relations of a repre-
sentation of C£„. See the beautiful book of Dirac [2] for a discussion of
the physics.
It is illuminating to consider this euclidean operator in low dimensions.
Let n = 1, so that С£х = V= C. Then we have
the generator of a basic semigroup of unitary operators on L2.
Let n = 2, so that C£2 = И= Н1=СфС. The decomposition of Hl
into С ® C is natural and corresponds to the Z2-grading C£2 =
C£2 ф C£2. With respect to this Z2-grading, the Dirac operator inter-
changes even and odd parts. In particular if we identify C£2 and C£2
with C by setting и 4- ve2er = it + iv = uex 4- ve2, then D = e^d/dxj 4-
e2(d/dx2) has the form
where d/dz = d/dx1 4- id/dx2. Thus, the Dirac operator on R2 = C, con-
sidered as mapping even to odd spinors, is exactly the Cauchy-Riemann
operator.
Let n = 3, so that C£3 = HI ф Hl and V = Hl. C£3 has two representa-
tions on Hl given as follows. Identify R3 with Im(H), and let {i,j,k} be the
standard basis of imaginary quaternions. Then the two representations
are generated by letting i,j,k act on either the right or the left in HL
Choosing multiplication on the left, we get the following expression for
the Dirac operator on H-valued functions
дхг J dx2 dx3'
If we re-express left quaternion multiplication, with respect to the basis
{l,i,j,k}, as 4 x 4 real matrices, then D becomes
where дк = d/dxk.
120
II. SPIN GEOMETRY AND DIRAC OPERATORS
Let n = 4, so that Cl4 = H(2) and V = H2 = И ф H. Here again the
splitting corresponds to a Z2-grading of the module F, and D interchanges
parts. To describe the full Dirac operator we consider first the following
quaternion analogue of the Cauchy-Riemann operator. Identify R4 with
H under the standard basis	and define the following operators
on functions from H to H:
£ = _L	JL JL k—	8 - 8	8	8	k d
dq SxQ^	dx1^^dx2^ dx3	dq dx0	1 dxr	dx2 ox3
Then the Dirac operator can be expressed with respect to the splitting
И ф HI as
Note the analogy with dimension two.
Note that left quaternion multiplication is always complex linear with
respect to the complex structure given by multiplication by i on the right.
Thus, left multiplication by ij and k on И = C2 could be represented by
complex 2x2-matrices a2 and a3 respectively. Under this conven-
tion, the operator d/dq becomes
d d d
dq dx0 + 671 dxx + °2 dx2 + dx3
The matrices can be chosen to be the classical Pauli matrices:
/i 0\	/0 -1\	/0 A
-i)’	a2-\l	0/’	0/
Note that these matrices generate the fundamental representation of C£3
in complex form.
We could continue this analysis. For general n, one can calculate
an enormous N x N-matrix whose entries are linear combinations of
d/dxu... ,d/dxn. Here N is on the order of 2". This matrix will have the
property that its square is Д • I where I is the N x N identity matrix. How-
ever, this explicit form of D is seldom, if ever, useful. It is always simpler
to use the structure of the Clifford module.
It is interesting to note that the concept of D as a generalized Cauchy-
Riemann operator is a useful one. Let us fix a dimension n, and let D
denote the euclidean Dirac operator acting on functions f: R" -► V where
V is some fixed C£„-module. Recall that the fundamental solution for the
§5. DIRAC OPERATORS
121
Laplace operator on R", for n > 2, is ф(х) = c„/||x||" 2 for an appropriate
constant cn. That is, we have
Дф = <50
where <50 is the Dirac (5-function at the origin. Since D2 = Д • 7, where I
is the identity on F, we have that
Ф = D(cpl)
is the fundamental solution for D. That is,
РФ = 6QI.
From this one can derive very pretty analogues of the Cauchy Integral
Formula, and the analysis of D becomes quite accessible.
We now examine Dirac bundles S over more general spaces. Let X be
an arbitrary riemannian manifold. Then there are two basic cases.
Example 5.8. (the Clifford bundle). Let S = C£(X) with its canonical
riemannian connection, and view C£(X) as a bundle of left modules over
itself by left Clifford multiplication. (Note that Property (5.4) was estab-
lished in Proposition 4.8.) The Dirac operator in this case is a square root
of the classical Hodge laplacian. Most of the remainder of this section is
devoted to a detailed analysis of this basic case.
Example 5.9. (the spinor bundles). Suppose X is a spin manifold with
a spin structure on its tangent bundle. Let S be any spinor bundle asso-
ciated to T(X). Then S is a bundle of modules over C£(X), and as shown
in §4, S carries a canonical riemannian connection which has property
(5.4) (see Proposition 4.11 and the discussion following Theorem 4.17).
The Dirac operator in this case was first written down by Atiyah and
Singer in their work on the Index Theorem. Finding this operator was a
major accomplishment, and for this reason we shall call it the Atiyah-
Singer operator.
Notation, For spin manifolds X of even dimension we shall denote the
(unique) irreducible complex spinor bundle by $c; and when dim(X) ф 3
(mod 4), we denote the irreducible real spinor bundle by $. In both cases
the Atiyah-Singer operator will be written
These basic examples each generate large families of new examples by
the following construction. Let S be a given Dirac bundle with connection
Vs over a riemannian manifold X, and let E be an arbitrary riemannian
vector bundle with connection V£ over X. Then the tensor product S ® E
is again a bundle of left modules over Ct(X), where for (p e Ct(X), о e S
and ее E, the module multiplication is given by setting
Ф • (a ® e) = (ф • a) ® e.
122
II. SPIN GEOMETRY AND DIRAC OPERATORS
It is clear that in the tensor product metric on S ® E we have that this
Clifford multiplication by unit tangent vectors on X is orthogonal (i.e.,
Property (5.3) is satisfied). Furthermore, we can equip S ® E with the ca-
nonical tensor product connection, V = Vs ® Vе, which is defined on sec-
tions of the form a ® e by the formula
V(a ® e) = (Vsa) ® e 4- a ® (VEe).
It is straightforward to verify that with this riemannian connection, the
bundle S ® E has the derivation property (5.4). Consequently, we have
proved the following.
Proposition 5.10. Let S be any Dirac bundle over a riemannian manifold X,
and suppose E is any riemannian bundle with connection. Then the tensor
product S ® E is again a Dirac bundle over X.
An interesting example of this construction is the following. Let X be
a spin manifold of dimension 8k, and let $ be the canonical (real) spinor
bundle of X. Then under the above construction the bundle $ ® $ is a
Dirac bundle over X in two canonical ways (by multiplication in the left
or the right factor). It follows from the representation theory (see 1.5.18
and IV.10.16-17) that
C£(X) $ ® $
where the two module structures correspond to multiplication on the left
and on the right (by the transpose) in C£(X).
Similarly, in all even dimensions we have
C£(X) ® C $c ® ft.
Remark 5.11. The construction above does indeed give rise to a large
number of examples. As we shall see later (Remark III.13.il), basically
every elliptic operator on a spin manifold can be constructed, up to homo-
topy and degree shift, as a Dirac operator of the form D: T(S° ® E) -►
T(Sl ® E) where S = S° ф S1 is the complex Z2-graded spinor bundle.
Note that in each of the basic cases above (Examples 5.8 and 5.9), the
Dirac bundles and their associated Dirac operators are canonically defined
in terms of the riemannian metric on X. Hence, any mathematical objects
constructed using these operators are invariants of the riemannian struc-
ture on X.
The remainder of this section will be devoted to an analysis of these
operators. We begin with the Clifford bundle C£(X).
Our first observation is that it is also possible to view C£(X) as a bundle
of right modules over C£(X) (by right Clifford multiplication). Property
(5.4) also holds for right multiplication. Hence, we can also define a “right-
§5. DIRAC OPERATORS	123
handed” Dirac operator D on CE(M) by setting
D<p=t (Vej<p) • Sj.	(5.9)
J=1
This operator is also elliptic and formally self-adjoint. The principal sym-
bol is just right multiplication by i^.
Recall now that there is a canonical isomorphism C£(X) s A*(T*(X)) =
A*(X). The bundle A*(X) also has two canonical first order operators,
namely the exterior derivative d:A*(X) -> A*(X) and its formal adjoint
d*:A*(X) -> A*(X). This adjoint is given by the formula
d* = (-l)"P+n+1*d*	(5.10)
on A₽(X), where *:A₽(X) -> A""P(X) is the linear map defined by the
condition that (p л = <<p,^>*l where *1 is the volume form.
The Dirac operators and the exterior derivative are directly related as
follows:
Theorem 5.12. Under the canonical isomorphism C£(X) s A*(X), the Dirac
operators of C£(X) satisfy the following equations:
D^d + d*	(5.11)
D = (— l)p(d - d*) on AP(X).	(5.12)
Consequently, since d2 — (d*)2 = 0, they also satisfy
D2 = D2 = dd* + d*d = A	(5.13)
DD = DD.	(5.14)
The operator A defined in (5.13) is called the Hodge laplacian.
Proof. Fix x g X and choose an orthonormal frame field (eb ... ,e„) in a
neighborhood U of x with the property that (Veje;)x = 0. We first observe
the following:
Lemma 5.13. The operators d and d* are given in U by the formulas
d = t eJ A Vej
d*= ~t eyLVe,
J = 1
where “l” denotes contraction in A*(X).
Proof. Both expressions are invariantly defined; that is, they are indepen-
dent of the choice of frame field (eb ... ,e„). To establish the first identity
it suffices to show that the operator on the right satisfies the following
124
II. SPIN GEOMETRY AND DIRAC OPERATORS
axioms for d:
(i)	d2cp = 0
(ii)	d(cp л (Д) = dcp /\ ф 4- ( — l)₽cp л d$
(iii)	df = grad(/)
for all smooth functions f and all smooth p-forms cp and <?-forms ф.
Property (iii) is obvious. Property (ii) is seen as follows:
d(cp лф)= £ ej л [(Vey<p) л ф + q> л (Ve/)]
J = 1
= dcp л Ф 4- (— Ip’cp л (1ф.
(Here we use the fact that V acts as a derivation on A*(X).) To prove prop-
erty (i) we first note that from the independence of the choice of frame
field, it suffices to verify this at the point x where we have (Ve.e7)x = 0.
Furthermore, by linearity it suffices to consider a cp of the form cp =
aex л ... л ep where a is some smooth function in U. Then one easily sees
that at x,
d2q> = L ([^A]a)ey л ek л л ... л ep.
P<j<k
Since [e;,ek]x = (Vejek — 7еке;)х = 0, property (i) is proved, and the first
equation is established.
For the second equation, we again consider ср = aer л ... л ep. Hence,
at x we have that
d(*<p) = £ (eja)ej л ep+i л ... л e„.
j=i
Consequently, at x
♦(d ♦</>)=£ (eja) ♦ (е; л ep+1 л ... л en)
j=l
= Д (~	‘(e/Ocj A ... A ej л ... л ep
= (- l)n(₽+ n £ (ejajej L (el л ... л ep)
j=i
= (-l)n(₽+1)
This completes the proof of Lemma 5.8. 
We now recall that under the canonical isomorphism C£(X) A*(X) we
have that
e' ср = e л cp ~ e L cp
cp • e = (— l)₽(e л cp 4- e L cp)
§5. DIRAC OPERATORS
125
for all e g A^X) and all p g A₽(X) (see the discussion at the end of §3 in
Chap. I). Equations (5.11) and (5.12) now follow directly from Lemma 5.13.
This completes the proof of the theorem. 
This theorem has the following important consequences. On X the space
of harmonic p-forms is defined to be
H = ф Hp = ker(A).
p = 0
Corollary 5.14. If X is compact and without boundary, then
ker(D) = ker(D) = ker(A)
In particular, under the isomorphism Ct(X) = A*(X), the kernels of D and D
correspond to the space of harmonic forms on X.
Note that by applying Theorem 5.7 to a non-compact complete mani-
fold we conclude that a differential form which is in L2 is harmonic if and
only if it is both closed and co-closed.
In the compact case the harmonic forms are related to the topology of
X as follows. Consider the so-called de Rham complex:
0-----► Г(Л°Х) -U Г(А’Х) Г(Л2Х) -U • • •.
Since d2 = 0 we can form the quotient:	= [ker(d)/image(d)]*. The
fundamental theorem of de Rham asserts that for each p = 0,..., n, 3tfp(X)
is isomorphic to H₽(X;IR), the pth singular cohomology group of X with
real coefficients.
Since we are given a riemannian metric, we can also consider tfc adjoint
sequence:
0 «---Г(А°Х) ЦЛ'У) Г(Л2Х) «-£----------------------
The fundamental result of harmonic theory is the following Hodge De-
composition Theorem (Corollary III.5.6):
Theorem 5.15. Let X be compact, and without boundary. Then there is an
orthogonal decomposition
Г(Л*Х) = H © Im(d) © Im(d*)
(where Im() denotes the image of the operator on Г(Л*Х)). In particular
there is an isomorphism
Hp s HP(X;R)
for each p = 0,... ,n,
A Note on Orientability, It is clear that orientability is not required for the
definition of D, and so the spaces H₽ can be defined for a non-orientable
126
II. SPIN GEOMETRY AND DIRAC OPERATORS
manifold X. Moreover, if X is compact the isomorphism Hp = HP(X;R) also
holds. This is proved as follows. Let n: X -► X be the two-fold, orientable
covering manifold, and let Г ={!,#} = Z/2Z denote the group of covering
transformations of X. Then Г acts naturally on A*(X). The subspace
A*(X) of Г-fixed elements is preserved by d, and its cohomology is just
the cohomology of X. (Orientability is not required for the de Rham Theo-
rem.) We claim that the inclusion of this cohomology into the cohomology
of X is injective. Suppose cp e Af(X) and cp = dip for some ф e A₽-1(X).
Set ф' = |((Д + д*ф). Then ф' e Af" \X) and since d commutes with g*,
<p = Дф'. This proves the injectivity.
Given a riemannian metric on X, we lift the metric to X. The harmonic
forms on X lift to Г-invariant harmonic forms on X, and every Г-invariant
harmonic form is such a lift. That is, H*(X)SH?(X). We now observe that
H?(X) corresponds to the subspace Я?(Х;К) с Я*(Х;К). To see this we
note that if cp is a harmonic form, so is g*cp. If cp is cohomologous to a
closed Г-invariant form, then cp and g*cp represent the same cohomology
class. Hence cp = g*cp. We conclude that H*(X) = H?(X) Я?(Х;К) s
Я*(Х;К) as claimed.
The fundamental identities (5.13) and (5.14), established above for the
operators D and /5, imply certain important identities for the curvature
tensor. In particular we have the following.
Theorem 5.16. Suppose X is a riemannian manifold, and let R denote the
Riemann curvature tensor acting by derivations on the bundle Ct(X) =
N*(X).Then for any element cp e C£(X),
X. {e.ejRe„ej(<P) ~ Яеье/ф)^.} = 0	(5.15)
X еЛ(,еД<р)е?- = 0,	(5.16)
where (e15... ,e„) is any orthonormal tangent frame at the point in question.
In particular, from (5.15) we conclude that
X efijRei,e^ = ~ X.	(5.17)
2 i<j
2 S ах1в<в^(Лвйв^(ф)).
Note. Since {ef- л е;}1<; represents an orthonormal basis of A2(X), the
formulas above could be easily re-expressed without the use of (en ... ,ел).
Proof. Let us fix a point x e X and choose a local orthonormal tangent
frame field (e19... ,e„) such that (Ve;)x = 0 for each j. Then for any section
§5. DIRAC OPERATORS
127
о e Г(С£(Х)), we have at x that
D2<P = £ e,Ve((e;V <p)
E ft e/P
= ~E Vei.Ve(<p + £ e.-e/V^ - VejVJ<p
= Ve,Ve,<P + E «.«Аое/ф)-
i	i<j
Similarly, we conclude that at the point x,
D2<P = ~E Ve(Vei<P + E Ле(,е>к/.-
i	i<j
Since D2 = D2 (see (5.13)), we conclude that (5.15) holds at x, and therefore
everywhere.
The second equation is proved similarly. We observe that at x,
DD<p = £ e,(Ve(V ej<p)e}
DD<p = £ ei(Veyei<p)ej.
ij
Subtracting these equations and recalling that DD = DD completes the
proof. 
Since equations (5.15) and (5.16) are for Clifford elements, each con-
stitutes 2" scalar equations. They include the Bianchi identities (consider
the A1-component.) They also include a large number of new identities
for the curvature transformation of the bundle A*(X). These identities
will prove useful in the Bochner-type vanishing arguments presented in
§8.
We now examine some of the basic operators on С£(Л) and analyse
their relationships with D and D. Recall that C£(X) carries a canonical
bundle mapping
a:C£(X)------► C£(X)	(5.18)
which is, on each fibre C£(X), the algebra automorphism extending the
map —1 on T/X), Since a2 = 1, we obtain a decomposition
C£(X) = C£°(X) ® C£x(X)	(5.19)
where C€°(X) and C£ x(X) are the 1 and — 1 eigenbundles of a respectively.
Under the isomorphism C£(X) A*(X), we have C£°(X) Acvcn(X) and
СР(Х) Aodd(X). For any non-zero vector e g Tx(X) left (or right) Clif-
ford multiplication gives an isomorphism erCf^W Л Cf^JV)- Hence,
if X admits a nowhere vanishing vector field, i.e., if the Euler characteris-
tic of X is zero, then the bundles С£°(АЭ and Cf x(X) are isomorphic.
128
II. SPIN GEOMETRY AND DIRAC OPERATORS
There is a second canonical bundle mapping
L:C£(X)------► C£(X)	(5.20)
which on C£x(X) is defined by the formula L(<p) = — Yj=i ej<Pej for any
orthonormal basis	of Tx(X). This map is globally diagonalizable
and yields the canonical bundle decomposition
C£(X) = ф A₽(X).	(5.21)
p = 0
In particular,
L = (- l)₽(n - 2p) on AP(X)	(5.22)
for each p = 0,... ,n (see Chap. I). Clearly a and L commute, and their
composition satisfies
a о L = L о a = (n - 2p) on AP(X)	(5.23)
for each p.
Finally, we consider the section co of A"(X) cz C£(X) given at each point
x by setting
co = er • • • en	(5.24)
where ... ,en is any positively oriented orthonormal basis of Tx(X).
Since co is independent of the choice of basis we may for any x e X choose
local fields ... ,en such that (VeJx = 0 for each i. This shows that
Vco = 0.	(5.25)
The section co satisfies the relations
n(n+ 1)
co2 = (-l)^—	(5.26)
coe = (-l)"-1eco	(5.27)
for any section e of T(X) c= C£(X).
We now define a canonical bundle map
4:C£(X)------->C£(X)	(5.28)
by setting
= <o • (p.
If n s 3 or 0 (mod 4), then = 1 and we have a decomposition
Ct(X) = Ct+(X) ф Ct ’ (X)	(5.29)
where Ct±(X) are the +1 eigenbundles of zw. From (5.27) we see that for
any non-zero vector e e Tx(X) at any point x, left Clifford multiplication
§5. DIRAC OPERATORS
129
gives isomorphisms:
e:Cf*(X)-----> Cl*(X)	if n = 0 (mod 4)	(5.30)
e: Cf*(X)----> Ct^(X)	if n = 3 (mod 4).	(5.31)
The bundles Cl ±(X) can be explicitly written as
Cf^X) = (1 ± o>)C£(X).	(5.32)
Each such bundle is evidently a submodule under right Clifford
multiplication.
The above construction is particularly important since it generalizes
immediately to any Dirac bundle S over X. Again we have a bundle
isomorphism
(5.33)
and a corresponding decomposition
S = S+ ®S~	(5.34)
where S* = (1 ± co) • S. Moreover, for any non-zero e e Tx(X), the for-
mulas analogous to (5.30) and (5.31) hold.
Note that for n = 1 or 2 (mod 4),	= —1. Hence if we complexify
S and consider the operator we obtain a splitting S ® C =
(S ® C)+ © (S ® C)“. In these dimensions, Ла defines a complex struc-
ture in S, and this splitting is the usual (1,0), (0,1 (-decomposition for the
complex structure.
Let us turn our attention back to the Clifford bundle C€(X) and ex-
amine some of the elementary properties of the operators defined above.
Lemma 5.17. The operators a, L and Лш satisfy the following relations
(i)	aL — La = 0
(ii)	аЛш + (- 1)"-1Лдаа = 0
(iii)	L4 + (- 1)ЧЛ = о.
Proof, (i) was observed above. For (ii) we note that a(co<p) = a(co)a(<p) =
(—l)"coa(<p). For (iii) we see that by (5.27), L(<o<p) = — £ efiMpej =
- (-1)" ~1 L озере, = (-1)" ~ 1 wL(<p). 
It follows from (iii) and (5.22) above that ^„(A^X)) = /\"~P(X). In fact
Ла is related to the «-operator (cf. (5.10)) as follows:
U)(p = ( — 1)
x	for g AP(X). (5.35)
<pCD = (—l)2
130	II. SPIN GEOMETRY AND DIRAC OPERATORS
To see this it suffices to consider (p = er • • • ep. Then *<p = x • • • en and
the computation is easy.
Lemma 5.18. The operators a, L and considered as sections of
Hom(Ct(X),Ct(Ar)), are globally parallel. That is,
[V,a] = [V,L] = [VX] = 0.
Proof. That a is parallel follows from the fact that —1 is parallel in
Hom(T(AXT(Aj) and V is a derivation. That is parallel follows from
(5.25). For L we fix a point x e X and choose a local orthonormal frame
field e19... ,en such that (Ve^)x = 0 for each j. Then at the point x, V{L(p) =
-V £ e^= -£ {^e^ej+e/y^ej+e^ej)} = -£ gj(V<p)e;=L(V<p).
This completes the proof. 
Corollary 5.19. The subbundles AP(X) and Ct,±{X) {when defined) are
preserved by covariant differentiation.
We now examine the relationship of these operators to the Dirac
operators.
Proposition 5.20. Let D and D be the Dirac operators on C£(X) defined
above. Then the following relations hold
(i)	Da + aD = Da + aD ~ 0
(ii)	D4 + (- IfAJD = D4 - 4D = 0
(iii)	DL + LD = 2D; DL + LD = 2D.
Corollary 5.21. The operator A = D2 = D2 satisfies the relations
[о. A] = [4, A] = [L,A] = 0.
In particular we have that АЮ:Н₽ Д H"-p. This is the Poincare Duality
Isomorphism.
Proof of Proposition 5.20. We shall consider only D. The arguments for
D are similar. Let q> be a section of C€(X). Then using the lemmas above,
we have
(i)	D(a<p) = £ eyVe/a<p) = £ eyx(Vej<p)
= -a^£	= -«(IM
(ii)	£>(«<?) = £ efl'fwp) = £
= ( -1)"-	£ ejVe><p = (-!)"' la>D(p
§5. DIRAC OPERATORS
131
(iii)	D(L<p) = % eyej(L(p) = £ ejL(Vej<p)
= “S eA(?ej<P>*
= “Ё (~ekej -
= -L(D<p) + 2D(<p).
This completes the proof. 
The above arguments carry directly over to the following case.
Proposition 5.22. Let S be any Dirac bundle over X. Then the Dirac opera-
tor on S satisfies the relation
Д4 = (-1ГЧЛ
We shall conclude this section with some remarks on the variation of
the Atiyah-Singer operator under changes of metric.
Notice that all of the standard bundles of riemannian geometry—the
tangent bundle, the cotangent bundle, and their tensor products—have
structure group GL„(R). They exist independently of any metric con-
siderations. In fact, the introduction of a riemannian metric amounts to
a (simultaneous) reduction of the structure group of these bundles to SO„.
When X is a spin manifold, the situation for the canonical spinor bundle
of X is completely different. The spinor bundle itself depends on the choice
of riemannian metric.
This last statement can be made precise as follows. Let PGL+(X) be the
oriented frame bundle of X and suppose that dim(X) = n > 2. Denote by
GL„+ (R) -► GL„+ (R) the 2-fold, universal covering group of GL„+ (R). Since
X is spin, there exists a principal GLn+ (R)-bundle Pgt+(X) with a GL„+ (R)-
equivariant bundle map P^(X) ->PGL+(X). Introducing a riemannian
metric gives a reduction of the structure groups and a commutative
diagram
Pspinffl-----> Pft-PO
Pso(*)

Now one could hope for the existence of a finite dimensional represen-
tation o-:GL„+([R) -> GL(F) whose restriction to Spin„ is an irreducible
132	II. SPIN GEOMETRY AND DIRAC OPERATORS
spinor representation. The associated vector bundle
S = P^(X) x, V
would then be the canonical spinor bundle, and choosing a metric on X
would induce a metric on S (as in the case of the tensor bundles.). This
would give us a fixed vector bundle on which we could consider a family
of Atiyah-Singer operators associated to each variation of the riemannian
structure on the base. Unfortunately, this hope cannot be fulfilled.
Lemma 5.23. The Lie group GL„+ (R) (n > 2) has no finite dimensional
representations other than those which descend to GL„+ (R).
Proof. Considerate subgroup SL„(R) <= GL +(R) and its 2-fold, universal
covering group SL„(R) c GL„+ (R). Since SL„(R) contains the kernel of the
homomorphism GL„+ (R) -► GL„+ (RJ^it will clearly suffice to prove the
assertion for this subgroup. Let q>: SL„(R) GLjy(R) be any Lie group
homomorphism. Let : $I„(R) -► gIN(R) denote the associated Lie algebra
homomorphism, and consider its complexification ® C:$I„(C)
qlN(C). Since SL„(C) is simply-connected, the elementary theory of Lie
groups tells us that ® C is induced by a homomorphism of Lie groups
4>:SL„(C) -> GLn(C). It follows that
Ф = ^|sLn(K)
in a neighborhood of the identity. By the uniqueness of analytic continua-
tion this identity holds everywhere, and so the representation descends
to SL„(R) as claimed. 
Note that the argument just given does not apply to the conformal
group C„ = {g g GLn(R):g = Ag0 for Я e R+ and g0 e SO„}. Indeed, con-
formal changes in the metric on the base manifold can be lifted to a fixed
spinor bundle, and one can study there the associated change in the
Atiyah-Singer operator. A basic and important fact is that the Atiyah-
Singer Dirac operator remains essentially invariant under all conformal
changes of the metric.
We now make this last statement precise. Fix a riemannian spin mani-
fold X with metric <•,•>, and consider the conformally related metric
<V>' = e2tt<V>
where и is some smooth function on X. Let X' denote this riemannian
manifold with metric <•,•>'. To each orthonormal tangent frame в =
{ei9 ...,еи} on X we can associate the orthonormal frame ^(<?) =
{e'i9... 9efn} on X'9 where e} = exp(—u)e7 for each j. This gives us an SO„-
equivariant map ф: Ps0(Ar) -► PsoPO which lifts to a Spin„-equivariant
§5. DIRAC OPERATORS
133
map
^:PSpin(X)—>PSpin(X')	(536)
between principal Spin„-bundles. (We have chosen the spin structure on
X' which is topologically equivalent to the one given on X.) For any
fixed spinor representation д: Spin„ -► SO(M) we have associated spinor
bundles 5 = PSPinW хд M and S' = PSpin(X') хд M, and the map (536)
gives us a bundle isometry
^:S------> S'.	(537)
We now modify this isometry by setting
(538)
The resulting map : S -► S' is a bundle isomorphism which is conformal
on each fibre. The basic result is the following:
Theorem 5.24. Let $:T(S) -► T(S) and 0':T(S') -► T(S') be the canonical
Atiyah-Singer Dirac operators defined over the conformally related riemann-
ian manifolds X and X' respectively. Then
0' = To^oxp-i	(5.39)
Corollary 5.25. Let $ and I/)' be as in 5.24. Then
dim(ker 0) = dim(ker 0').
In other words the dimension of the space of harmonic spinors remains con-
stant under pointwise conformal changes of the riemannian metric.
Note. Suppose we have a decomposition S = S + ®S" defined by a canon-
ical volume element as in the next chapter. Then Corollary 5.25 applies to
each of the operators and that is, dim(ker 0±) = dimfker^')*).
Proof of Theorem 5.24. We have two metrics <•,•> and <•/>' defined
on the same vector bundle T = TX = TX'. Let V and V' respectively de-
note the associated canonical riemannian connections. It is straight-
foward to verify that for vector fields V and W we have
VvW =	+ (Vu)W + (W-u)V - <F,jy>grad(u)
where the gradient is taken in the metric <•>•>. (Check the axioms.) Sup-
pose now that £ — {el9... 9e„} and = {e'l9... 9e'„} are local ortho-
normal frame fields for <•,• > and >' respectively, and let = (Ve^ej)
and co'jt = (V'e'i9e'j)' be the associated 1-forms. One easily finds that for
134
II. SPIN GEOMETRY AND DIRAC OPERATORS
any tangent vector V,
<4,(К) = «ЫП + (et • uKV,ej> - (e} 
The local tangent frame field S = {eu... ,e„} determines a local frame
field if = {ct15 ... ,<rN} for S. Similarly, S' = {e'1,... ,e'„} determines a
frame field if' = {a\,... ,cr'N} for S' where a'j = фц(о^ for each j. From
(4.35) we see that the induced connections on S and S' are related as
follows:
= i £ <о;((ИИе>'в
b’J	J
= 4 •Agfc + (e««X^ej> - (М<И’е.»е,е/7.1
vJ	J
= oa + i (grad(u)- V - V • grad(u))aa|.
Since grad(u)- V = — K-grad(u) — 2<grad(u), V), we conclude the
following.
Lemma 5.27. Let Vs and Vs denote the riemannian connections on S and
S' respectively. Then
Vs' = Ф* ° |Vs - i F grad(u) - (F- u)| о -
Corollary 5.28.
0' = Фц °	+ j (n - l)grad(u)| о Ф -1
We now observe that for any constant, a,
|>(e“utr) = e“u(0a + a £ (e;u)e/r)
\ j J
= е*“(|>ст + agrad(u) • a),
and therefore
И- 1	/ и - 1	\
'Po^o'p-1 = e 2 |Дво^о1е2 ф ~1 j
= Фц 0 (& + | (n - l)grad(u) j о ф;1 = J>'. 
§6. FUNDAMENTAL ELLIPTIC OPERATORS	135
§6. The Fundamental Elliptic Operators
In this section we shall use the Clifford bundle and its modules to sys-
tematically derive the fundamental elliptic operators in riemannian geom-
etry, that is, the Euler characteristic operator, the signature operator,
and the Atiyah-Singer Л-operator.
The basic construction is the following. Let S be a Dirac bundle with
Dirac operator D over a riemannian manifold X, and suppose that S is
Z2-graded. This means that there is a parallel decomposition
S = S°®S1	(6.1)
so that CP(X) • SJ £ for all ij e Z2. From the definition (5.1) of the
Dirac operator it is clear that D is of the form
where
D°: T(S°)----► ns1) and D1: ЦХ1)-------------► T(S°).	(6.3)
Since D is self-adjoint, we see that D° and D1 are adjoints of one another.
The ellipticity of D established in §5 implies that each operator Dk is
also elliptic. In fact, the principal symbol of D* at a cotangent vector £
is simply Clifford multiplication by that is,
a^Dk) = %•: Sk----► S*+1 for к e Z2	(6.4)
(see Lemma 5.1). Since £•<!;= —1|£||2, we see that is an isomorphism
for £ 0. It is a fact that over a compact manifold, the kernel and cokemel
of an elliptic operator P are of finite dimension, and a basic invariant of
P is its index which is defined as
ind P = dim(ker P) — dim(coker P).	(6.5)
Since D1 is the adjoint of D°, there is an isomorphism ker(Dx) s coker(D°),
and so we have that
ind D° = dim(ker D°) — dim(ker D1).	(6.6)
From (6.2) it is clear that
kerD = ker D° © kerD1.	(6.7)
In particular, if D is injective, then ind D° = 0.
Whenever X is oriented and even-dimensional, we recall (cf. (3.1 l)ff) that
there is an important method for introducing a Z2-grading on any Dirac
bundle S. Let coc be the complex volume element, given in terms of a posi-
tively oriented orthonormal tangent frame (el9... ,e2m) by
wc = ime1 • • • e2m	(6.8)
136	II. SPIN GEOMETRY AND DIRAC OPERATORS
where 2m = dim(X). This is a globally defined section of C£(X) =
C£(X) ® C with the following properties.
Vcoc = 0,	(6.9)
= 1,	(6.Ю)
coce = — ecoc for any	ее TX,	(6.11)
Property (6.9) follows easily from the derivation property (4.26) of V. (Fix
x e X and choose e/s with (Ve7)x = 0.). For the rest, see Proposition 3.3
of Chapter I.
Suppose now that S is a Dirac bundle over X (complex if m is odd).
Then 5 has a decomposition
S = S+®S~	(6.12)
into the -ь 1 and — 1 eigenbundles for multiplication by coc, These bundles
can be simply expressed as
S1 =(1 ±coc)S.	(6.13)
From (6.9) we see that this decomposition is parallel, and from (6.11) we
see that for any e e TX,
e-S±	(6.14)
This means that after identifying S° with S+ and S1 with S“, the decom-
position (6.12) gives a Z2-grading on S.
We now examine some important examples of this construction.
Example 6.	1 (the Euler characteristic operator). Let X be a compact
riemannian manifold and consider the basic case where
S = C£(X) = C€°(X) © C€x(X)
(cf. (3.2)). By Theorem 5.7 we see that under the canonical isomorphism
C£(X) A*(X), the operator П0:Г(С£°(Х))	Г(С€х(Х)) corresponds to
the operator
d + d*:F(Aeven(X))-----► ЦЛ^^Х)).
Consequently, we have
ind D° = dim Heven - dim Hodd
= the Euler characteristic of X,
Example 6.	2 (the signature operator). Let X be a compact, oriented
riemannian manifold of dimension 4/c and consider again the basic case
5 = C£(X) = C£+(X) © C£'(X)
§6. FUNDAMENTAL ELLIPTIC OPERATORS
137
where the Z2-grading is now given as above by the complex volume ele-
ment coc = (— l)*co (see (6.12) and also (5.26)ff). There is a corresponding
decomposition
ker D = ker D+ © ker D~,
Since coc is parallel, it preserves ker D and the subspaces ker D± are just
the ± 1 eigenspaces under multiplication by coc on ker D. That is,
ker D± = (1 ± coc)ker D.
Now, under the canonical isomorphism C£(X) s A*(X) we know that
ker D H = H° © • • • © H4*, the space of harmonic forms (see Corollary
5.9). Furthermore, under this isomorphism, left multiplication by coc cor-
responds to the Hodge ^-operator, that is, for (p g Ap(X),
k + p(p~ i)
®сФ = (-1)	2	(6.15)
(see (5.35)). Consequently, for each p = 0,... ,2k we have an isomorphism
coc:H'H4*-'.
This, in turn, implies that the space H(p) = H₽ ф H4*_₽, for p < 2k, has
a decomposition
H(p) = H+(p) ф H’(p)
where the subspaces H*(p) = (1 ± coc)H(p) are of the same dimension. Since
ker D* = H1 = H*(0) ф • • • ф H±(2k - 1) ф (H2*)*, where (H2*)* »
(1 ± coc)H2*, we conclude that
ind D+ = dim(H2*)+ — dim(H2*)-
= sigW-
For this last statement we recall that the signature of X, denoted sig(X),
is defined to be the signature of the quadratic form
g(<p,t/0 = (p л ф s ([<p] и МХИ)
on H2* s H2k(X;R). Since * = wc in dimension 2k and since
£ <P л *<P = ||ф||2,
we see that this signature is just the difference of the dimensions of the
4-1 and — 1 eigenspaces of ♦ on H2*.
Example 6.3 (the Atiyah-Singer A-operator). Let X be a compact rie-
mannian spin manifold of dimension 4k and consider the complex spinor
bundle $c with Dirac operator 0. We split	© $c under the
138
II. SPIN GEOMETRY AND DIRAC OPERATORS
complex volume element as above. Then it is a consequence of the Atiyah-
Singer Index Theorem in Chapter III that
ind(0+) = Л(Х)
where A(X) is a rational Pontryagin number of X called the Л-genus. We
shall examine this invariant in detail in §11 of Chapter III; however, some
discussion of it is in order here.
The Л-genus has an important multiplicative property like that of the sig-
nature. If X and Y are compact oriented manifolds. Then
A(X x У) = Л(Х) x Л(У).	(6.16)
The Л-genus is, in general, not an integer. For example, for a compact
4-manifold X9 it is a fact that
-l(X) = |sig(X) = lP1W	(6-17)
where pr(X) is the first Pontryagin number of X. In particular, for the
complex projective plane P2(C), we have H2(IP2(C)) Z and so the sig-
nature is 1. It follows that Л(1Р2(С))	—1/8.
The index of an elliptic operator is, of course, an integer. Hence, we
conclude the following result (cf. Atiyah-Hirzebruch [1]).
The A-genus of a compact spin manifold is an integer. (6.18)
Note also that from (6.17) we retrieve the basic fact that the signature of
a spin 4-manifold must be a multiple of 8 (see Corollary 2.12 and the fol-
lowing discussion).
An important set of spin manifolds with non-zero Л-genus is provided
by the hypersurfaces V2n(d) of complex projective space P2" + 1(C) (see
Example 2.7). Recall that the manifold V2n(d) is spin if and only if the
degree d is even. It is a fact that (cf. Lawson-Michelsohn [1])
-	2~2nd n
Л(И2"(<П) = Й7ТТЙ П (d2 - W).	(6.19)
Thus, each of the spin manifolds V2n(2d), for d > n, has non-zero Л-
genus. Taking products of these gives further examples by (6.16).
Each of the fundamental examples above gives rise to a family of as-
sociated operators by the process of taking “coefficients in a bundle.” This
works as follows. Let S = S° © S1 be a Z2-graded Dirac bundle as before,
and let E be any riemannian bundle with connection over X. Then the
bundle
S ® E = (S° ® E) ® (S1 ® E)
is again a Z2-graded Dirac bundle over X (see Proposition 5.10).
§7. C£rLINEAR OPERATORS
139
Example 6.4	. Let S = C£(X) = C£+(X) © C£"(X) be as in Example
6.2. Then for any bundle E over X we can construct the twisted signature
operator :Г(С£+(Х) ® E) -► T(Ct“(X) ® E) whose index is
sig(X;E) = {ch2E • L(X)}[X].
This formula is explained in detail in Chapter III.
Example 6.5	. Let $c =	be the complex spinor bundle of Ex-
ample 6.3. Then for any bundle E over X we can construct the twisted
Atiyah-Singer operator : П^с ® E) -> Г(5ё ® E) whose index is
A(X;E) = {ch E • A(X)}[X].
Again see Chapter III for details.
§7. Ctfc-Linear Dirac Operators
There is a variation of the constructions given above, due to Atiyah and
Singer, which has proved to be very important in riemannian geometry.
To introduce the concept we return to a point mentioned earlier (in
Example 3.7). Let PSpin(20 be the principal Spin„-bundle of an n-di-
mensional spin manifold X. Then from the representation : Spin„ -►
Hom(CtR,Ct„) given by left multiplication, we have the associated vector
bundle
$(X) = PSpin x,C£„.	(7.1)
Since right multiplication commutes with we see that there is a right
action of the algebra Ct„ on the bundle $(X) which preserves the fibres.
This action makes $(X) a bundle of rank-1 Ct„-modules.
The idea now is to construct elliptic operators and an appropriate index
theory which take into account this action of Ct„.
We shall begin with the construction of such an operator in the basic
case of Ф(Х). Note first that the action of Ctn on $(X) clearly commutes
with Clifford multiplication by elements of Ct(X).
Since $(X) is associated to PSpin(X), it carries the canonical riemannian
connection, and as such it is clearly a Dirac bundle over X. In fact, as a
vector bundle Ф(Х) is simply a direct sum of irreducible (real) spinor bundles
of X. (This comes from the decomposition of Ctn into irreducible modules
under left-multiplication.)
The right action of Ctn on $(X) is parallel in the riemannian connec-
tion, i.e., for any section о g Г(Ф(Х)) and any element <p g Ct„, we have
V(a • <p) = (Ver) • <p. (This is evident since the holonomy in Ф(Х) is left
multiplication by elements of Spin„. It can also be seen directly from the
methods of §4.)
140	II. SPIN GEOMETRY AND DIRAC OPERATORS
From the definition (7.1) we see that the decomposition C£„ = C£° ф
C£„ gives rise to a parallel decomposition
$(X) = $°(X) ® SV)	(7.2)
which is not only a Z2-grading over the bundle Ct(X) but also over the
free Ct„-action. That is, we have
SU)-C£isSi+V)	(7.3)
for all i, j g Z2.
Since S(X) is a Dirac bundle it carries a canonical Dirac operator
which, with respect to the decomposition (7.2) is of the form
Furthermore, this operator commutes with the action of Ctn. To see this,
note that Ф(а<р) = £ eyVej(a<p) = £ (e7Veja)(p = Ф(а)ф, since multiplica-
tion by <p is parallel and commutes with multiplication by elements from
Ct(X). This operator $ is called the Ct„-linear Atiyah-Singer operator of
X.
We now proceed as above and consider the restricted operator
t)°:T(S°(X))------------------------->W(X)).
From (7.3) and the paragraph above we conclude the following:
Lemma 7Л. The operator ©° is a real, elliptic first-order operator which
commutes with the action of Ct J s Ctn_x on <S(X) = SV) ф SV)«
This construction is sufficiently important that we shall axiomatize it.
Definition 7.2. By a Ctfc-Dirac bundle over a riemannian manifold X
we mean a real Dirac bundle S over X, together with a right action
Ctk Aut(S) which is parallel and commutes with multiplication by
elements of Ct(X).
This can be thought of as a Dirac bundle which carries “scalar multi-
plication” by Ctk. Notice that a Ctr- or Cl2-Dirac bundle is just a com-
plex or quaternionic Dirac bundle respectively.
Definition 7.3. A Ctk-Dirac bundle S is said to be Z2-graded if it
carries a Z2-grading 8 = 8° ф S1 as a Dirac bundle, which is simulta-
neously a Z2-grading for the Ctk-action (that is, (7.3) is satisfied).
Any C£k-Dirac bundle S has a canonically associated Dirac operator
© which commutes with the C£k-action. If S is Z2-graded, then © de-
§7. COLINEAR OPERATORS
141
composes as in (7.4), and we get an elliptic operator
Ф°:Г(®°)----►HS1)	(7.5)
which commutes with the action of Ct° £ Ctk-i.
When X is compact, we can directly define an analytic index for such
operators as follows. Since ©° commutes with Ct° s the kernel
of ©° is a finite dimensional C£k_x-module, and thereby ker©0 deter-
mines an element in the Grothendieck group ®lk-i of such modules.
Consider now the residue class of this element in _ 1/1*9ЭТк, where i* is
induced by the homomorphism i*:C£k_i -► Ctk determined by the in-
clusion map i: Rk“1 Rk. Recall (from Chap. I, §9) that these quotient
groups are naturally isomorphic to the KO-groups of a point.
Definition 7.4. Let S = S° ® S1 be a Z2-graded C£k-Dirac bundle
over a compact manifold. Then the analytic index of the Dirac operator
©°:F(S°)	HS1), denoted by indk(©°), is the residue class
[ker ©°] g ®lk_i/i*®?, KO’k(pt).
We recall that the groups KO *(pt) are the same as the stable homotopy
groups of the orthogonal group, that is,
fZ
KO"k(pt) = ^Z2
к = 0 (mod 4)
к = 1 or 2 (mod 8)
otherwise.
(7.6)
0
к
A standard argument shows that this index is constant under deforma-
tions of the operator. One of the deepest aspects of the work of Atiyah
and Singer is the computation of this index topologically (see III. 16). The
applications of this result are among the most far-reaching in all of dif-
ferential geometry. For this reason we devote the remainder of this section
to a detailed examination of such operators particularly in the cases of
fundamental interest in geometry.
We begin with a remark which will be useful when studying the multi-
plicative properties of the operator. Recall from Chapter I (Proposition
5.20) that there is a natural equivalence between the category of (ungraded)
modules over С1к-г and the category of Z2-graded modules over Ctk.
This equivalence induces a natural isomorphism
(7.7)
where as before ®lk denotes the Grothendieck group of finite dimensional
Z2-graded R-modules over Ctk. The map from the graded to the ungraded
case is given by taking the even part.
Given now a Dirac operator ©: T(S) -► T(S) as above, we see that
ker(©) is a Z2-graded Ctk-module. The even part of ker(©) is exactly
142
II. SPIN GEOMETRY AND DIRAC OPERATORS
кег(Ф°). This gives the following:
Alternative Definition 7.4. Let S = S° © S1 be a Z2-graded Ctk-
Dirac bundle over a compact manifold. The analytic index of the Dirac
operator Ф of S is the residue class
[ker Ф] e + 1	KO"*(pt).
Via the isomorphism (7.7) this index coincides with the index [ker Ф0]
given in Definition 7.4.
This second definition is actually more natural and is important for
understanding the multiplicative properties of the index. With this second
definition we see that the Clifford index generalizes the classical one in
that ind0( ) = ind( ) = dimR(ker ) — dimR(coker). To see this, note first
that Ct0 = R and = C. A Z2-graded Ct0-module is just a pair of real
vector spaces V° © K1. Now [V © 0] = — [0 ф V] in	because
V ф V V ® C extends to be a graded Ctrmodule. Consequently,
ind0(D) = [ker ®° © ker®1]^ [ker ®° © 0]- [ker Ф1 © 0] as claimed.
Remark 7.5. All of these constructions could be carried out in the
complex category. One could consider complex Ctk-Dirac bundles, etc.
Here the index will be valued in

if к is even
if к is odd.
Unfortunately, this leads to nothing essentially new, so we have con-
centrated our attention on the real case.
Some examples are in order. The first and most illuminating one comes
by taking the “Ctk-ification” of an ordinary elliptic operator.
Example 7.6	. Let S = S° © S1 be an ordinary real Z2-graded Dirac
bundle over a compact manifold X, and let
D =
0 J
be its Dirac operator. We now consider an irreducible real Z2-graded
module V = V° © K1 over the Clifford algebra Ctk, and take the tensor
product
S = S® V
where V is here considered as the trivialized bundle V x X X. This
bundle is, in a natural way, a Z2-graded Ctk-Dirac bundle. The grading
S = S° © S1 is given by
S° = (S° ® K°) © (S1 ® K1) and S1 = (S° ® V1) © (S1 ® K°).
§7. C£r LINEAR OPERATORS
143
Of course, multiplication by Clk takes place on the К-factor. The asso-
ciated Dirac operator Ф on S is simply the extension of D, i.e.,
S = D®IdK.
Consequently, we have that
ker Ф0 = ((ker D°) ® K°) © ((ker D1) ® K1),
and therefore
{ker Ф0} = bQ{V°} +bi{Vi}eWlk_i
where th = dimR(ker 1У). In passing to the quotient	we see
that [K°] = — [K1] since V° © K1 is a Ctk-module. (Caution: one must
show that K° © K1 can be made a Ctk-module in such a way that V°
and V1 are invariant subspaces under the subalgebra Ctk_x c= Ctk. This
requires a case by case check modulo 8.). Hence,
indk(S)°) = (b° - bl)[VQ] = ind(D°)[K°] g KO"*(pt),
and since [Ko] generates this group for each к (see 1.9) we conclude that
find D°
indkD° = < ind D° (mod 2)
0
if к = 0 (mod 4)
if к = 1 or 2 (mod 8)
otherwise.
(7.8)
This formula shows that, as one would expect, nothing essentially new
can be found by this trivial construction. The interesting examples are
those where the Ctk-structure is more intrinsic to the geometry. This is
the case in the following:
Example 7.7	(the Kervaire Semicharacteristic). Let X be a compact
oriented (riemannian) manifold, and take S to be the Clifford bundle
Ct(X) = Ct°(X) ф С1'(Х) (considered as a Z2-graded Dirac bundle as
in Example 6.1). If X has dimension 4^ 4- 1, then S is naturally a Ctr
Dirac bundle as follows. Consider the oriented volume form co =
’ еы+1 °f This form is parallel and satisfies co2 = — 1. Hence, right
multiplication by co in Ct(X) makes Ct(X) a Ct i-Dirac bundle. Since co
is of odd degree, right multiplication by co interchanges Ct°(X) and
Ct1 (A"). Hence, Ct(X) is Z2-graded as a Ct x-Dirac bundle. Hence, the
operator ©° — D° (d + d*)|Aeven has an index in KO~ \pt) £ Z2. To
compute this index we want to find the residue class of ker Ф0 if 3M0/ix.
Since Ctx s C and Ct0 s R, we easily identify ®l0 as the Grothendieck
group of equivalence classes of real finite dimensional vector spaces, and
SWi as the complex analogue. Hence i*®lx is given by spaces of even
144	II. SPIN GEOMETRY AND DIRAC OPERATORS
dimension, and we clearly have
indjC®0) = dimR(ker D°) (mod 2)
= dimR(Heven) (mod 2)
21
= Y b2j(X) (mod 2)
j=o
where bJX) denotes the ith Betti number of X. In other words, ind^©0)
is exactly the Kervaire semicharacteristic of X, a basic cobordism invariant
of an oriented (4^ 4- l)-manifold.
Note that in order that the volume element co have square — 1 and be
of odd degree, one is restricted to dimensions 4^ 4- 1.
It should be pointed out that an exactly analogous construction fails
for the signature complex (Example 6.2), since the subbundles Cl±{X) =
(1 ± coc)Ct(X) are each invariant under right multiplication by any
Clifford element. Hence, this is never a grading for right multiplication.
We now pass to an example which illustrates this construction.
The Fundamental Case (the Atiyah-Milnor-Singer Invariant). Let X
be a compact spin manifold of dimension n, and let Ф(Х) = Ф°(Х) Ф
^(X) be the Z2-graded Ct„-Dirac bundle given by (7.1). It is a conse-
quence of the Atiyah-Singer theorem that ind„ = ind„($°) is a spin cobor-
dism invariant of X, and in fact gives a graded ring homomorphism
QSpin KQ-*(pt).	(7.9)
This homomorphism coincides with the one defined homotopy-theoreti-
cally in (3.22). We shall return to this in Chapter III.
Because of their fundamental nature, it is useful to examine these Ct„-
spinor bundles $(X) in some detail. We begin with the simplest case where
n = 8/c.
Consider an irreducible real left module VSk for the Clifford algebra Ct8k.
Let V3k denote the right Ct8k-module obtained from VSk by simply
multiplying by the transposed element, i.e., by setting v • cp = <pr • v for
Ф g Ct8k and v g K8k. Then there is an isomorphism of bimodules
C£8k = VSk ® V3k.	(7.10)
To see this we recall from Proposition 5.18 of Chapter I, that in all even
dimensions we have the complex bimodule isomorphism
(7.11)
where V2k denotes the irreducible complex Ct2k-module and where
Ct2k = C£2k ® R C. The assertion (7.10) follows from (7.11) since in dimen-
sions 8/c the complex case is simply the complexification of the real one.
§7. CO LINEAR OPERATORS	145
We conclude from (7.1) that in these dimensions
ФР0 = J’spin x c Ct8k = PSpin x д8к® i (Vsk ® VSk) (7.12)
= (J’spin ХД8к ^8k) ® ^8k
= £(X) ® V*k
where $(X) denotes the real spinor bundle of X, and where V8k now be-
comes the constant (trivialized) VSk bundle.
Recall now that VSk has two inequivalent Z2-gradings, obtained one
from the other by interchanging the factors. Either one gives the same
Z2-grading on the tensor product V8k ® VSk9 and the bimodule isomor-
phism (7.10) is a Z2-graded one.
We are now exactly in the situation of Example 7.6. That is, Ф =
0 ® IdK, where 0 is the Atiyah-Singer operator on spinors. We can there-
fore read off from (7.8) that
ind8k($°) = ind(0°) = A(X).	(7.13)
The situation in dimensions 8k + 4 is very much similar. The principal
difference is that in these dimensions the irreducible real module K8k+4
for Ct8k+4 is also the irreducible complex module; that is,
^+4 =	(7-14)
(Recall that C£8k+4 is a quaternion, and hence also a complex, matrix
algebra.) From (7.11) we know that K8)k+4 ®c V^k+4. s C£8t+4 s
C£8t+4 ®r C, which yields the real bundle isomorphism
2C(m+45 ^*+4®c^*+4.	(7.15)
This implies, as above, that there is an isomorphism of real Z2-graded
C£8t+4-bundles
2^(X) Jfc(X) ®c P£k+4	(7.16)
where $cW is the complex spinor bundle of X and K8k+4 is the tri-
vialized bundle. Hence, we have that 2ф = 0C ®c Id^c. Using (7.14) and
the arguments above, it is not difficult to see that
ind8k+4($°) == f Л(Х).	(7.17)
Notice that implicit in equation (7.17) is the fact that in these dimensions
the Л-genus of a spin manifold is an even integer. A direct proof of this can
be given by observing that the Atiyah-Singer operator in these dimensions
is quaternion linear, and so the complex dimensions of its kernel and
cokernel are even. This fact, in dimension four, is just Rochlin’s Theorem
2.13.
146	И. SPIN GEOMETRY AND DIRAC OPERATORS
We now examine this index in the interesting cases where it is an integer
modulo 2. We begin with dimension 8k 4- 1 and the observation that from
the classification of Clifford algebras we have
C£8k + 1 C£8k ® C C£8k ® i C£8k.	(7.18)
In fact, the element i in this algebra corresponds exactly to the volume
form о =	• • • e8k + 1, which is clearly central and has co2 = — 1. Thus,
the decomposition in (7.18) gives the Z2-grading on Ct8k+1. In particular,
we have
C^ek + i = Ct8k	and Ct8k + 1 = о Ct8k+1 = i Ct8k. (7.19)
These isomorphisms show that
^8к + 1 = ^8к ® C = VSk ® iK8k	(7
И1к+1 = ^8к аП<1 ^8k+l = ^HJk + l = ^Sk-
Combining this with (7.11), we find that
Cf 8k + 1 = ^^8k = ^8k ®C
= (И8к® C) ®c(K8k® C)
= №+1 ® C) ®c(Kgk+1 ®C)
= (H!k+1 ® C) ®K i^8k+l
= ^8к + 1 ®K ^8k + l-
This implies as above that
®(X)^^(X)®RPgk + 1	(7.22)
and that Ф s 0 ® Idp0. It follows immediately that ker($°) =
H° ® И8к+1 where H° = ker($°). Thus we have that
ind8k + ЛФ0) = dimRH° (mod 2).	(7.23)
This index can be reinterpreted in more elementary terms as follows.
Observe that from the discussion above, we have in these dimensions
that the spinor bundle is the complexification of a real bundle, i.e.,
$(X) = $°(X) ® co $°(X). We can construct from the Dirac operator
0°: W)) - Г(№))> an operator
P: Г(J°(X))--> r(J°(X))	(7.24)
by setting
P = co • 0°.	(7.25)
This operator is elliptic and skew-adjoint. To prove the skew adjointness,
we note that since co is parallel, commutes with 0 and satisfies co2 = — 1,
§7. C£rLINEAR OPERATORS	147
ve have (Ра,т) = (а)0°а9т) = (со2^>°а,сот) = — ($°сг,сот) = —(а90°а)т) =
- (а, со^>°т) = — (a, Pt).
It is an elementary fact that the parity of a real skew-adjoint Fredholm
jperator is conserved under deformations (cf. III. 10). Thus for a skew-
idjoint elliptic operator P on a compact manifold, the mod-2 index
indZ2(P) = dim(ker P) (mod 2)	(7.26)
is well defined. The result (7.23) can be reexpressed as
ind8t + 1($>°)= indZ2(P)	(7.27)
here P is given by (7.25).
In the final case of dimension 8/c + 2 there is a strongly analogous
situation. Here we have from the classification of Clifford algebras that
Ct8k + 2 C£8k ® H	and	Ct8k + 2	C£8k ® C,	(7.28)
F8k + 2 = V6k ® H	and	Kgk + 2	V8k ® C.	(7.29)
Using (7.11) one can deduce from here that
Ctek + 2 = ^8к + 2	F8k + 2
= F8k + 2 ®c K8k + 2-
It follows as before that
$(X) #X) ®c Kgk + 2	(7.31)
with respect to which ф 0 ® Idy. Consequently, ker (Ф0) s
ker(f>°) ®c P8k + 2, and so
ind8k + 2(ф°) = dimjker 0°) (mod 2).	(7.32)
Now since the representation F8k + 2 is H-linear, the bundle $(X) carries
a parallel quaternion structure, i.e., there are parallel endomorphisms /,
J, К of $(X) which satisfy the standard quaternion relations: I2 = J2 =
K2 = -1JJ + JI = IK + KI = JK + KJ = 0. The Z2-grading on $(X)
can be written in terms of these as
$(X) = $°(X) © J($°(X)).	(133)
We can then define a skew-hermitian operator
P = J о 0°
on the bundle $%¥), and as before we have
ind8k+2($°) = indZ2(P)	(7.34)
where ind2z(P) = dimc(ker P) (mod 2).
148	II. SPIN GEOMETRY AND DIRAC OPERATORS
We shall now examine this mod 2 index of the Atiyah-Singer operator
in low dimensions. The computations here are of some importance since,
as we shall see, the map ind* :Q*pin -► KO ”*(pt) is a ring homomor-
phism, and the non-zero element g KO-1(pt) has the property that: rjx
and rfx are not zero whenever x is an odd multiple of the generator in
degree 8k.
Example 7.8 (the circle as a spin manifold). Consider the circle S1 with
a riemannian metric. The oriented orthonormal frame bundle is canon-
ically diffeomorphic to the circle itself, i.e., P^S1) = S1, since there is
exactly one oriented unit tangent vector at each point. A spin structure
on S1 is a 2-fold covering Psp^S1) -► PsoC^1) of the circle. As we noted
in Chapter I, there are two such coverings, one connected and one with
two components; and it is the non-connected covering S1 x Z2 -> S1
which is not spin-cobordant to zero. We shall refer to this as the interesting
spin structure on S1.
Notice that since = C we have that
^(S1) = S1 x C
where the product structure gives the connection. Of course C£? = R
and the Z2-grading on C£x is the standard decomposition C = R © iR.
Thus, ®°(S1) = S1 x R and ^1(S1) = S1 x iR. Sections of $ are just
complex-valued functions f(s) on the circle, and the Dirac operator is
simply
$ =	(7-35)
as
where s is arc-length on S1.
The kernel of Ф°:Г(^°) -► ЦФ1) is the set of real-valued constant
functions on S1. The dimension of this space as a C£° = R module is one.
Hence, we have that for the interesting spin structure on S1,
ind^S1) / 0,	(7.36)
i.e., ind^S1) is the generator of KO-1(pt) Z2.
Exercise Show directly that indx = 0 for the “uninteresting” spin struc-
ture on S1. Show also that the connected sum of the intersecting spin
structure with itself is the uninteresting one.
Example 7.9 (the torus as a spin manifold). Let T = S1 x S1 be the flat
square torus. The oriented orthonormal frame bundle is canonically
trivialized
PSO(T) = T x S1.
§7. COLINEAR OPERATORS
149
The spin structure on T given by squaring the interesting spin structure
on the circle is the covering:
Pspin(n = т X S1	T X S1 = PSO(T).
Since C£2 = M and C£2 = C, we see as before that £(T) = T x H
and = T x C. The kernel of$° is just the complex-valued constant
functions, and we conclude that
ind2(S1 xS^O	(7.37)
where S1 carries the interesting spin structure.
Exercise. Compute ind2 for the remaining spin-structures on T.
Exercise. Prove directly that ind2(S2) = 0.
It is a good time to summarize what we have established so far. To
each riemannian spin manifold X, we have associated the canonical bundle
£>(X), given in (7.1). This is a Z2-graded C£„-Dirac bundle and its Dirac
operator has an index, which we denote indfc(X) in KO"fc(pt). We have
proved the following.
Theorem 7.10. Let X be a compact spin manifold of dimension n. When
n = 1 or 2 (mod 8), let H = ker(0) denote the space of real harmonic spinors,
that is, the kernel of the Atiyah-Singer operator on the irreducible real spinor
bundle of X. Then
dimcH
 A / V\ _
indJX)
Л(Х)
(mod 2)
(mod 2)
if n = 1
if n = 2
if n = 4
if n = 0
(mod 8)
(mod 8)
(mod 8)
(mod 8)
Furthermore, for the interesting spin structure on S1 and for its square on
Sl x S1, this invariant is non-zero.
We shall now investigate the multiplicative properties of ind*. To begin
we recall the ring structure on KO”*(pt) which was discussed in Chapter
I, §9. We set
(7.38)
rj = the generator of KO~ x(pt) Z2
у = the generator of K0“4(pt) Z
x = the generator of KO-8(pt) s T.
Then the multiplicative structure on KO"*(pt) is given by the following
(cf. 1.9):
KO *(pt) s Z[w,x]/<2>b г)3, ПУ, y2 - 4x>
(7.39)
150	II. SPIN GEOMETRY AND DIRAC OPERATORS
where the grading is fixed by the requirement that deg(rj) = 1, deg(y) = 4
and deg(x) = 8.
Our next result is that ind* is, in fact, a ring homomorphism. More
specifically, we mean the following. The homotopy invariance of the index
shows that ind„(X) is independent of the choice of riemannian metric on
X. (Any two metrics are smoothly homotopic.) We consider then the free
abelian group Mjpin generated by spin-structure preserving diffeomor-
phism classes of connected spin manifolds. The sum M^,pin = M^pin
is naturally a graded ring under the direct product of manifolds.
Theorem 7.11. The mapping
ind*: M*pin----► KO ~ *(pt)
is a surjective graded-ring homomorphism.
Proof. The map ind* is additive by definition. To prove that it is
multiplicative, we use the Alternative Definition 7.4 of the index. Let X ,
and X2 be compact riemannian spin manifolds of dimensions пг and n2
respectively. Let : Г(фк) -» Г(фк) be the Atiyah-Singer operator for
the Cf„k-Dirac bundle = $(A\) defined in (7.1), for £ = 1,2. Let $>:
T((jB) -> Г(ф) be the corresponding object for X = XV x X2 with the
product riemannian and spin structure. For this structure we have
^SpinC^l X % 2)	PSpinC^l) x Z.2
where Z2 acts by (-1,-1) on the product. From here one sees easily,
using (7.1), that
$ =	® $2	(7.40)
the exterior Z2-graded tensor product. This is a C£ni +(tI = (C£B1 ® C£n,)-
Dirac bundle, and one can straightforwardly verify that with respect to
(7.40), the Atiyah-Singer operator of $ can be written as
Ф = $>i ® Id2 + eq ® $2
where aj :C£„, -> Cl„t is the automorphism extending —Id on Rni. Since
<*!$! + Ф1а1 = 0, it follows that
Ф2 =	® Id2 + Idt ® $22.
Each of the operators $)2, $)2 and $>2 is non-negative, self-adjoint and
elliptic over a compact manifold. It follows from standard spectral theory
that ker($2) = ker($)j) ® ker($2). Since ker($;) = ker($J), we conclude
that
кегф) = кег($>1) ® кег(Ф2)
(7.41)
§7. CO LINEAR OPERATORS
151
where the tensor product in (7.41) inherits the structure of a Z2-graded
tensor product of Z2-graded Clifford modules from the structure of (7.40).
This graded tensor product is exactly the multiplication in =
9W+/i*9W+ + 1. Thus, we have established that
indn,+n2($>) = indn,(?>i)ind„2(t)2)
and so ind* is a ring homomorphism.
To see that ind* is surjective we need only show that it maps onto the
set of multiplicative generators {^,y,x} given in (7.38). Example 7.8 shows
that the circle with interesting spin structure maps onto r]. The КЗ-surface
Y given in Example 2.14 is a compact spin 4-manifold whose Л-genus is 2.
Hence, ind4(T) = |Л(У) = у. Finally, to produce a spin 8-manifold X with
ind8(X) = x, we must find one such that A(X) = 1. For this one could take
X = Mq, the almost parallelizable 8-manifold of index 224 constructed
in Kervaire-Milnor [1]. Alternatively, one could take the 4-disk bundle
E over S4 with %(E) = 1 and p^E)2 = 900. Then dE = S7, and the mani-
fold X = E uS7 D8, obtained by attaching an 8-disk along this boundary,
is a closed spin 8-manifold with A(X) — 1 (see Milnor [7]). This completes
the proof of Theorem 7.11. 
Remark 7.12. To prove the index theorem for this fundamental case, it
remains only to prove that ind* is a spin-cobordism invariant. It will
then descend to a ring-homomorphism ind*:Q8pin -► KO”*(pt), which
can be seen to agree with the map (3.22) directly. (For the torsion part,
this requires some argument.)
It is worth noting that all of the above could be carried out with
coefficients in a vector bundle. Let X be a compact riemannian spin mani-
fold of dimension n, and let E be a real vector bundle over X with an
orthogonal connection. Then the bundle £(X) ® E is naturally a Z2-
graded C£„-Dirac bundle with a Dirac operator which we shall denote by
We shall denote
indF(X) = ind„(3M g KO'"(pt).	(7.42)
Now we also have the fundamental real spinor bundle $ over X and we
can take the tensor product $ ® E in the spirit of Example 6.5 above.
We denote the Dirac operator on $ ® E by I/)E. Then arguing exactly as we
did above proves the following.
Theorem 7.13. Let X be a compact spin manifold of dimension n, and let E
be a real vector bundle over X with an orthogonal connection. When n = 1
or 2 (mod 8), let HE = ker($E) denote the space of real harmonic E-valued
152
II. SPIN GEOMETRY AND DIRAC OPERATORS
spinors. Then
indE(X) =
dimcH£ (mod 2)
dimMHE (mod 2)
{{Ch E • A(X)}[X]
{ch E • A(X)}[X]
if n = 1
if n = 2
if n = 4
if n = 0
(mod 8)
(mod 8)
(mod 8)
(mod 8)
A topological computation of this index can be given as follows. Embed
Xn cz S" + 8k for к sufficiently large, and identify the normal bundle v of X
with a tubular neighborhood of the embedding. The spin structure on X
determines a unique spin structure on v (via Proposition 1.15 and the
uniqueness of the spin structure on S" + 8k). Let $+(v) and $ "(v) denote the
canonical real spinor bundles of v (whose dimension is 8fc). Lift ^±(v) to
the total space of v by the projection it: v -► X, and at each non-zero
vector nev consider the isomorphism nn:7t*$+(v)	given by
Clifford multiplication by n. Then the difference element
tv = [rc*$+(v),7i*$"(v);/z]
represents a class in the relative KO-group KO(v, v — X), where X c v
is the zero-section. This is the KO-theory Thom class of v.
Given a real bundle E over X, we can consider the class
tv(E) = Tv • [Я*Е] G KO(v, v - X).
Since v is embedded as a domain v cz S"+8k, we have an excision isomor-
phism j:KO(v9 v — X) = KO(S" + 8k, S" + 8k -^X). Composing this with the
natural map i^KO(Sn+Sk9Sn^k - X) - KO(Sn+Sk) and applying Bott
Periodicity p:KO(Sn + sk) s KO(Sn) = KO""(pt), we obtain a class
Л(Х) = pо io7(Tv(E)) e KO-"(pt)
One assertion of the Atiyah-Singer Ctk-Index Theorem, proved in III. 16,
is that
Theorem 7.14.
ind£(X) = j/£(X).
The “cobordism invariance” in this case asserts that this map ind = j/
determines a transformation
st*: Q8pin(BO)----► KO"*(pt)
where Q8pin(BO) denotes the spin-bordism of the space BO.
Remark 7.15. It is interesting to note that in dimensions 1 and 2 (mod 8),
the index can be changed even by twisting with a flat bundle. (This is not
§8. VANISHING THEOREMS AND APPLICATIONS	153
true of the usual index.) For example consider the non-trivial real flat line
bundle t over the circle (the Mobius band). Using the analysis of Example
7.8, one easily finds that ^(S1) ® s / © i/. Since the kernel of =
i(d/ds) consists of locally constant sections, we see that ker ($,) = {0} and
ind/S1) = 0.
Recall that ind^S1) 0.
This sensitivity to flat bundles is a reflection of the interesting fact
that these (mod 2)rinvariants are not local, i.e., they cannot be computed
by universal formulas involving only the local data of the operator, as
in the Gauss-Bonnet and Chern-Weil theorems. Another indication is
that the (mod 2)-invariants are not multiplicative under coverings (see
Atiyah-Singer [4]).
§8. Vanishing Theorems and Some Applications
Suppose X is a compact riemannian manifold and let D2 be the Dirac
laplacian on the bundle Ct(X). One of the major results in riemannian
geometry was the following discovery of S. Bochner. There exists a second,
naturally defined laplacian V*V on Ct(X). It is self-adjoint, non-negative
and has the same symbol as D2. The difference, D2 — V*V, which is nec-
essarily an operator of order gl, is, in fact, of order zero and can be
expressed in terms of the curvature tensor of X. Using harmonic theory,
Bochner was thereby able to conclude the vanishing of certain Betti num-
bers of X under appropriate positivity assumptions on the curvature
tensor.
Following Bochner’s original paper, arguments of this kind have ap-
peared repeatedly in real and complex geometry. They are known gene-
rically as “Bochner’s method.”
In this section we shall give a systematic derivation of Bochner-type
formulas on general Dirac bundles. This will include the classical formu-
las on C£(X), although even here the algebra is vastly simplified by using
Clifford multiplication. It will also include the Lichnerowicz formula for
the Dirac laplacian on spinor bundles, and generalizations of this to spinors
with coefficients in an arbitrary vector bundle. This latter result is very
useful in the study of manifolds of positive scalar curvature (cf. Gromov-
Lawson [1], [2], [3]). Everything will follow from a single elegant formula
which applies to any Dirac bundle.
We begin with the definition of the operator V*V. Let E be any
riemannian vector bundle over X, and assume E has a riemannian con-
nection with covariant derivative V. Then to any pair of tangent vector
154	II. SPIN GEOMETRY AND DIRAC OPERATORS
fields V and W on X, we associate an invariant second derivative V^,w:
Г(Е) Г(Е) by setting
(8.1)
(Here, VKW is the riemannian covariant derivative on X.) At any point
x g X, the operator depends only on the values Vx and WX9 i.e., it
is tensorial in these variables. This is evidently true for Vx since it is a
general property of the covariant derivative. That it is true for Wx follows
from the identity'
^V,W ~ ^W,V = &V,W	(8-2)
where R is the curvature tensor of E. Equation (8.2) is an immediate con-
sequence of the fact that VKW — VWV = [K,W].
Now given a smooth section (p of E, we see that V?.cp is a section of
T* ® T* ® E\ that is, at each point it defines a bilinear form on the tan-
gent space with values in E. The connection laplacian
V*V:T(E)------► Г(Е)
is defined by taking the trace, i.e.,
V*Vcp = — trace(V?.<p).	(8.3)
In terms of a local orthonormal tangent frame field (e19... ,en) on X,
v*v<p = - £ v^ej(p.
J= 1
It is easy to see that the symbol of V*V at a cotangent vector £ is
fff(V*V)=№	(8-4)
and so V*V is elliptic. We shall now show that it is also symmetric and
^0.
Recall that the inner product (,•) on Г(Е) is defined by integration:
(ф, ф) =	<ф, ф~). Similarly, we define
(?Ф,^) = £<Уф,^>	(8.5)
where <7ф, Vi//> is defined in terms of local orthonormal tangent frames
(ei>... ,e„) by the expression <7ф, V«A> = X; <Ve/p, Ve^>.
Proposition 8.1. The operator V*V:T(E) Г(Е) is non-negative and for-
mally self-adjoint. In particular,
(V*V<p,^) = (V<p,ViA)	(8.6)
for all (р,ф g Г(Е) provided that one of (p or ф has compact support.
§8. VANISHING THEOREMS AND APPLICATIONS	155
IfX is compact, then = 0 if and only ifVcp = 0 i.e., if and only if
cp is globally parallel.
Proof. Fix x g X and choose a local orthonormal tangent frame field
(en ... ,en) with the property that (Ve7)x = 0 for all j. Then we have at the
point x that
<V*V<p, </*>=- Z <yejVej<p,	(8.7)
j
= -Z	~ <Vej<P,Ve>>}
= — div(V) + <V<p, Vi/*>
where V is the tangent vector field on X defined by the condition that
= (Уц,(р,ф) for all W e T(X). The last line of (8.7) is proved as
follows. At x,
div(K) = £<Ve/,e;>
j
= Т.е1<Уе^>-
Equation (8.6) now follows by integration of (8.7). The remainder of the
proposition is a consequence of formula (8.6). 
Using arguments similar to those given for Theorem 5.7, one can show
that on a complete riemannian manifold the operator V*V is essentially self -
adjoint, i.e., it has a unique self-adjoint closed extension on L2(S). Further-
more, the kernel ofV*V on L2(S) consists of the parallel sections of S, i.e.,
those which satisfy Va = 0. If X has infinite volume, then no such sections
exist except a = 0, since ||a|| is constant.
We suppose now that S' is any Dirac bundle over X, and we define a
canonical section 91 of Hom(S,S) by the formula
«(<p) f e/er^Jcp)	(8.8)
M=1
where (e1?... ,e„) is any orthonormal tangent frame at the point in ques-
tion, where Rv w is the curvature transformation of S, and where the dot
denotes Clifford multiplication.
Theorem 8.2 (the general Bochner Identity). Let D be the Dirac operator
and V*V the connection laplacian for any Dirac bundle S. Then
D2 = V*V + W
(8.9)
156	II. SPIN GEOMETRY AND DIRAC OPERATORS
Proof. Fix x e X and choose a local orthonormal frame field (elf... ,e„)
such that (Vcj)x = 0 for all j. Then using (8.2) we have at x that
~ X ej	ek
J,к
j,k
= — £ ^ej,ej + S ej ’ ek ’ (^ej,ek ~ ^ek,ej)
j	j<k
= v*v + «.	
Recall that the Ricci transformation of T(X) = Ax(X) is defined by the
formula
Ric(<p) = - £ Rej^ej)	(8.10)
j = i
where R is the curvature transformation of T(X). This determines a bilinear
form, called the Ricci curvature form, by setting
Ric(cp,^) = <Ric(cp),^>.	(8.11)
From the fundamental identity (4.44) for the Riemann curvature tensor,
the form Ric is seen to be symmetric.
A consequence of Theorem 8.2 applied to the bundle S = Ct(X) is the
following
Corollary 8.3. Let A be the Hodge laplacian and V*V the connection lap-
lacian of the tangent bundle T(X). Then
A = V*V + Ric .	(8.12)
Proof. Consider T(X) = A\X) c Ct(X). Since D2 = A it suffices to com-
pute the right hand side of (8.9) for vectors cp e A\X). Note that since
[L,V] = [L,A] = 0 (cf. Lemma 5.18 and Corollary 5.21), both of the
operators V*V and A preserve the subbundle Ax(X). Hence, so does the
operator 91. Therefore, using the identities (4.43) and (4.44), we have
W) = 2 X eieJRe,.ej((P)
ij
= i E eieJ<Ret.ej(<p),ek)ek
ij,k
§8. VANISHING THEOREMS AND APPLICATIONS	157
= 6 X <Яе,.е#к) + Rek.e^j) + Rej.e^il
i*j*k*i
+ 2 X	(<p), ej^Cj + |	(</>), e(>e(
i.i	ij
=	-Y<R<p.ej(e^eJ>e‘
ij	ij
= -X <W = -Y RejtV(ej)
ij	J
= Ric(cp). 
Note. As pointed out above, we know from (8.9) that ^(A^X)) c: Ax(X).
However, efijek g A3(X) if i j # к # i. Hence, the third line of the com-
putation above constitutes a proof of the first Bianchi identity (4.43). In
Theorem 5.16 we saw more sophisticated examples of curvature identities
that also followed from operator identities and Clifford multiplication.
Corollary 8.3 has the following important consequence:
Theorem 8.4 (Bochner). Let X be a compact riemannian manifold without
boundary. If Ric > 0, then the first Betti number bJX) is zero. The con-
clusion also holds if Ric 0 and >0 at one point.
Proof. Suppose bJX) = dim	> 0. Then there exists a non-zero
harmonic 1-form (p gH1 H^XjR) by Theorem 5.15. By Corollary 8.3
and Proposition 8.1 we have that
j*x Ric(p,p) = -(V*V<p, <p) = - ||V<p||2.	(8.13)
If Ric 0, we conclude that Vcp = 0, i.e., cp is parallel. In particular, ||ф||
is constant. Hence, if at some point we have Ric > 0, then J Ric(cp,(p) > 0
and we have a contradiction. 
Note that this argument also proves that when Ric 0, every harmonic
1-form is parallel. Under the metric correspondence T*X £ TX, the
parallel 1-forms become parallel vector fields. Thus we can conclude the
following
Theorem 8.5. Let X be a compact riemannian manifold of non-negative Ricci
curvature. Then br(X) equals the dimension of the space of parallel vector
fields on X. In particular,
MX) dim(X)
with equality if and only if X is a flat torus.
Proof. Let к = MX). Then by the argument above, there are к linearly
independent parallel vector fields on X. Parallel vector fields are linearly
158
II. SPIN GEOMETRY AND DIRAC OPERATORS
independent if and only if they are linearly independent at each point.
Hence, к dim(X), and к = dim(X) if and only if X has a globally parallel
framing. That X must then be a flat torus can be seen as follows. We may
choose parallel vector fields ... ,Ek which are pointwise orthonormal.
Since [£<,£;] = VEiEj — VEjEf = 0 for all ij, these vector fields generate
a locally free Reaction. Since к = dim(X), we see that X is an orbit of
this action, i.e., X Rk/A, where A is a lattice in Rk. The metric on X
clearly agrees with the usual one on Rk. 
Any parallel vector field generates an isometric flow, and its integral
curves are geodesics. Thus even when к = bt(X) < dim(X) (and Ric 0),
we get a locally free action of Rk by isometries on X with totally geodesic
orbits. We can also consider the dual basis .. 9cpk of parallel 1-forms.
Integration of ср = (<p15... ,<pk) gives a riemannian submersion J :X
Tk = Rk/A where A is the lattice in Rk generated by the “periods.” That
is, A = {2 g Rk: 2 = Jy cp for some closed curve у in X}. By construction,
the map J is a covering map on each orbit. With a little more work one
can show that the universal covering space X of X splits as a riemannian
product X = Rk x Xo where Xo is compact. The original manifold X is a
(possibly twisted) riemannian product of Tk with Xo.
Remark. The first statement in Theorem 8.4 was considerably improved
by Myers [1]. Much stronger results for non-compact complete manifolds
with Ric 0 were proved by Cheeger and Gromoll [2]. It is a deep
theorem of Gromov [1], that there exist explicit a priori bounds, depend-
ing only on dimension, for all the Betti numbers of a compact manifold
of non-negative sectional curvature. Such bounds do not exist under the
weaker hypothesis Ric 0 (except for degree 1) by examples of Sha and
Yang [1].
As one might guess, theorems similar to 8.4 can be proved for the higher
Betti numbers. For each p, there is a positivity assumption on the curvature
tensor which guarantees that bp(X) = 0. For detailed statements of these
results the reader is referred to Bochner and Yano [1] and to Goldberg
[1]. However, one of the more quotable results of this type can be proved
rather easily using the Clifford formalism. We present this now.
Recall that the curvature tensor ^з, F4> is antisymmetric in
(Vl9 V2) and in (V3, V4) and symmetric under the interchange of these pairs
(see (4.44)). Hence, R can be considered as a symmetric endomorphism
R: Л2(Х) -► A2(X). We call this transformation the curvature operator and
say that it is positive (or non-negative) if all of its eigenvalues are < 0
(or 0 respectively).
Theorem 8.6 (Gallot and Meyer [1]). Let X be a compact riemannian n-
manifold (without boundary) with the property that its curvature operator
§8. VANISHING THEOREMS AND APPLICATIONS	159
s positive at every point. Then all the Betti numbers, bp(X) are zero for
7 = 1,... ,n — 1, i.e., X is a real homology sphere. The same conclusion
holds if the curvature operator is ^0 and >0 at some point.
Proof. It will suffice to prove that positivity of the curvature operator
implies that
<9t(<p),<p> > 0 for all non-zero (p e A₽(X), p = 1,... ,n — 1. (8.14)
The argument then proceeds as in the proof of Theorem 8.4. From the
curvature identity (5.17) we have that
< W),<P> = £ <е,е;Ке(>еу(ф), <p>
i<j
= i.Z <k-ejA(e>)].<P>
= -2 E.	[е(еу,ф]>.
From Theorem 4.16 we know that the curvature transformation RytW:
C£(X) -► C£(X) can be written as Rv w = | £i<; (R^foXe^ad^.
Hence,
<<R(<p),<p> =	£ <Re.e.(el[),^><ade(ej(<p),adeke/<p)>	(8.15)
= “1 E <R(eiej)>e^Xadeie/<P),ade)ce/<p)>.
i<J
k<t
Observe now that the elements {eiej}i<j form an orthonormal basis of
A2(X) c= C£(X). The last expression in (8.15) is clearly independent of the
choice of orthonormal basis. That is, we can write
< W ф> = -1 E <ща S,Xa<W ad^)>	(8.16)
where {£а}л is any orthonormal basis of Л2(Х) c C£(X). We choose a
basis that diagonalizes R. Let {Xa} be the eigenvalues of R and set Aa =
Then (8.16) becomes
<<й(ф),Ф> = E 4||ad4d!(<p)||2
a
where Aa > 0 for all a. This proves that < SR(<p), <p> 0 and equality holds
iff ad^cp) = 0 for all £ g A2(X). Thus, it remains only to prove the following:
Lemina 8.7. Consider a form (p e A₽(R") c= C£(R") for 1 p n — 1. If
ad^cp) = 0 for all £ e A2(IR"), then (p = 0.
Proof. Recall that the representation ad^ on A₽(IR") is just the standard
representation of the Lie algebra so(n) = A2(R"). This lemma is therefore
160	II. SPIN GEOMETRY AND DIRAC OPERATORS
equivalent to the well-known fact that these representations have no fixed
vectors for p / 0 or n. However, since an elementary proof is possible we
shall give it.
Let ... ,en be the standard basis of R" and write (p = £|/|=p a^. By
assumption, [e^, <p] = 0 for all i < j. One can see easily that
f 0	if both ij ф I
[e^, £/] = < 0	if both i, j g I
12eie/€/	otherwise.
If i ф I and j g I, then = ±e/kj{0 _ {j}. Therefore, [e^, <p] = 0 implies
that at = 0 whenever i g I or j g I but not both ij g I. Applying this for
all i < j shows that (p = 0, provided that p / 0 or n. This completes the
proof of the lemma and the theorem. 
We now take up the case of spinor bundles. From now on we assume
that X is a compact spin manifold with a fixed spin structure on its tangent
bundle. Let S' be any spinor bundle for T(X) endowed with its canonical
riemannian connection.
Before stating the vanishing theorem in this case we recall one of the
simplest invariants of the riemannian curvature tensor, the so-called scalar
curvature. This is a function к: X -► R defined by setting
к = trace(Ric) = — 2 trace(R).
In terms of an orthonormal tangent frame (en... ,e„) at a point x g X,
* = - t	(8.17)
i,j= 1
When X has dimension two, к coincides with the classical Gauss cur-
vature function.
Theorem 8.8 (A. Lichnerowicz [1]). Let X he a spin manifold and suppose
S is any bundle of spinors over X endowed with the canonical riemannian
connection. Let $ denote the Atiyah-Singer operator and V*V the connec-
tion laplacian on S. Then
02 = v*v + iK	(8.18)
This formula has the following striking consequence. We say that a spin
manifold X has no harmonic spinors, if ker 0 = 0 for any spinor bundle
associated to T(X).
Corollary 8.9. Any compact spin manifold of positive scalar curvature admits
no harmonic spinors. In fact, the same conclusion holds if the scalar cur-
vature is ^0 and >0 at some point.
§8. VANISHING THEOREMS AND APPLICATIONS	161
Proof of Corollary 8.9. This follows from formula (8.18) as before. Suppose
a g Г(£) satisfies tyo = 0. Then by integration of (8.18) we find
pll<= — (V’Vct.ct) = -IlVall2.
If к 0, then we must have Va = 0. Hence ||a|| is constant, and if k(x) > 0
for any point x, then J к||а||2 > 0 and we have a contradiction. 
Note that we have also proved the following:
Corollary 8.10. On a compact spin manifold with к = 0, every harmonic
spinor is globally parallel.
Proof of Theorem 8.8. We need to compute the curvature term in equa-
tion (8.9) for the canonical spinor connection. In Theorem 4.15 we es-
tablished that for all V,W e Tx(X), the curvature transformation Ry^:
Sx -> Sx is given by the formula Rsyw = |	<Kr >Иг(ек),^>ек^, where
Rv w: Tx(X) -► Tx(X) is the curvature transformation of X and where
(eb... ,en) is any orthornomal basis of Tx(X). Consequently, using the
identities (4.43) and (4.44) we see that:
я = 1ЕеЛ<,
ij
= 8 E
i,j,k,S
zLj ) zLj <^eiej(^k) 4"	4" Rejekfeih е/У^^^к
t (л
distinct
+ E
ij	i,J	J
= i E <Ле(е>),^>е^
= {k. 
As a consequence of the Atiyah-Singer Index Theorem applied to the
fundamental spin complex (Example 6.3), we have the following:
Theorem 8.11. Let X be a compact spin manifold of dimension 4k. If X
admits a metric of positive scalar curvature, then Л(Х) = 0.
More generally, applying the same argument to the Dirac operator Ф
of the bundle (7.1), we find that in any dimension, positive scalar curvature
implies that the analytic index ind„($°) must vanish (see Definition 7.4).
162	II. SPIN GEOMETRY AND DIRAC OPERATORS
From the C£k-Index Theorem, this implies the following:
Theorem 8.12. Let X be a compact spin manifold. If X admits a metric of
positive scalar curvature, then ^(X) = 0.
By Theorem 7.10 we know that this result implies Theorem 8.11 above.
However, it gives new information in dimensions one and two (mod 8).
Recall that d(X) is an invariant of the spin-cobordism class of X. The
above results therefore explicitly present large classes of manifolds which
cannot carry metrics of positive scalar curvature. In particular, from
Theorem 2.8 we have the following striking result:
Theorem 8.13 (N. Hitchin [1]). In every dimension n = 1 or 2 (mod 8)
where n > 8, there exist compact differentiable manifolds which are homeo-
morphic to the n-sphere but which do not admit any riemannian metric with
positive scalar curvature. In fact, they admit no metrics such that к 0
but ^0.
Such spheres are hardly Platonic. This theorem can be perhaps best
appreciated in light of the current positive results. The standard metric
on the standard n-sphere S" = {x g R"+1: ||x|| = 1} is of course the most
uniformly positively curved manifold. Its curvature transformation is given
by the formula — Rv w = V /\ W, i.e.,
— RVW(U) = <V, U>W - <JV, Uyv	(8.19)
for all U,V,We Tx(Sn). Hence, Ric(V) = (n - 1)F and к = n(n - 1). The
most symmetric exotic spheres (see Hsiang and Hsiang [1]) are the Kervaire
spheres which have dimension 4k + 1. They can be constructed by taking
the boundary of the manifold obtained by plumbing together two copies
of the tangent disk bundle of S2k + i:
§8. VANISHING THEOREMS AND APPLICATIONS	163
However, Brieskorn has proved that these manifolds can be described
algebraically as follows (see Hirzebruch and Mayer [1]). For each integer
d, consider the complex polynomial
Pd(z0,... ,z„) = Zq 4- z\ + z\ + ... + z2+1.
Let V(d) = {zg C"+2:pd(z) = 0}; S2"+3 — {z g C" + 2:||z|| = 1}; and set
M2n + 1(d) = V(d) n 52и+3.	(8.20)
For n = 2k and d = 3 or 5 (mod 8), M2n + 1(d) is a Kervaire sphere.
In this thesis Hernandez [1] proved that the Brieskorn manifolds
M2"+1(d) carry metrics of positive Ricci curvature. Moreover, Gromoll
and Meyer [2] have proved that a certain exotic 7-sphere carries non-
negative sectional curvature which is strictly positive on an open subset.
It is therefore something of a surprise that in dimension nine there exist
exotic spheres which carry no metric of even positive scalar curvature!
Manifolds which carry positive scalar curvature are not hard to find.
Any homogeneous space G/H, where G is a compact Lie group, carries a
metric with к 0. Furthermore, it carries к > 0 unless G/H is a torus.
More generally one has the following:
Theorem 8.14 (Lawson and Yau [1]). Let X be a compact manifold which
admits an effective differentiable action by a compact, connected, non-abelian
Lie group. Then X admits a metric of positive scalar curvature.
Corollary 8.15. Let X be a compact spin manifold such that d(X) / 0. Then
the only compact, connected Lie transformation groups of X are tori. In
particular, this conclusion holds for any exotic sphere which does not bound
a spin manifold.
In dimensions 4k, it is a result of Atiyah and Hirzebruch [3] that a
compact connected spin manifold X with A(X) / 0 does not even admit
an S^action! (See IV.3.)
Remark 8.16. We point out that the results above definitely require
that X be a spin manifold. We know from (6.19) that the complex pro-
jective spaces IP2k(C) have
-	/2k\
Л(Р>2к(С)) = (- l)k2"4k	.	(8.21)
\ к /
These spaces are all homogeneous; IP"(C) = U(n 4- 1)/U(1) x U(n — 1).
Hence, they have metrics with к > 0 and large non-abelian Lie trans-
formation groups. However, as we saw in §2, IP2k(C) is not a spin manifold.
Compact manifolds of positive scalar curvature are now rather well
understood. A thorough discussion will be given in Chapter IV.
164	II. SPIN GEOMETRY AND DIRAC OPERATORS
We conclude this section with an important generalization of the
Lichnerowicz formula (Theorem 8.8) to the case of “twisted” spinor
bundles. This result is also quite useful in applications as we shall see in
Chapter IV.
Let X be a compact riemannian spin manifold, and let S' be a spinor
bundle for X with the canonical riemannian connection. Let E be any
vector bundle over X equipped with an arbitrary orthogonal connection.
Then the bundle S ® E, equipped with the tensor product connection, is
again a Dirac bundle over X (see Proposition 5.10). Here the Clifford
multiplication takes place in the “S-factor”. We now define a smooth,
symmetric bundle endomorphism
<RE:S®E------>S® E
by the formula
<RE(a®e) = | f (W)®(RE.ek£)	(8.22)
j.k=i
on vectors a ® £ of simple type. Here RE denotes the curvature of the
bundle E, and, as usual, (eb... ,e„) denotes an orthonormal tangent frame
to X at the point in question. The sum in (8.22) is essentially the trace of
a bilinear object defined on A2TX.
Note. In the case that S is a complex spinor bundle, we may assume E
to be complex and endowed with a unitary connection. The tensor product
of S with E can then be taken over the complex numbers.
Theorem 8.17. Let X be a riemannian spin manifold with scalar curvature
k9 and let S ® E be any twisted spinor bundle over X as above. Then the
Dirac operator 0E and the connection laplacian V*V of S ® E satisfy the
identity:
0| = V*V + |k+9Ie
(8.23)
Remark 8.18. Note that the operator 9iE depends linearly and uni-
versally on the components of the curvature tensor RE of E. Thus, if E is
flat, 9tE = 0, and if RE is small, then 9iE is correspondingly small by an
estimate depending only on dimension.
Proof. Recall that the covariant derivative of the tensor product con-
nection on S' ® E acts as a derivation, i.e.,
V(a ® £) = (Vsa) ® £ + a ® (Ve£)
§8. VANISHING THEOREMS AND APPLICATIONS	165
where Vs, V£ denote the covariant derivatives on S' and E respectively.
The commutator of two derivations is again a derivation. This fact (or
direct verification) shows that the curvature transformation of S ® E is
also a derivation, i.e.,
R(a ® e) = (Rs(f) ® £ 4- a ® (REe)
where Rs and RE denote the curvature transformations of S and E
respectively.
We now wish to compute the curvature term 91 in the general Bochner
Identity (8.9). It is given by
<R(a ® e) = | f ejekR (o ® e)
j.k=l
= 2 Z	® £ + ° ® (Rej.ek£)}
J,к = 1
= ( 2 t, ejekR^ a) ®	|	® (R& eie).
\ j\k—1	/	j,k=l
Now from the Lichnerowicz calculation (see the proof of Theorem 8.8),
we know that the first term in the last line above is just Hence from
(8.22) we conclude that
9i(a ® e) = |k(ct ® e) + 9iE(a ® e),
and the proof is complete. 
CHAPTER III
Index Theorems
In this chapter we shall present the analytic underpinnings of the subject of
spin geometry. In particular we shall formulate and prove various forms of
the Atiyah-Singer Index Theorem. This will include the classical theorem
and its consequent cohomological formula for the index. It will also in-
clude the Index Theorem for G-Operators, the Index Theorem for Families,
and the Index Theorem for C£k-Linear Operators. This last result is one
of the deepest in the theory. It involves indices in KO-theory which are
in general not locally computable on the manifold. Our exposition of this
result differs somewhat from that which currently appears in the literature.
There are now in existence many beautiful and illuminating proofs of
the more classical index theorems which use the asymptotics of the heat
kernel. This is a method that was pioneered by Gilkey and Patodi in the
late 1960s. We have elected to present here instead the arguments which
originally appeared in Atiyah-Singer [l]-[5]. This is in part because the
Atiyah-Singer methods lead to the non-local results just mentioned (and
these results are not accessible by heat equation techniques). It is also,
however, because their arguments, which proceed in the spirit of Grothen-
dieck, are really quite beautiful and simple. The essential idea is this. One
observes that the index is an insensitive object, unperturbed by rather bru-
tal changes in the analytic data. Furthermore, the index is a “functorial”
object, which transforms nicely with respect to global operations such as
the embedding of one manifold into another, the addition and multiplica-
tion of operators, etc. Using an appropriate form of К-theory, one then for-
mulates a topologically defined index possessing the same transformation
properties as the analytic index. By performing manipulations allowed in
the theory, everything can be reduced to the trivial case where the mani-
fold is a point. Here the analytic and the topological indices are easily
seen to coincide, and it follows that they must coincide in general.
Our ambition has been to make the presentation in this chapter rea-
sonably self-contained. All the requisite material on pseudodifferential op-
erators is developed assuming only a knowledge of elementary Fourier
analysis. Along the way a proof of the generalized Hodge Decomposition
Theorem is given. Most of the material necessary for the derivation and
computation of the cohomological formulas is also presented in detail.
§1. DIFFERENTIAL OPERATORS
167
The exposition in this chapter owes much to the writings of Atiyah and
Singer and also to Gilkey and Nirenberg. The reader is encouraged to con-
sult the excellent literature on the subject of pseudodifferential operators
and index theory which has appeared over the past twenty-five years.
§1. Differential Operators
This section presents the basic notion of a linear elliptic differential opera-
tor over a manifold X. We begin by fixing notation. For an n-tuple of non-
negative integers a = (ab ... ,a„), we set |a| = £k ak, and for each e R" we
set £* =	‘ ln local coordinates (xn ... ,x„) on X we define the
differentiation operators Da by i'a'Da = д'а.'/дха = ff^/dx^dx^2 • • • dx“n.
Recall that for a smooth vector bundle E on X, the symbol Г(Е) denotes
the space of smooth (i.e., C00) cross-sections of E.
Definition 1.1. A differential operator of order m on X is a linear map
P: Г(Е) -► T(F), where E and F are smooth complex vector bundles over
X, with the following property. Each point of X has a neighborhood U
with local coordinates (x1?... ,x„) and local trivializations: E\v Д U x Cp
and F\u U x C*, in which P can be written in the form:
а,л|
p= L
|a|^m OX
where each Aa(x) is a q x p-matrix of smooth complex-valued functions
and where Aa 0 for some a with |a| = m.
A real differential operator of order m is defined similarly with C re-
placed by R.
Observe that if we make a change of the local trivializations of E\v and
F^ by smooth maps gE:U -► GLP(C) and gF : U -> GLq(C) respectively,
then in these new trivializations P has the form
/ д|л| \
P=9f\ E
d|a|
|<x|£m VA
where the Aa’s are again p x ^-matrices of smooth functions of x and where
4“ = дРА“дЁ1 for W = w.	(1.2)
If we make a change of local coordinates x = x(x) on U, then using the
fact that
(1.1)
S " дхк d
x— = /. ~x~ xxx for each j,
dXj к =i dxj dxk
168
III. INDEX THEOREMS
we find that P again takes the form
„ ~ da
P= L AW-
|a|<m	OX
where
Aa = X Afi
101 = m
dx a
for |a| = m
(13)
and where [dx/dx]* denotes the symmetrization of the mth tensor power
of the Jacobian matrix [(dxk/dx^).
Equations (1.2) and (1.3) together imply that the coefficients {fM*}|e( =w
represent a well-defined section a(P) of the bundle (OmTX) ® Hom(E,F)
where Q denotes symmetric tensor product.
Definition 1.2. The section a(P) g T((QWTX) ® Hom(£,F)) is called
the principal symbol of the differential operator P.
Recall that for a vector space K, the space QmV is canonically isomor-
phic to the space of homogeneous polynomial functions of degree m on
K*. Hence, for each cotangent vector £ g TJX, the principal symbol gives
an element
Ъ(Р):ЕХ---->FX.
(1.4)
If we fix local coordinates and trivializations as in Definition 1.1, we find
that for £ = Y£kdxk
a^P) = im X W-
Ia| =m
(1.5)
It is now possible to present one of the fundamental concepts of this
chapter.
Definition 1.3. Let P be a differential operator of order m over a mani-
fold X. Then P is elliptic if for each non-zero cotangent vector £ g T*X,
the principal symbol a5(P): Ex -► Fx is invertible.
Example 1.4. Let E = F be the trivialized line bundle and consider
the Laplace-Beltrami operator A : C°°(X) —> C°°(X) of a given riemannian
metric on X. In local coordinates (xn ... ,x„) we have
= E 9Jk ч—о--------b lower order terms
j,k^ 1 OXj oxk
§1. DIFFERENTIAL OPERATORS	169
where %9jkdxjdxk is the metric tensor and where ((gJ*)) = ((д#))”1- For
a given cotangent vector £ = Y£k dxk we find that
^=-S^=-l№
which is certainly invertible (as a linear map С -> C) for 0 0.
Example 1.5. Let S be a Dirac bundle over a riemannian manifold X
(see II.5.2), and consider the associated Dirac operator D: T(S) -» T(S). It
is straightforward to show that
^(D) =
where “£•” denotes “Clifford multiplication by £.” Since £ • <!;• = — ||£||2Id,
this map is certainly invertible for / 0.
Example 1.6. Let S be as in Example 1.5 and consider the Dirac
Laplacian D2:T(S) -> T(S). Then one has that
^O2) = ||^||2Id.
This shows, from the discussion in Chapter II, §5, that the Hodge La-
placian on exterior p-forms is an elliptic operator. The proof of the fol-
lowing statement is an easy exercise.
Proposition 1.7. Let P: Г(Е) -► Г(Г), P': T(E) -► T(F) and Q: Г(Г) -► T(L)
be differential operators over X where P and P' have the same order. Then
for all £ g T*X and for all t,t' g R, one has that
a^tP 4- t'P') = ta^P) 4- f^(P')
and
° P) = ^Q) ° <?№)
This proposition says that the symbol is a rather nice object when con-
sidered to live on the cotangent bundle. Given Р:Г(Е) -► Г(Г), we pull
back the bundles to T*X via the projection n: T*X -► X and consider
the principal symbol as a bundle map
o(P):n*E---->7i*F.	(1.6)
If P is elliptic, this map is an isomorphism away from the zero section,
and we can assign topological data to the operator as follows. Fix a metric
on X and set DX = {£ g T*X: ||£||	1}. Then via the construction given
in Chapter I, §9, the symbol of P defines a class
i(P) = [7i*E,7i*F; o(P)] g K(DX9 5DX).	(1.7)
Important Fact 1.8. Suppose X is a spin manifold of even dimension
and that P is the Atiyah-Singer operator on complex spinors. Then the
170	III. INDEX THEOREMS
class f(P), when restricted to any fibre, becomes the element t] which gen-
erates the Bott periodicity mapping in K(D2\ S2k~l) = K(S2k) = K(pt) (see
1.9). This means that on an even-dimensional spin manifold, the principal
symbol of the Atiyah-Singer operator gives a К-theory orientation on the
cotangent bundle, i.e., a generator of the Thorn-isomorphism (see Appen-
dix C).
Similarly, on a spin manifold of dimension 8k, the principal symbol of
the real Atiyah-Singer operator gives a KO-theory orientation on the co-
tangent bundle.
§2. Sobolev Spaces and Sobolev Theorems
Let E be a hermitian vector bundle with connection V on a compact
riemannian manifold X. Given иеГ(Е) we have УиеГ(Т*Х ® E),
and using the tensor product connection on T*X ® E, we have Wu g
Г(Т*Х ® T*X ® £). This process continues, and for any к we can de-
fine the norm
IMI*2 = f fjyv — yul2,	(2.1)
J=Ov
j times
called the basic Sobolev к-norm on Г(Е). An easy exercise shows the equiv-
alence class of this norm to be independent of the choice of metrics and
connection. The completion of Г(£) in this norm is the Sobolev space
L*(E). It is straightforward to verify the following:
Proposition 2.1. A differential operator P : Г(£)	T(F) of order m extends
to a bounded linear map P:I$E) -►	for all k^.m.
Ultimately we shall see that if P is elliptic then these extensions have
finite dimensional kernel and “cokernel” which consist of smooth sections
and are independent of k.
Our aim at present is to establish some analytical tools. This is best
done using Fourier transform methods. To this end we select a good sys-
tem of trivializations of our bundle E. To start we choose a finite cover-
ing of X by closed coordinate balls yfi: Ufi Bn = {y g : |y|	1}, /? =
1,... ,N. Over each ball Ufi9 we choose a smooth trivialization of E
E\v, U„ x C'
which possesses a smooth extension to an open neighborhood of U„. We
further assume that the open balls of radius 1/^/2 cover X, i.e., X =
U«=i Bp where Bp = {peUp: ]ур(р)|2 < {}.
§2. SOBOLEV SPACES AND SOBOLEV THEOREMS	171
We now change each coordinate yp to a local coordinate xf by setting
Note that xfi: Л R" and that x^Bfi) = B" = {x g R": |x| < 1}. Further-
more, under the given trivialization over Ufi9 any smooth section of E re-
stricts to become a bounded function u: R" -► C₽. In fact the function
|Deu(x)|(l -I- |x|)|e| is bounded for any a.
Let’s now choose a smooth partition of unity i subordinate to
the covering x. Any section и g Г(Е) can now be written as и = £ ufi
where up = %fiu. In our system of coordinate trivializations, each ufi be-
comes a smooth function with compact support in the unit ball Bn.
Definition 2.2. Any system of local coordinates for X and local triv-
ializations for £, together with a partition of unity, all chosen as above,
will be called a good presentation of E. Good presentations of each of a
family of vector bundles over X having the same local coordinates and
the same partition of unity, will be called a good presentation of the family.
Using a good presentation of £, we can reduce the study of Г(£) to the
study of smooth C₽-valued functions with compact support in Bn. Here
we can apply the classical Fourier transform
ЙЮ = (2я)-"/2 e~i<x,i>u(x)dx	(2.2)
whose elementary properties we summarize (see Taylor [2] as a basic ref-
erence). We assume all functions to be C₽-valued, and define the Schwartz
space = {u g C°°(R"): Va,/c,3CaJt such that |Dau(x)| = СаД(1 -I- |x|)"* on
R"}. (Recall that Da = i~]a'd'a'/8xa.)
The Fourier transform defines an isomorphism (•)"’.«^->«^ whose
inverse is given by the following “Inversion Formula”
U(x) =	(2л)-"'2	(2.3)
аде) = eafi(e).	(2.4)
^e) = аде).	(2.5)
(u,v)L2 - (u9v)L2 (Plancherel’s Formula)	(2.6)
where (u,d)L2 = J <u,u> is the usual L2 inner product.
Definition 2.3. For s g R and и g $f9 the Sobolev s-norm is the norm
||< given by the formula
ii«il2=J*(i + |e|)W)l2^e.	(2.7)
The completion of Sf in this norm is the Sobolev space L2.
172
III. INDEX THEOREMS
Remark 2.4. Let s be a positive integer. Then there are constants
and c2 so that ct(l + |£|)2s	1 +	+ ... + |^|2s c2(l + |^|)2s. It fol-
lows from formulas (2.4) and (2.6) that there are constants Ct and C2 so
that
Cjlullf < X (\Dau(x)\2dx^C2\\u\\2.	(2.8)
This means that the norm (2.1) (for the trivialized C₽-bundle with the triv-
ial connection) is equivalent to the norm (2.7).
For any integer к 0, let Ck denote the space of к-times continuously
differentiable functions on R" equipped with the uniform C*-norm, defined
for и g Ck by
||l4|& = SUp |°1м|2-	(29)
|a|^k
Our first main result is the following:
Theorem 2.5 (The Sobolev Embedding Theorem). For each real number
s > (n/2) -I- k, there is a constant Ks such that
Hulled Ks||u||s	(2.10)
for all ue£f. Consequently there is a continuous embedding
L2sc:Ck	(2.11)
for each such s.
Proof. We begin with к = 0. For x e R", formula (2.3) gives
|u(x)| (2n)-"/2 j*|u(£)|d£
= (2л)-"/2 J*(l + |£|)-2s(i + KDWH-
Since (1 + |£|)_2s is integrable if 2s > n we have by the Schwarz inequality
that
|u(x>l2 (2л)-- J (i + |e|)-2sde j*(i + |e|)2s|rf(£)l2^
= Ks2||u||2.
Repeating this argument for each derivative Dau with |a| < s — (n/2) and
then summing, proves the result. 
Observe that since (1 + |^|)2s' (1 + |^|)2s if s' < s we have that
IK' Hulls V s' < s.
Hence there is a continuous inclusion L2 cz L2 for all s' < s. When re-
stricted to functions with support in a fixed compact set, this inclusion is
“compact.”
§2. SOBOLEV SPACES AND SOBOLEV THEOREMS
173
Theorem 2.6 (The Rellich Lemma). Let {uJ}JL1 be a sequence offunctions
with support in Bn such that ||иД C for all j. Then for any s' < s there
is a subsequence which is Cauchy in the norm || ||s and therefore converges
in
Proof. We begin by recalling another elementary fact concerning the
Fourier transform. Let cp be a smooth C-valued function with compact
support in R". Then for all integrable functions u,
<pu = ф*й and cp*u = фй	(2.12)
where denotes the convolution product given by
<д.и(х) = JV(x - y)u(y) dy.	(2.13)
Suppose now that supp и c= Bn and <p = 1 on B”. Then и = (pu and so
u = <p*u. Taking derivatives gives the formula
£>ай(0 =	- rf)u(ri)dti,
and applying the Schwarz Inequality then gives
|O“«(^)|2 j*(l +	- tfidri j*(l + M)2s|ufr)|2
= к.ю|142
where Ka(£) is the continuous function defined by the first integral.
Applying (2.14) to the given sequence shows that the sequence
{D^UjijL x is uniformly bounded on compact subsets of R" for any a. In
particular the sequence [nJ f= x is uniformly equicontinuous on compact
subsets, and by the Arzela-Ascoli Theorem there is a subsequence which
is uniformly Cauchy on compact subsets.
Fix r > 0 and split the integral
ii«j - «*11? = £i>ru + isi)2sW) -
+£йД1 + и)2,ж-
For |^| > r, we have that (1 + |^|)2s'	(1 + r)_2(s-I,(l + |^|)21, and so the
first integral is ||и; — «k||2/(l + r)2(s-’') 2C/r2<1-, ). Hence, for any
given e > 0 we can make the first integral less than e/2 for all j and к by
choosing r sufficiently large. The second integral is then bounded above
by a constant multiple of
sup - “*(£)|2-
Hence, by the previous paragraph there is a J so that the second integral
is e/2 for all J,к J, and the sequence r is Cauchy in L2 as
claimed. 
174	Ш. INDEX THEOREMS
Combining the theorems above gives the following:
Corollary 2.7. Let {uj}f= t be a sequence in L? with supp uy <= B" and
||u7||s c for all j. If s> (n/2) + k, then there is a subsequence which con-
verges to a function ueCk0 in the uniform Ck-norm.
We denote by z • w = Yf)=! z?w, the standard C-bilinear pairing on C₽.
Theorem 2.8. The bilinear function
(u,v) = ]*«(£) • r(£)d£
is a perfect pairing on L2'x L2_s, that is, it identifies L2_s with the dual of
L2,for any s g R.
Proof. For u9v g we have
(m,v) = рю(1 + |e|)s • txexi + |e|)_s^,
and so by the Schwarz Inequality,
IM II4IHI-,.
Hence, the bilinear function has a continuous extension to Lf x LLS. In
particular, for any v e Lls, we have that
sup |(«,v)| |H|_S.	(2.15)
I|m|I>= 1
It remains to establish equality. For this we choose и so that u(£) =
v(O(l + |^|)”2s. Then we find |fi|2(l + |£|)2s = |0|2(1 + |^|)-2s so that
||u||s = IMI-s- Furthermore,
(M = J|p|2(i + |^|)-2s^ = |MI2-s.
Hence (u,u)/||u||s = ||v||_s and (2.15) can be made an equality. We have
established the isomorphism L2_s = (L2)*. 
Corollary 2.9. Let -► and T*:Sf be linear maps such that
(Tu, r) = (u, T*v)for all u,v g If for some s g R and some constant c, the
map T satisfies the condition
||T«||S c||m||s for all ueif,	(2.16)
then T* satisfies the condition
||TM-s = ФИ -s f°r aW V e Sf.
§2. SOBOLEV SPACES AND SOBOLEV THEOREMS	175
In particular, if T extends to a bounded linear map T.Lk -» for all
positive integers k, then T* extends to a bounded map T* :L2_k L2_kfor
all negative integers —k.
Proof. Given u,veSf we have |(T*v,n)| = |(v, Tn)|
c||v||_s||u||s. Hence, by Theorem 2.8 we find ||T*v||_s = sup{|(T*v,i/)|:
||u||s= l}^c||v||_s. .
There is a stronger conclusion possible here which is proved by the
interpolation methods of Calderon. Since we shall only need || • ||k for к e IL
in our work here, we simply state the result
Theorem 2.10. Let T\$f be a linear map such that (2.16) is satisfied
for s = sr and s = s2. Then T also satisfies (2.16) for all values of s between
sr and s2.
Corollary 2.9 gives the following key to transferring our local results to
global results on a compact manifold.
Proposition 2.11. Let A be a smooth matrix-valued function on R" so that
|De4| is bounded for all a. Then the map T .£f given by Tu = Au
extends to a bounded linear map T:Lf -► L* for all s e R.
Proof. By formula (2.6) we see that T*u = A*u where A\x) denotes the
transpose of the matrix A(x). For any integer к 0, the maps T and T*
are clearly bounded with respect to the classical norm
11* = | Z
which is equivalent to ||u||fc (see 2.4). The result now follows immediately
from 2.9 and 2.10. 
For any open set ft c= R", let L2n denote the ||• ||s-closure of Co (ft) =
{u g : supp и c= ft}.
Proposition 2.12. Let ft, Q' be bounded open sets with smooth boundary in
R", and let Ф: ft -► O' be a diffeomorphism. Then the map T: Cq (Q') -►
Co (ft) given by Tu = и ° Ф, extends to a bounded linear map T: L2n, L^n
for all s.
Suppose similarly that ®:R" -► R" is a diffeomorphism which is linear
outside a compact subset. Then the map T .L-f given by Tu = и ° Ф
extends to a bounded map T:Lj -► Lj for all s.
Proof. We begin with the second statement. Note that (Tu,v) = (u,T*u)
where T*v = ;(Ф)и оф-1 where ДФ) denotes the Jacobian determinant of
Ф-1. As above we easily see that T and T* are bounded for the norm
176
III. INDEX THEOREMS
|| ||i ~ || ||k for any integer к 0. Hence, 2.9 and 2.10 apply to give the
result.
The first statement is proved similarly using the straightforward exten-
sions of 2.8, 2.9 and 2.10 to the case ofQ c R". 
Proposition 2.13. Let P = £|e|^m Aa(x)Da be a differential operator of order
m on R" whose coefficients are bounded as in Proposition 2.11. Then
P:SP -► Sf extends to a bounded linear map P:Lj -► Lj_mfor all s.
Proof. Consider P = Da. Since |Dau|2 = |£а|2|й|2, we find immediately that
llD“«lb2 = f (1 + k|)21d2^)|2^ = ll“ll2+i«l for апУ a and s- Applying the
triangle inequality and (2.11) completes the proof. 
We are now in a position to globalize. Let E be a smooth vector bundle
over a compact manifold X and fix a good presentation for E with coor-
dinates xfi: Ufi R", ft = 1,... ,N, and with partition of unity {%fi} sub-
ordinate to the covering {Bfi = x£ X(B”)} (see 2.2).
Since Yip = h any u G Г(£) can be written as и = where ufi = %fiu.
For s g R, a Sobolev s-norm can be defined on Г(Е) by setting
N
ii4» e на	(2.17)
Д=1
where \\ufi\\s is defined in terms of the presentation: E\Ufi R" x C₽. The
following is a direct consequence of Propositions 2.11 and 2.12 which
assert the bounded effect of changes of coordinates and trivializations.
Proposition 2.14. For any s e R and any smooth vector bundle E over a
compact manifold, the equivalence class of the norm ||- ||s is independent of
the good presentation chosen to define it. Furthermore, when s = к is a non-
negative integer, the equivalence class of || • ||k is the same as that defined by
(2.1) for any choice of metric or connection on E.
This justifies our use of the symbol || ||s without subscripts to indicate
the presentation chosen. It is now straightforward to see that our main
theorems, 2.5-2.8 and 2.13, can be globalized. We summarize the result
here.
Theorem 2.15. Let E and F be smooth vector bundles over a compact mani-
fold X of dimension n.
(1)	For each integer к 0 and each s > (n/2) + k, there is a continuous
inclusion Lg(E) cz Ck(E). Furthermore, every sequence {Uj}f= i which is
bounded in the Ц-Цгпогт, has a subsequence which converges in the uniform
Ck-norm.
§3. PSEUDODIFFERENTIAL OPERATORS	177
(2)	For any riemannian volume measure p on X, the bilinear map on
Г(Е) x Г(Е*) given by setting
(«,«♦) = j*x u*(u)dn
extends to a perfect pairing Ц(Е) x Lis(£*) for all s.
(3)	Multiplication TAu = Au by any element A e F(Hom(E,F)) extends
to a bounded linear map TA :L^(E) -► L^(F) for all s.
(4)	Any differential operator Р:Г(Е) -► T(F) of order m extends to a
bounded linear map Р:Ц(Е) -► L^.^F) for all s.
§3. Pseudodifferential Operators
The concept of a pseudodifferential operator has its roots in the following
observation. Let P = £4a(x)Da be a differential operator on R" acting
on functions и with, say, compact support. By Fourier Inversion (2.3) any
such и can be written as
u(x) = (2л)-"'2 Jе‘<х’{>й(№.
Applying P we find that
Pu(x) = (2л)-"'2 J*е‘<х-<>р(х,ем№	(3.1)
where
p(x,^= £ Л’(х){‘	(3.2)
|a| $ m
is the (total) symbol of P. Replacing p by a more general function of x
and £ defines a pseudodifferential operator. Note that in (3.2) the order of
P corresponds to the degree of p as a polynomial in In the general case
one must be careful with growth in the variable.
Definition 3.1. Fix me R. A smooth (matrix-valued) function p(x,^) on
R" x R" is said to be a symbol of order m if for each a,a' there is a constant
such that
|ВД'р(хЛ)| саД1 +|ф"”,а'1	(3.3)
for all x,£. Let Symm denote the space of these symbols.
Proposition 3.2. To each p e Symw the formula (3.1) defines a linear opera-
tor P:£F -► If p has compact x-support, this operator has a continuous
extension Р:1^+т -► L* for all s.
178	III. INDEX THEOREMS
Proof. If ueif, then й e if. For any integer N > 0, we have
|x|2NPm(x) = (- 1)N J [А* ?<х,4>]р(хЛ)м(0^<
= (-1)" J Д*[р(х,<Ж)] di.
The second integral is bounded by (3.3) and growth properties of u. Hence,
PueSf.
To prove the second part note first that integration by parts gives
i“ j*е‘<х>{>р(хЛ)</х = j*el<x,^Dxp(x,i)dx.
Since p has compact x-support, (3.3) then implies that
jel<x^p(x,i)dx < C,(l + ЮТ + |C|)-'
for each t e Z+. It follows that
ЧШ)= $e‘<x’<-»p(x,i)dx (1 + |£|)
-t + |S|
where Ct and Ct are constants that depend on t. In particular there exists
a constant C such that
for all £ and Using the pairing from (2.8) we now compute that
(Pu,v) = fpu(t}) -v(t})dt}
= Я |р‘<х’4 ’>p(x,0dx|fi(0- vlrfididt].
Setting (7(£) = й(£)(1 + |£|)s+m and F(ij) = £(>/)( 1 + HI)-1, we And that
|(Pu,v)| < JJ V^didt,
£ {Яdiчшж2(ч)адр
< C||<+WIH-,.
Applying duality and interpolation completes the proof. 
The operators P given in Proposition 3.2 are called pseudodifferential
operators of order m on R", and the space of all such is denoted by TDOm.
§3. PSEUDODIFFERENTIAL OPERATORS
179
We shall study some of the properties of these “local” operators before
globalizing them to bundles on manifolds.
Note that a pseudodifferential operator can have order — m < 0. Such
an operator is said to be smoothing of order m. A linear map т: Sf -► Sf
which extends to a bounded linear map t:Ls -► Ls+m for all s and m is
called an infinitely smoothing operator. Note that by the Sobolev Embed-
ding Theorem 2.5, we have r(LJ c= C00 for all s.
Two pseudodifferential operators P and P' will be called equivalent if
P — P' is an infinitely smoothing operator.
We want now to examine what happens to the symbols of pseudodiffer-
ential operators under the operations of composition, of taking adjoints
and of changing coordinates. This is referred to as the “symbol calculus.”
To this end we make the following definition.
Definition 3.3. Let P be a pseudodifferential operator with symbol p.
Then p is said to have a formal development
00
p~ E pj
J=1
(each pj g SymWj for some if for each integer m, there is a К so that
p - i Pj g Sym"w for all к K. (Hence, the corresponding operator is
m-smoothing for all к K.)
The utility of such formal developments is evident from the following
result.
Proposition 3.4. Any formal series pj9 where р}е SymWj and -►
- oo, is the formal development of a pseudodifferential operator. This oper-
ator is unique up to equivalence.
Proof. By grouping terms we can assume mj+1 < т} for all j. Fix a smooth
function (p:R+	[0,1] such that <p(t) = 0 for t < 1 and <p(t) — 1 for
t > 2. For any sequence of radii {rj}f=i with lim rj = oo the symbol
p(x£)= E ф(|£|/г,)рАЛ)
j=i
is well defined since the sum is finite for each (x,^). Set </>(£) = <p(|£|) for
i, e R", and for each j define = (j + 1 )"||ф||с>- Recall that for each a,a'
and j there is a constant C^j such that |DJD£'p7| g CMj(l + |£|)mj-l“ I Let
Cj = max{COTj: |a| j and |a'| < j} and choose r} > mj2JC}. Then for any
к > j and for |a| < j, |a'| < j, we have
1ВД c/1 +
m:ZJ
180
Ш. INDEX THEOREMS
for all |f| гр Setting <pfg) = <p(|^|/ry), we see that
for all ^аЦа'^ j < к and for all £. It follows easily that p = £ <PjPj e Sym"1
and, moreover, that for all к
P- Z PjeSymmk+‘.
J = 1
This proves the existence of the operator. Its uniqueness up to infinitely
smoothing operators is obvious. 
Before beginning the symbol calculus it is useful to note that up to
equivalence any pseudodifferential operator can be made “local” in the
sense that Pu always has support in a neighborhood of supp u. For this
and many other basic calculations we shall need the following:
Workhorse Theorem 3.5. Let a(x,y£) be a smooth matrix-valued function
on R" x R" x R" with compact x- and у-support. Fix m e R and assume
that for each а,Ду, there is a constant Cafiy such that |D*D^D£a|
Саду(1 + |£|)m’lyl. Then the operator -► given by
(Ku)(x) = (2л)-" JJe‘<x-^>a(X,y^u(y)dyd£	(3.4)
is a pseudodifferential operator whose symbol к has asymptotic development
Mx4)~Z^-(W)(x,x,e).	(3.5)
Proof. Note that the у-integral in (3.4) is a Fourier transform. Using the
rule uv = u*v (and neglecting constants (2л)~"/2 which will take care of
themselves), we find that _
(Ku)(x) = ^^^-ri^drid^
= j*e,<x’’> j*- ’>а(хЛ — q,G) d£ fi(tj) dr)
=f j* e'<x,’’>k(x,>?)fi(>?) dr)
where a denotes Fourier transform in the second variable. The interchange
of integration is allowed since for each integer / we have
|а(хЛ-^)| |M < СД1 + |^)m(l + |£ -	+ И)"',
§3. PSEUDODIFFERENTIAL OPERATORS
181
and the right hand side is integrable for / sufficiently large. Formula (3.6)
shows that К is pseudodifferential with symbol
k(x,Yj} =	— r]£)dl;
= p'<x’{>d(x,C,C + '0C
For each integer / we have the Taylor expansion in the third variable:
,-i«i
«£+>/) = £ -(D;d)(x(^)C + RAxU+n)
a!
where the remainder is given by the formula
RAx,U; + ri) = V+V)i'+1 £ 1 fpD?d)(x£tC+^(l-t)'dt
|д|=<г+1 д! J0
Recalling that (•)* denotes the Fourier transform in the middle variable,
we find that
J	= J e‘<x^am(x,i;,t))d(
=/е'<^>(5да(х,сЛ)^
= (DJDJa)(x,x,>j).
Consequently, the symbol of К can be written in the form
,-i«i
k(x,tl)= £ — (Dyty)(x,x,T)) + rAx,q)
И!
and by Proposition 3.4 it will suffice to show that
rAx,n) = j* е'<х’{>КДх,СС+if) dC e Symm {e+1 ’
for each /.To prove this we first show that for each a, /?, and к there is
a constant Cafik such that the inequality
g слДк(1 + IC-F^|)т"|д|(1 + |C|)’fc
is satisfied for all x,£,^. To establish this we first note that from our basic
assumption on a we have
|СРдах,«+>1)| = jc^aAx^+^e-^dy
J (PxD*Dfa)(x,y£+ti)e ~	dy
^C^(£.supp)<e/y)(i + ic + ^-1'1
= ад + IC +
182	Ш. INDEX THEOREMS
After setting £ = £, the inequality follows easily. Using this inequality and
the above formula for Re, we calculate that
|£>5DgRXx(« + f?)|
= ('+!) Z л C	- tfdt
с«дЛ £ (1 + K + r)\)m-^+ 1,-IW(1 + |Ф"‘|С|'+ ‘(1 - t/dt
саДА(1 + W)'"-^+1>-^(i + |C|/+1-‘.
From this inequality it follows that there are constants Cafi so that
iD^rXx^)) С.Д1 +
Hence, rf e Symw-('+1} and the theorem is proved. 
Theorem 3.5 has the following easy consequences:
Observation 3.6. If the function а(х,уЛ) of Theorem 3.5 vanishes for all
(x,y) in a neighborhood of the diagonal, then the corresponding operator К
given by (3.4) is infinitely smoothing.
Proof. By (3.5) we have к ~ 0. 
For A g R" and e > 0, we set Лг = {x g R": distance(x,A) e}. An op-
erator P: if -► if is said to be e-local if for all и e Cq
supp Pu cz (supp u)£.	(3.7)
Corollary 3.7. Given P e '¥DOm whose symbol has compact x-support, and
given any e > 0, there exists P£ e '¥DOm which is equivalent to P and is
e-local.
Proof. Choose a smooth real-valued function ф on R" x R" such that
ф(х,у) = 1 in a neighborhood of the diagonal, and ф(х,у) = 0 if |x — y| e.
Let p be the symbol of P. Then the operator
(P8M)(X) = (271)-" JJei<x-^>ф(х,у)р(х^)и(у)dy d£	(3.8)
is clearly e-local. By Theorem 3.5 P£ is pseudodifferential with symbol
p£ ~ p. Hence, PE is equivalent to P. 
Corollary 3.8. Let / — (/pfc) be a pair of real-valued functions with com-
pact support on R". Then for any P e ^PDOm the operator Px given by
Px(u) = Z1P(Z2u)	(3.9)
is also in 4'DOm.
§3. PSEUDODIFFERENTIAL OPERATORS	183
Proof. The operator P* can be expressed in the form (3.4) with а[х,у£) =
Xi(x)z2(y)P(x£). 
Theorem 3.9	. Let P be a pseudodifferential operator and и a function in its
domain (in Lj for some s, say). Then for any open set U a R",
Proof. Suppose u^ is smooth and fix x e U. Choose x = (Xi^Xz) as above
so that: x e supp Xi c supp /2; Xi = 1 near x; and x2 = 1 in a neighbor-
hood of supp Xi- Since x2« g Co , we have Х1ЛХгм)G C°°- Furthermore, by
3.6 we have that XiPm — Х1ЛХ2М) = Х1Л(1 — Xz)M) G Consequently,
XiPu g C00 and so Pu is smooth near x. 
The reader can probably see the potential usefulness of the above corol-
laries in trying to define and study pseudodifferential operators on general
manifolds. These considerations motivate the following definition.
Definition. An operator P g is said to have support in a com-
pact set K, if supp(Pu) <= К for all и g Cq , and if Pu = 0 whenever
supp и n К = 0. The linear space of such operators is denoted 4*DOK
We now begin the local symbol calculus.
Theorem 3.10	. Given P g and Q g m with symbols p and q
respectively, the composition P о Q g 4/DOK^+m has symbol with formal
development
;l«l
8ут(Рое)~£-(^р)(ад.	(3.10)
а Я!
Given P g ^DOK m, we define its formal adjoint P* by setting
(Pu,d)l2 = (u,P*d)l2	(3.11)
for all u,v g Sf with support in K.
Theorem 3.11	. Given Pe4*D0K m with symbol p, its formal adjoint P* g
has symbol p* with formal development
i|e|
(312)
where (•)' denotes the transposed matrix.
Theorem 3.12	. Let ф:У -► V be a diffeomorphism between open subsets of
R". Then for each compact subset К с У,ф induces a map ф*: 4*00K m -►
184
III. INDEX THEOREMS
ЬУ setting
(ф^Р)(и) = Р(иоф)оф-\	(3.13)
Proof of Theorem 3.11. Choose u,v e C00 with support in K, and note that
(Pu,v)Li = Це‘<хЛ><р(х^ЖО, v(x))d^dx
= JjJ е,<х-,,л><р(х^)м(у), фс)> dyd£ dx
= j]]*<u(y),	dx dt, dy
= (U,P*V)L2
Fix a real-valued function ф e Cq such that ф = 1 on K, and note that
since фи = и, we can write
(P*r)(y) = jj e‘<y-^y)^Yv(x)dx d£.	(3.14)
This operator satisfies the conditions of Theorem 3.5 and therefore has a
symbol p* with formal development
р*(хл)~1-ода(х)р(ул)'
a a.	x = y
fl«l _______
where we use the fact that фр* = p* because p has x-support contained in
K. 
Proof of Theorem 3.10. Note that
(PQu)(x) = /е'<х«>р(хЛ)О^а^
and so we need a reasonable expression for Qu(^). To find this, note that
Q — (6*)* an(l so from equation (3.14)
(Qu)(x) = JJ e'<x-^ W£j'u(y) dy dt,	(3.15)
for x e K, where g* = sym(Q*). Now (3.15) is just an inverse Fourier trans-
form. Hence,
(МФ = J e-'^'^y^yydy
where r = (g*)f. It follows that
(PQu)(x) = У[е<<х-^«>р(х^)г(у^)и(у)^^
§3. PSEUDODIFFERENTIAL OPERATORS
185
to which Theorem 3.5 applies. We conclude that PQ is pseudodifferential
with formal development:
dal
sym(Pe) ~ - О?£>;р(хЛИуЛ)
a i*'	x = y
»i«i a’
M + Ы
-e e
a fi + y = a р'У'
.Wl	/ fl7l \
1Л
~E-^№(D?q)
fi P-
where the last line is a consequence of Theorem 3.11. 
Proof of Theorem 3.12. Write x = </>(x) and x = ф-1(х) = ф(х). We note
that
x-y
= Ф(х) - ф(у) = £ ^(tx + (1 - t)y)dt
= T(x, У) ’ (x - У)
where 'Pfoy) is a smooth matrix-valued function. Since T(x,x) = W/3x)x
and ф is a diffeomorphism, the matrix ^(x,^) is invertible for (x,y) in a
neighborhood О of the diagonal. We choose / g C®($) such that / = 1
in a (smaller) neighborhood of the diagonal.
Let J = det(3^/3x) denote the Jacobian determinant of ф, and note that
[(</>*P)«](x) = [P(u ° </>)](x)
= jj " 5л>р(хЛ)«(фу) dy d£
= JJ ei<x-^4’,'x-rt«>p(^(x)^)u(y)J(y)dydt
Let J denote the integrand in this last integral, and write	+
(1 - /)«/. By 3.6, the integral of (1 - /)</ represents an infinitely smoothing
operator. In the integral of we can make a change of coordinates
£ = [Ч"(х,у)]"Ч = ®(x,y) ‘ C and find that, modulo infinitely smoothing
operators, we have
1(Ф*Р)и](х) s ^ei<x~y-^a(x,y£)u(y)dydi;
where
a(x,y,C) = x(x,y)J(y)|det 0|p(^(x), 0(x,y)C).
186
III. INDEX THEOREMS
By the workhorse Theorem 3.5 we conclude that ф*Р is pseudodifferential.

Applying formula (3.5) and recalling that / = 1 near the diagonal, we
find that
dal
sym(</>*P) ~ — D’D’J(y)|det ®\р(ф(х),®£)
a Л*	x = y
= P ( x(x), H-И n (mod SynT"r)
since 0(x,x) = [(<ty/5x)‘]"1 = (dx/dx)' and |det 0(x,x)| = J-1(x). Equa-
tion (3.16) states exactly that, modulo symbols of lower order, the symbol
of a pseudodifferential operator transforms like a function on the cotangent
bundle. More precisely, let the coordinates for the Fourier transform (x,f)
be considered as standard coordinates for 1-forms co = £ on x-space.
Definition 3.13. Let P g ^DOm have symbol p g Symw. Then the prin-
cipal symbol of P is the residue class a(P) = [p] g Symw/Symw" x. Our dis-
cussion above shows the following:
Corollary 3.14. The principal symbol a(P) transforms under diffeomorphisms
like a function on the cotangent bundle of R".
We are now in a position to consider global questions. Let X be a com-
pact n-dimensional manifold, and let E and F be smooth complex vector
bundles over X. We say that a linear map Р:Г(Е) -► T(F) is infinitely
smoothing if it extends to a bounded linear map P : L^(E) -► Lf+m(F) for
all s,mG R. This implies by (2.15) that P(L$(E)) c T(F)( = C°°-sections)
for all s. It is a straightforward exercise to show that given a riemannian
volume measure p on X, any infinitely smoothing operator can be written
as an integral operator:
P«(x) = K(x,y)u(y)dM(y)	(3.17)
where K(x,y) g Homc(Ey,Fx) varies smoothly on X x X.
Definition 3.15. A linear map P: Г(Е) -► T(F) is called a pseudodiffer-
ential operator of order m if modulo infinitely smoothing operators P can
be written as a finite sum P = £ Pa where each Pa can be expressed in
some system of local coordinates xa: Ua -► R" and smooth bundle trivi-
alizations as a pseudodifferential operator of order m with compact sup-
port. The linear space of all such operators is denoted TDOm(E,F). Two
such operators are called equivalent if they differ by an infinitely smoothing
operator.
Any differential operator P of order m is pseudodifferential.
§3. PSEUDODIFFERENTIAL OPERATORS
187
Remark 3.16. Given a riemannian metric on X and P e ^DOm(E,F\
e see from 3.7 that for any e > 0 there is an operator P£ e 4'D0m(E,F)
hich is equivalent to P and is e-local. (Of course, a differential operator
. 0-local.)
Using a good presentation of the bundles E and F (cf. 2.2), and patching
jgether local pseudodifferential operators with a partition of unity, one
an manufacture interesting elements in TDOm(E,F) for any m e IR.
Given a riemannian volume measure p on X, we can associate to any
perator P: Г(Е) -► T(F) a formal adjoint P*: T(F*) -► Г(Е*) by setting
<Pu, v> du =	<m, P*t>> du	(3.18)
эг и g Г(Е) and v g T(F*). The following theorem is an immediate con-
equence of the previous results of this section.
Theorem 3.17. Let EJF and G be smooth vector bundles over a compact
manifold X and fix operators P g ^PDOm(E,F) and Q g ^PDO^F.G). Then the
oilowing statements hold:
(i)	P extends to a bounded linear map P.Ll(E) -► Lf.^F) for all s.
(ii)	For any open set U cz X,
м|<, is С00	=> Рм|и is C°°.
(iii)	QPe'PDOm+f(E,G\
(iv)	P* 6 '¥DOm(F*,E*)for any ц.
(v)	A diffeomorphism ф :Х -► X induces a linear map
ф*:Ч'ВОт(ф*Е,ф*Р) 4D0m(E9F)
by the formula ф*[(ф*Р)и] = Р(ф*и).
We now discuss some elements of the symbol calculus in the global
setting. Let л*Е and n*F denote the pull-backs of E and F respectively
to the cotangent bundle я: T*X -► X.
Definition 3.18. Let p be a smooth cross-section of the bundle
Hom(n*E, tt*F) on T*X. Then p is a symbol of order m if in a good
presentation of E and F, p defines an element of Symw in each local co-
ordinate chart of the presentation (where the variables (f x,... Лл) are
canonically identified with the coefficients of cotangent vectors in the basis
{dxi,... ,dx„}.
It is easy to see that this definition is independent of the good presen-
tation that is used.
We let Symm(E,F) denote the vector space of all such symbols of order m.
From Corollary 3.14 and Theorem 3.10 we have the following conclusion.
188
III. INDEX THEOREMS
Theorem 3.19. Each P e 4'DOm(E,F) has an associated “principal symbol”
o(P) defined in the quotient space Symm(E,F)/Symm-1(E,F).
It would be useful to be able to canonically construct for each ele-
ment p g Symw(E,F) an associated operator P g TDOm(E,F) with the same
principal symbol as p. This can be done after introducing a riemannian
metric on X and a connection V on E. We proceed as follows. Fix p > 0
sufficiently small that the exponential map expx: TxX -► X gives a smooth
embedding of the p-disk at each x. Fix a smooth “cut-off” function
ф: [0,p] -> [0,1] with ф = 1 near 0 and ф = 0 near 1. The riemannian
metric determines a Lebesgue measure in each fibre of TX and of T*X9
and we can define a “Fourier Transform” (•)*: Г(Е) -> Г(л*Е) as follows.
For £ g TJX, we set
e-^U(V)dV	(3.19)
v 1 хЛ
where U(V) = ф(\K|)u(expxK) and where u(y) denotes the parallel trans-
late of u(y) along the (unique, shortest) geodesic ray joining у to x, when
distance (y,x) < p. For |K| p we set U(V) — 0. The function u(£) lies in
the Schwartz class of each fibre, T*X.
Given a symbol p g Symm(E,F), we now define an operator P = p(V) as
follows. For и g Г(Е) we set
Ри(х)^(2я)-" f p(W)^.	(3.20)
V 1 JcA
We leave as an exercise the proof that P g TDOm(E,F). A symbol cal-
culus for operators defined in this way has been worked out in Bokobza-
Haggiag [1] and Widom [1].
If we take X to be flat euclidean space, and E = F to be the trivialized
line bundles, and if we choose p(x,^) = £ Aa(x)g\ then a direct calculation
shows that Pu(x) = £ /T(x)Dau(x).
§4. Elliptic Operators and Parametrices
Recall that a differential operator Р:Г(Е) -► T(F) over a compact mani-
fold is called elliptic if its principal symbol a^(P) is invertible at all non-
zero cotangent vectors £ In this section we shall prove the fundamental
result that modulo infinitely smoothing operators, an elliptic operator is
invertible.
To this end we consider the “local” case of pseudodifferential operators
on R" which map C*-valued functions to themselves.
Definition 4.1 An operator P g 4*DOm with symbol p is said to be
elliptic if there exists a constant c > 0 such that for all |^| c the matrix
§4. ELLIPTIC OPERATORS AND PARAMETRICES	189
inverse of p(x,<J) exists and satisfies
|Ax,O-1| £c(l +	(4.1)
For example, an operator P whose symbol is of the form p(|£|)Id, where
p(t) is a polynomial with constant positive coefficients, is elliptic.
Remark 4.2. It is straightforward to verify that if P : Г(Е) -► T(F) is an
elliptic differential operator, then the local representations of P in a good
presentation of E and F are elliptic in the sense of 4.1.
Theorem 4.3. Let P e ^PDOm be elliptic. Then there exists an operator
Q g TDO_W, unique up to equivalence, such that
PQ = Id~S' and QP = Id - S	(4.2)
where S and S' are infinitely smoothing operators.
Proof. Let p be the symbol of P and let c be the constant in Definition 4.1.
Set q0(x, = х(|£|)р(х, f)"1 where z:R+ -► [0, 1] is a smooth function
with z(t) = 0 for t < c and /(t) = 1 for t 2c. 
Lemma 4.4. ^oeSym-m.
Proof. We must show that for each a and Д there is a constant Cafi so
that|D“D^0| Ca/?(1 4- |f|)"w",z*1. Fora = Д = 0, this follows immediately
from (4.1). For higher derivatives we first note that <?0| is estimated
uniformly in x by derivatives of p-1. Taking derivatives of the equation
pp-1 = p-1p = I (for | c) and applying (3.3) and (4.1), we see that
|5p-1/5fj| = |p”1(5p/dfJ)p“1| Cj(l 4- |f|)"w-1 for each;. Taking further
derivatives, using (3.3), and applying straightforward induction complete
the proof of the lemma. 
Note that qop — 1 = pq0 — 1 = 0 for |f | 2c. Consequently these func-
tions lie in Symm for all m and the corresponding operators are infinitely
smoothing. Unfortunately, pointwise multiplication of symbols does not
give rise to composition of operators. We have instead the complicated
formula (3.10):
:lal
sym(QP) = £-(£^)(D£p).
a Л:
Placing q0 in this formula, we find from the above observation that at
least sym(g0P— l)GSyrnm-1. This suggests proceeding inductively to
define a formal development
£ Qk	(4-3)
k = 0
190
III. INDEX THEOREMS
where qk e Symm k is defined by
*-i
J = o
i,e|	1
% -(Dlq^D'p) q0.
— L fit.!
(4.4)
By Proposition 3.4 there is an operator Qe^DO_m, unique up to
equivalence, whose symbol has the formal development (4.3)-(4.4). Since
the composition of any pseudodifferential operator with an infinitely
smoothing operator is infinitely smoothing we are free to replace P and
Q with any operators equivalent to them. In particular, by Corollary 3.7
we may assume P and Q to be 1-local. Theorem 3.10 now applies to prove
that QP — I is an infinitely smoothing operator.
A completely analogous argument proves the existence of an operator
Q' e^PDO_m such that PQ' — I is infinitely smoothing. Note, however,
that Q ~ Q(PQ') = (QP)Q' ~ Q', that is, Q and Q' are equivalent. 
The operator Q constructed in Theorem 4.3 is called a parametrix for
P. Its existence proves many of the basic important facts concerning elliptic
operators. Here is an example:
Theorem 43. Let P e ^PDOm be an elliptic operator and choose и e for
some s. Then on any open set U R", it is true that:
Pu is C°° on U => и is C°° on U.	(4.5)
Furthermore, if Pu = Xu for some X e C and if m > 0, then и is smooth.
Proof. Choose a parametrix Q with Id = QP + S as above. By 3.9, if Pu
is smooth on U, then и = QPu + Su is smooth on 17.
If P is elliptic, so is P — X Id for any X g C provided m > 0. Hence, by
the above, if (P — X)u = 0, then и is smooth. 
We pass now to the global case. Let E and F be smooth vector bun-
dles over a compact manifold X and consider an elliptic differential
operator P : Г(Е) -► T(F) of order m. Choose a good presentation of the
bundles E,F with coordinates : Up -► R" where /J = 1,... ,N, and with
partition of unity {фр} subordinate to the covering {Bp} where Bp =
{p g Up: |x/p)| < 1} (see Definition 2.2). In the /Jth system P defines an
elliptic operator Pfi g 4*DOm for which there is a parametrix Qp g KPDO_m
satisfying
P pQp = Id	S'p and QpP p = Id Sp,	(4.6)
where Sp and S'p are infinitely smoothing. By (3.7) we may assume that
Qp and, therefore, also Sp and S'p are 1-local. Now observe that for any
1-local operator R in the /Jth coordinate system, the operators ij/pR and
§4. ELLIPTIC OPERATORS AND PARAMETRICES	191
Рфр have compact support in 2Bfi = {xfi: |x^| < 2}, and therefore define
global operators in T£)O#(E,F). (In particular, if и e T\E), we have
фрРи = фрЩсри) + фрйЦ — д>)и) = фрЩсри) where <р is any smooth cut-
off* function with support in Up and with (p = 1 on 2Bp.)
We now define global operators Q,Q' g ^DO_m(E,F) and S,S' g
TDO.^F) by setting
Q —	Ф&р>	Q' = £ 6/^/ь
S = £ ФрЗр9	S' = £ З'рфр.
It follows immediately from (4.6) that PQ'u = £ PpQptypu) = £ фри —
Z S/r(^4w) = u-S'u since £ фр = 1. Similarly, one finds that QP = Id — S,
and therefore also that Q ~ Q(PQ') = (QP)Q' ~ Q', that is, Q and Q' are
equivalent. We have proved the following main result:
Theorem 4.6. Let Р:Г(Е) -+T\F) be an elliptic differential operator of
order m over a compact manifold. Then there is an operator Q g
TDO_m(F,E), unique up to equivalence, such that
PQ = Id - S' and QP = Id - S	(4.7)
where S and S' are infinitely smoothing operators.
The operator Q is called a parametrix for P.
Notice that the equations (4.7) imply that PQP = P — S'P = P — PS
and QPQ = Q — QS' = Q — SQ, and consequently that
PS = S'P and QS' = SQ.	(4.8)
Furthermore, it is evident that
S|kerP = Id and S'|kerQ = Id.	(4.9)
Remark 4.7. Theorem 4.6 carries over to pseudodifferential operators.
An operator P g TDOm(E,F) is said to be elliptic if its principal symbol
o(P) g Symw(E,F)/Symw" \E,F) has a representative p which is pointwise
invertible outside a compact set in T*X and satisfies the estimate
|р(£)”11 < C(1 4- |£|)-w for some constant C and some riemannian metric
on X. A straightforward adaptation of the arguments above shows that
Theorem 4.6 remains valid if the word “differential” is replaced by “ pseudo -
differential.” This fact will not be used here, so we leave the details to the
reader.
Remark 4.8. Given a differential operator Р:Г(Е) -► T(F) of order m
and a riemannian volume measure on X, let P*: T(F*) -► Г(Е*) be the
formal adjoint given by (3.18). Then P* is again a differential operator of
order m whose principal symbol a(P*) is the pointwise transpose of a(P).
In particular, P is elliptic if and only if P* is elliptic.
192
III. INDEX THEOREMS
§5. Fundamental Results for Elliptic Operators
In this section we shall prove some basic theorems for elliptic operators.
These will include the fundamental elliptic estimates, the classical “Hodge
Decomposition Theorem,” the spectral decomposition for self-adjoint el-
liptic operators, and estimates for the growth of eigenvalues, used later
to establish the strong convergence properties of the heat kernel.
Throughout this section E and F will denote smooth vector bundles
over a compact n-dimensional manifold X. We shall prove our results
here for differential operators but many of them carry over to pseudo-
differential operators of positive order.
To begin we recall some basic concepts concerned with a bounded
linear operator T:HY -► H2 between Hilbert spaces. The kernel of T is
the subspace ker(T) = {ve H1:Tv = 0}9 and the range of Tis the subspace
Im(T) = {Tv e H2:v e Ht}. The cokernel of T is the quotient space
coker(T) = H2/kn(T) by the closure of the range. The operator is called
Fredholm if its kernel and cokernel are finite dimensional and its range is
closed. Its index is then defined to be the integer ind(T) = dim(ker T) -
dim(coker T).
At the other extreme from Fredholm operators are compact operators.
A bounded operator T:HY -► H2 is said to be compact if the image of
each bounded sequence from Hr has a subsequence which converges in
H2. By the Sobolev Embedding Theorem, the inclusion Lj(E) c L,(E)
for s' > s is compact. In particular any infinitely smoothing operator
S: Lj(E) -► Lj(E) is compact. The Fredholm operators are exactly those
which are invertible modulo the compact ones.
Lemma 5.1. Let T:Hr -► H2 and Q :H2 -► H± be bounded linear maps
such that QT = Id — St and TQ = Id — S2 where Sx and S2 are compact.
Then T and Q are Fredholm operators.
Proof. Since Silk^T) = Id and Sx is compact, ker(T) must be finite dimen-
sional. Taking adjoints, we find Q*T* = Id — SJ, and since SJ is compact
we conclude as above that dim(ker T*) = dim(coker T) < oo. It remains
to prove that Im(T) is closed. By restricting to (ker T)1 we may assume
that T is injective. Let vk = Tuk9 к = 1,2,..., be a sequence such that
vk -► v in H2. We want to show that v = Tu for some и e Hr. We note
first that the sequence {uk}k=t is bounded. Otherwise by passing to a sub-
sequence, we can assume that ||uk|| -► oo and so Т(и*/||м*||) = vt/||uj| -+ 0.
Since QT = I — Sx and Sx is compact, we may assume by passing to a
subsequence that limOVHuJI) = lim SJUk/HuJI) = w where ||w|| = 1. How-
ever, by continuity Tw = 0, and since T is injective, w = 0. We conclude
that {uk}k=i must be bounded.
§5. FUNDAMENTAL RESULTS
193
Consider now the convergent sequence Qvk = QTuk = uk — S^uJ -►
Qv. Since {uk}®=1 is bounded and Si is compact we may assume, after
passing to a subsequence, that S{(uk) -► ию. Applying T to the line above
we find lim(7uk — TSruk) = v — Tu^ = TQv. Hence, v e Im(T) and we
have proved that Im(T) is closed. Therefore T (and by symmetry Q also)
is Fredholm. 
We now state our first main theorem:
Theorem 5.2. Let P:T\E) -► T(F) be an elliptic operator of order m over
a compact manifold X. Then the following is true:
(i)	For any open set U g X and any и e Lj(E)9
Pu\veC- =>	«I^C»
(ii)	For each s9 P extends to a Fredholm map P: Ls(E) -► Lj_m(F) whose
index is independent of s.
(iii)	For each s there is a constant Cs such that
114 Q||u||,_m + ||Pu||s_m)
for all и e L*. Hence the norms ||• ||, and || - ||e_m + ||P-||,_w on L* are
equivalent.
Proof. Part (i) is a restatement of Theorem 4.5. For part (ii), note first
that by Proposition 2.13, P extends to a bounded linear map P: Lj(E) -►
L,_m(F). That this extension is Fredholm follows immediately from the
existence of the parametrix (Theorem 4.6) and the lemma above. By part
(i), ker P consists of smooth sections and its dimension is therefore inde-
pendent of s. Similarly, the cokernel of P is isomorphic to the kernel of
the adjoint
LS2-„(F)*	L2(£)*
?ll	?ll	(5.1)
L2-s+m(F*)	b2-s(£*)
which is easily seen to be the natural extension of the formal adjoint of
P. Since P* is elliptic (see 4.8), dim(ker P*) is also independent of s. This
proves part (ii).
For part (iii), let Q e 4'D0_m(F,E) be a parametrix as in Theorem 4.7.
Then и = QPu 4- Su, and since S is infinitely smoothing, llulL llQPulL +
c(||Pw||s_m + 114-n). 
The above proof shows the following. Let P: Г(Е) -» Г(Р) be an elliptic
operator and P*: T(F*) -» Г(£*) its formal adjoint (defined using any
194
III. INDEX THEOREMS
volume form on X\ Define the index of P to be
ind(P) = dim(ker P) - dim(ker P*).	(5.2)
Corollary 53. The index of an elliptic operator P equals the index of any
of its Fredholm extensions P:Lj(E) -► Lj_m(F\
Part (iii) of Theorem 5.2 is called a “fundamental elliptic estimate.”
Notice that if we solve the equation Pu = v, it allows us to estimate the
ll’Hj-norm of и in terms of the ||-||s_m-norms of и and v. It will be useful
in later discussions to write down an important local consequence of this
result.
Theorem 5.4. Let P be an elliptic differential operator (of order > 0) defined
on an open subset 12 o/R". Then for every compact subset К c 12 and every
integer к 0, there is a constant CKk such that for all solutions и of the
equation Pu = 0, one has that
||W||x,Ck = ^K,k||Wlln,L2	(5.3)
where Ц-Ц^с* denotes the uniform Ck-norm on К and ||-||aL2 denotes the
L2-norm on Q.
Proof. Choose (p e Cq (12) with <p = 1 on K. Observe that P((pu) = (pPu +
£ aa(x)(D*u)(x) where the sum is over |a| < m = order(P) and where the
coefficients aa depend only on P and (p. Assume Pu = 0. Then the funda-
mental elliptic estimate 5.2(iii) applies to give
||«|k,Li <	C(|W|n.Li-„, + IMliMj. J
C'||w||n,Li-1
Taking a sequence К с c Qx с c Q2 с с - c c Qv = Q, applying the
argument repeatedly, and then using the Sobolev Embedding Theorem
2.15(1) completes the proof. 
It is often useful, when studying an operator P: Г(Е) -► T(F) to consider
non-negative operators P*P and PP* where P*: T(F) -► Г(Е) is the formal
adjoint defined via bundle metrics. For this reason we shall assume from
this point on that E and F are equipped with hermitian inner products and
unitary connections, and that X is furnished with a riemannian metric.
All connections will be denoted by V. The Sobolev norms || • ||k with к e T*
will be given explicitly by (2.1).
Given an mth order differential operator P: T(E) -► T(F) and bundle
metrics as above, we define the formal adjoint of P to be the map P* :
T(F) - Г(Е) such that (Pu,d)l2 = (u,P*d)l2 for all и e Г(Е) and all v e T(F).
Integration by parts shows that P* exists and is also a differential operator
§5. FUNDAMENTAL RESULTS	195
of order m. Furthermore, a(P*) = a(P)* and so P* is elliptic if and only
if P is elliptic.
A differential operator P:T(E) -► Г(Е) is then said to be self-adjoint if
P = P*. The Dirac operator of a Dirac bundle is always self-adjoint, and
so, of course, are its powers (see И.5.З.). An important example for rie-
mannian geometry is the Dirac operator D on the Clifford bundle Ct(X).
Under the canonical isomorphism Ct(X) s A*(X) we have D = d 4- d*,
and so D2 = dd* 4- d*d = A, the Hodge Laplacian (see II.5.12).
Theorem 53. Let Р:Г(Е) -► Г(Е) be an elliptic self-adjoint differential op-
erator over a compact riemannian manifold. Then there is an L2-orthogonal
direct sum decomposition:
Г(Е) = ker P ф Im P	(5.4)
Proof. With respect to the decomposition L2(E) = ker P ф (ker P)1 any
element и e Г(Е) c L2(E) can be written as и = u0 4- ur with Pu0 = 0.
Since и and u0 are smooth, so is
Consider now the Fredholm map P:L2(E) -► Llm(E) whose adjoint
P*: L2(E) -► L2(E) (cf. 5.1) is the natural extension of P itself. We clearly
have that (ker P)1 = Im P* = P(L2(E)). Hence, we can write u± = Рйг for
Hi g L2(E). Since Uj is smooth, so is by Theorem 5.2. 
Consider the special case E = A*X and P = A = dd*4-d*d (see II.5.12
forward). Clearly, Im A = Im d 4- Im d*. However, since (d*(p,d^)Lz =
(<p,d2^)L2 = 0 for all (p,i/s this sum is L2-orthogonal. Letting H* = ker A
denote the harmonic forms we have the following:
Corollary 5.6 (The Hodge Decomposition Theorem). Let X be a compact
riemannian n-manifold. Then there is an L2-orthogonal direct sum decom-
position of the smooth p-forms
Г(ЛРХ) = № ф Im d ф Im d*	(5.5)
for p — 0,... ,n.
Theorem 5.5 has another important consequence:
Corollary 5.7. Let P : T(E) -► T(E) be a self-adjoint elliptic operator over a
compact riemannian manifold, and let H: Г(Е) -► ker P be the orthogonal
projection given by the decomposition (5.4) Then there is an operator G :
Г(Е) -► Г(Е) of degree — m, called the “Green's operator," such that
PG = GP = Id- H.
Proof. The map P: Im P -► Im P is an algebraic isomorphism and has
an inverse P~ \ Let G = P"1 ° (/ — H). G obviously extends to a bounded
196
III. INDEX THEOREMS
map G: Lj(E) -► Ц+т(Е) which is a Hilbert space isomorphism on (ker P)1
for all s. 
We now consider the spectral theory for a self-adjoint elliptic operator
Р:Г(Е) -► T(F). For Я g C we consider the Я-eigenspace of P given by
Ел = ker(P - Я/),	(5.6)
and we say that Я is an eigenvalue of P if dim Ел > 0.
Theorem 5.8. Let P:T\E) T(E) be a self-adjoint elliptic differential op-
erator of order m > 0 over a compact riemannian n-manifold. Then each
eigenspace of P is finite-dimensional and consists of smooth sections. The
eigenvalues of P are real, discrete and tend rapidly to infinity in the fol-
lowing sense. If
d(A)sdim( © еД	(5.6)
\U|SA /
Then there is a constant c such that
d(A)^cAn(n+2m+2)/2m.	(5.7)
Furthermore, the eigenspaces of P furnish complete orthonormal systems for
L2(E), i.e., there is a Hilbert space direct sum decomposition
1?(Е)=фЕ;.	(5.8)
Proof. The first statement follows immediately from the fact that P — Я/
is elliptic. To see that each eigenvalue is real, note that if Pu = Яи, then
Ж12 = (Pu,u)L2 =-(u,Pu)L2 = A||u|l2.
To prove the estimate (5.7) we proceed as follows. Given £ > 0, we say
that a subset А с X is £-dense if for each point x e X there is a point
a g A with dist(x,a) < £. For each £ > 0, let N(e) be the minimal possible
number of elements in an £-dense subset of X. One sees easily that for
some constant C.
N(£)^C£-" for all £ > 0.	(5.9)
Consider now a function и g Ел and note that for each integer к > 0,
P*u = Я*и. Hence by the elliptic estimates 5.2(iii) we see that there is a
constant Ck, independent of u, such that
Q||U||0 + ||P*u||0).	(5.10)
Set E(A) = @|A| s л Ел and write и e E(A) as и = £ axux where ux e Ед has
uA| =1. Then Pku = X axAkux, and since ux 1 ид for Я / p, we have that
Pu |o = £ |ал|2|Я|2* £ Y. |ал|2Л2* = A2*||w||o. Consequently, by (5.10) we
lave that
||«||и* C*(l + Л*)||ы||о
§5. FUNDAMENTAL RESULTS
197
for all и e E(A). If we assume that mk > (n/2) + 1, the Sobolev Embedding
Theorem 2.15 gives us a constant Ck such that
sup |Vw| G(1 + A*)||w||o	(5.11)
X
for all и g E(A). Suppose now that for a given £ > 0 we have d(A) =
dim E(A) > N(e). Then there will be an element и g E(A) with ||u||0 = 1
such that и = 0 on an £-dense subset of points in X. It then follows from
(5.11) that
sup |tz| £Ci(l 4- A*).
x
X
However, this is impossible if eC'k(l + A*)vol(X)2 < 1 since J |u|2 — 1. We
conclude that d(A) g N(eA) where e^1 = 2C'k(l 4- A*)vol(X)2 C"A\
Consequently, by (5.9) we have d(A) Cea" ckA"*. Choosing к =
[(л 4- 2m 4- 2)/2m] completes the proof of the estimate (5.7).
It remains to prove (5.8). Let Ks denote the closure in Lf(E) of the
subspace consisting of finite linear combinations of eigensections of P.
Note that by 5.2(ii) the map P:L2(E) -► L2_m(E) is an isomorphism on
(ker P)1. Hence, P: -► is a Hilbert space isomorphism for all s.
We want to show that Vq = {0}. Let’s assume Vq / {0} and consider
/z2 = inf{(P2u,u) = \\Pu\\%:ueVi and ||u||0 = 1}.
Note that V2m is dense in and so /z2 < oo. (This density follows from
the fact that L2m-orthogonality implies Lo-orthogonality, V2m is dense in
VQ and Ljm is dense in Lq.)
Choose a sequence {h*}*°=i <= П) with ||мл||о = 1 for aH Л and
lim(P2uk,uk) = /z2. Since ||tzk||o 4- ||Pw*||o is uniformly bounded we know
from Theorem 5.2(iii) and the compactness of the embedding L2(E) c
Lq(E), that there exists a subsequence, also denoted {uk}k= b such that
uk -► и in Lq(E). We claim that
((P2 - /z2)iz, v)0 = (iz, (P2 - /z2)r)0 = 0	(5.12)
for all v g V2m = (^-2m)*« If n°t, then there exists v g V2m with ||v||0 = 1
and ((P2 - /z2)tz, v) = -a<0. For teR, consider the vectors uk(t) =
uk 4- tv and note that
0 < ((f2 -	“t(0)o_____> ~2at +f2 0
||**fc(0||o	1 + 2f(M’r) + f2
for |t| sufficiently small. This proves (5.12), which in turn proves that
P2u = /z2tz. Since P2 is elliptic, the /z2-eigenspace is finite dimensional and
P-invariant. Consequently it contains an eigenvector of P in contradiction
with the definition of Kq. This completes the proof of Theorem 5.8. 
1%
III. INDEX THEOREMS
A self-adjoint operator P is said to be positive if (Pu,u)0 0 for all
и g Г(Е). Theorem 5.8 can be reformulated in the positive case as follows:
Corollary 5.9. Let P: Г(Е) -► Г(Е) be a positive self-adjoint elliptic differ-
ential operator of order m > 0 over a compact manifold. Then there is a
complete orthonormal basis {u*}*L j. of L%(E) such that
Luk — Лкик for all к	(5.13)
where 0 < Лк A2 ... -► oo. In fact for some constant c > 0,
Лк ck2m/n(n+2w+2) for all k.	(5.14)
Proof. The first statement is clear. For the second note that d(Ak) = k. 
§6. The Heat Kernel and the Index
Let P: Г(Е) -► Г(Е) be a positive self-adjoint elliptic differential operator
of order m over a compact riemannian и-manifold X. In this section we
shall explicitly construct the heat operator е"гР:Г(Е) -► Г(Е), for t > 0,
which is an infinitely smoothing operator with the property that if ц =
e~tpu9 for some и g Г(Е), then ut satisfies the equation
d
+ P“t = 0-	(6-1)
We shall define e~tp as an integral operator of the form
(e-"u)(x) = Jx K,(x,y)u(y)dy	(6.2)
where K^x9y): Ey -► Ex is a linear map depending smoothly on x,y and t.
(Hence, К is a smooth section of the obvious bundle over IR x X x X.)
This kernel К is called the heat kernel for P and is defined as follows. Let
{u*}®=i be a complete orthonormal basis of L2(E) consisting of eigen-
sections of P with Puk = ikuk and 0 Л2 . -► oo as guaranteed
by Corollary 5.9. We set
*ч(х,у) = £ е-Лк'ы4(х) ® ul(y)	(6.3)
*=1
where p* g E* denotes the element such that u*(u) = <u,p>y for all и e Er
Lemma 6.1. For any r 0 and any closed interval I <= (0,oo), the series
(6.3) converges uniformly in the Cr-topology on I x X x X.
Proof. Fix a positive integer s with ms > (n/2) + r. By the Sobolev Em-
bedding Theorem 2.15 and the Fundamental Elliptic Estimates 5.2(iii) we
have constants c and c'9 depending only on s, such that each uk satisfies
§6. HEAT KERNEL AND THE INDEX
199
the inequality:
||«*||c- C'||«*||ms CdlU*l|o + Iloilo) = C(1 +
where || Цр- denotes the function Cr-norm on X. By Corollary 5.9, Лк satis-
fies the inequality
where у = 2m/n(n + 2m + 2). This implies that
e~*ktXsk (e~kvt)(ksy)
for all к > (s/t)1/y. The uniform Cr-convergence of (6.3) now follows di-
rectly from the convergence of the integral J® e~txxsdx. 
This lemma has the following immediate consequence:
Theorem 6.2. For each t >0 the operator e~tP'.T\E) -► Г(Е) is infinitely
smoothing. Furthermore, given и e Lf(E), for any s, the section u(t,x) =
(e~tpu)(x) is C00 on R x X and satisfies the “heat equation” du/dt = —Pu.
Proof. The first two statements are an immediate consequence of the fact
that К is C00. To see that du/dt = — Pu, note that
Щх, у) = -£ е~21Чкик(х) ® uf(y)
=—PxKt(x,y). 
For a given operator P as above, we introduce the following important
concept.
Definition 6.3. The trace of the heat kernel for P is the function
tr(e_,/>) = j** tracex[K,(x,x)] dx
= £ e~^
k = l
which is defined and analytic for all t > 0.
Note that tracex(uk(x) ® u*(x)) = |ut(x)|2 and so the second equality
follows from the fact that ||uk||2 = 1 for all k.
Example. Let X be the flat cubical torus R72nZ", where R" carries the
standard metric, and let P = — A = — £ d2ldQk acting on functions. The
normalized eigenfunctions are given by uN(0) = (2K)“"/2ei<N,e> for N =
(N1(... ,N„) g Z". Note that — AuN = |N|2U;v, and so
K,(0,0')= £ e-Dvi2'+i<v.e-e->
NeZn
III. INDEX THEOREMS
200
and
tr(erA) = £ a(k)e kt
k = 0
where a(k) denotes the number of elements NeZn with |N|2 = k. For n = 1,
we find
tr(e'4) = f e-*2’ 7^5- х/я/t.
— 00
Let us return now to the case of a general elliptic differential operator
P: Г(Е) -► T(F) over a compact riemannian manifold X. We assume E
and F are equipped with bundle metrics, and we consider the “Laplace
operators”:
P*P: ЦЕ)-------► Г(Е) and PP*: T(F)---------------► T(F).
Since (P*Pu,u)0 = ||Pu||2 and (PP*d,d)0 = ||P*u||2, we see that these
elliptic operators are self-adjoint and positive. It is furthermore evident
that ker P*P = ker P and ker PP* = ker P*. Consequently, we have
(cf.(5.2)) that
ind(P) = dim(ker P*P) — dim(ker PP*)	(6.5)
On the other hand, the operators P*P and PP* have exactly the same
sequence of non-zero eigenvalues. To see this, set Ел = {u g Г(Е): P*Pu =
Au} and Fa = {ver(F):PP*v = At;} for Ag R. Observe that if ugEa, then
(PP*)(Pu) = P(P*Pu) = A(Pu); that is, P(EJ <= FA. Similarly, we find that
P*(FJ с Ел. Since P*P = A Id on Ел, we conclude that P: Ел Л FA is an
isomorphism for all A / 0.
LetO < Ai A2 A3	00 denote the common non-zero eigen-
values of P*P and PP* (listed to multiplicity). Then taking the difference
of the traces of the heat kernels, we get enormous cancellation^.):
tr(e-cp*p) - tr(e"rPP*) = (dim Eo + E e~*kt) “ (dim Fo + E e~Xkt)
= dim Eo — dim Fo.
This proves the following:
Theorem 6.4. LetP: Г(Е) -► T(F) be an elliptic differential operator over a
compact riemannian manifold, and let P*: T(F) -► Г(Е) be defined via inner
products in E and F. Then
ind(P) = tr(e-rP*p) — tr(e"rPP*)
for all t > 0.
Observe that as t -► 00, the operator e~tp*p converges strongly to the
orthogonal projection He :L2(E) -► ker P. Consequently the direct sum
§7. TOPOLOGICAL INVARIANCE OF THE INDEX	201
Dt = e~tp*p ® ( — e~tPP*) acting on Г(Е ® F) can be thought of as a “ho-
motopy” of operators which, as t -► oo, converges to HE ® ( — HF), the
“difference of harmonic projections.” We see that for all t, Dt is of trace
class and its trace, tr(Dr) = ind(P), is time independent. It is useful to con-
sider the operator Dt at the other end of the “homotopy,” as t 4 0.
When deg(P) = 1, it turns out that as t \ 0, the heat kernel for P*P
has an asymptotic expansion
tracexKt(x,x) ~ f рк(х)г(‘_",/2	(6.6)
k = 0
Where pk(x) are densities on X which are locally and explicitly computable
in terms of the geometry of X and P. Theorem 6.4 says that the index of
P depends only on the coefficient pn(x) for the operators P*P and PP*.
Careful computations of these terms yields a proof of the classical Atiyah-
Singer Index Theorem.
§7. The Topological Invariance of the Index
The index of an elliptic operator is a quite stable object. It remains con-
stant during continuous perturbations of the operator and, in fact, it can
be. shown to depend only on the homotopy class of the principal symbol.
A proof of this fact is one of the main objectives of this section. In succeed-
ing sections we shall consider elliptic operators with additional structure,
such as operators which are Ctk-linear for some к or operators which
commute with a given action of a compact Lie group. In each case we
shall define a refined index and examine its elementary homotopy invari-
ance properties.
The index of an elliptic operator P: Г(Е) -► T(F) is the index of any of
its Fredholm extensions P:Lj(E)	For this reason we begin
with a discussion of Fredholm operators.
Let and H2 be separable Hilbert spaces and let = &\HVH2)
denote the Banach space of bounded linear maps from to H2 with
norm given by ||T|| = sup{||7Y||:||v|| < 1}. It is elementary to verify that
the subset cz & of linear isomorphisms is open in the norm topology
(cf. Palais [1]). We are interested here in the subset 3 = 5(Hn H2) of
Fredholm operators from to H2 (see §5). To each Те g, we have defined
the index of T:
ind T = dim(ker T) — dim(coker T).
Proposition 7.1. The map ind: $ -► Z is locally constant on 3, and induces
a bijection
ind: л0(3)
between the connected components of 5 and the integers Z.
202
III. INDEX THEOREMS
Proof. We begin the proof with a lemma which will be useful later on.
Lemina 7.2. Fix To e S and let V<^ Нг be a closed subspace of finite codi-
mension with Vr\ (ker To) = {0}. Then there is a neighborhood of To in
& such that for all T in we have:
(i)	V (ker T) = {0},
(ii)	TV is closed in H2,
(iii)	the subspace W = (TqF)1 c H2 projects isomorphically:
H2/TV.
Furthermore, the isomorphisms of (iii) assemble to give that:
(iv)	The family {Jtg^(^2/TV) -► <4l, topologized as a quotient oftftx
H2, is equivalent to the trivial bundle x W -> <%.
Proof. To each Те we associate the bounded linear map
T:W® V--------► H2	(7.1)
given by setting f(w,v) = w + Tv. This correspondence T->t defines a
continuous map -> &(WQ) V,H2) in the norm topology. Since f0 is
an isomorphism, so is T for all T in a neighborhood of To. This estab-
lishes (i)-(iv).	
Corollary 73. g is open in
Proof. Choose V = (ker To)1 in Lemma 7.2 and note that c g by (i),
(ii), and (iii). 
Corollary 7.4. The index is constant on connected components of g.
Proof. Fix To e g, set V = (ker To)1, and let Ш be given by Lemma 7.2. It
will suffice to show that ind T = ind To for all T e Fix T e <4l and con-
sider the (non-orthogonal) direct sum decomposition Hr = (ker T) ©
Z © V where Z = (ker T © У)1 (see 7.2 (i)). By the Open Mapping Theo-
rem, T induces an isomorphism between Z © V and the closed subspace
TZ © TV. Setting coker T = (TH J1 = (TZ © TV)1, we obtain the fol-
lowing factoring of T:	~
Hj------------T~--------> H2
*11	*11	(7.2)
ker T © Z © V -------> coker T © TZ © TV
Dividing by V = (ker To)1 gives an isomorphism
ker To = ker T © Z.	(7.3)
§7. TOPOLOGICAL INVARIANCE OF THE INDEX
203
Dividing by TV and using 7.2(iii) (with W = coker To) gives an isomor-
phism
coker To coker T ф TZ coker T ®Z.	(7.4)
It follows immediately that ind To = ind T. 
We shall examine this argument again when the maps have more struc-
ture. The proof of Proposition 7.1 is completed by the following:
Lemina 7.6. Any two operators To, 7\ e 5 with the same index lie in the
same connected component of 8-
Proof. Since ind(T*) = —ind T, it suffices to consider the case where
ind TQ = ind Tx 0. To begin we note that any Те 8 with ind T 0 is
homotopic in g to a surjective map. To see this, choose any linear surjec-
tion L. ker T—►* coker T = (Im T)x, and consider the homotopy T +
tL, t 0. We may assume therefore that To and 7\ are both surjective. Set
Kj = kerfTj) and consider the decomposition = Kj ф Kj for each j.
We have the isomorphism В = T^TqiKq Kj, and since Ko and
Kx have the same finite dimension, we can choose an isomorphism
4:K0 -► Kv The direct sum С = А ф В:К0 ф Kq -► Кг ф is an
automorphism of such that To = ТгС. Suppose we can find a contin-
uous family of isomorphisms Ct :H1 -►	0 t <. 1, with Co = C and
Ci = Id. Then Tt = TiCt connects To to Tt. Therefore we are done when
we have proved the following:
Lemma 7.7. The set = £P*(H9H) of isomorphisms of Hilbert space is
connected.
Proof. We fix Ce^x and show that C can be connected to the identity.
To begin we put C in polar form C = UA where A is the positive square
root of the positive self-adjoint operator C*C, and where (since U*U =
A ~1 C*CA ~1 = A ~1X24 “1 = Id), U is unitary. Since the positive bounded
self-adjoint operators form a convex set, we see that C is homotopic to
17. Now U has a spectral decomposition of the form U = fo* eaditi and
can be written as U = eiT where T = fJ* Л dnx. The homotopy Ut =
е*т9 0 <; t <, 1, completes the proof of Lemma 7.7 and so also of Lemma
7.6 and Proposition 7.1. 
Note. When = H2 = H, the composition of operators makes я08 a
semigroup. Interestingly, the map ind: л08 -> Z is a group isomorphism.
To prove this we need only observe that
ind(S ° T) = ind(S) + ind(T)	(7.5)
204
III. INDEX THEOREMS
for all S,T g g. To begin let n: H -► (ker S)1 be orthogonal projection and
note that since T is homotopic to я ° T in g (by a linear homotopy) we
can assume that Image(T) c (ker S)1. The assertion is now easily checked.
Let us now return to the case of elliptic operators over a compact
manifold X. A family Pt: T(E) T(F), 0 < t < 1, of such operators is said
to be continuous if in any good presentation of the bundles the coefficients
of the local representations Pt = Aa(x,t)Da are jointly continuous in x and
t. Under this hypothesis the order of the operators must be a constant,
say m. Furthermore, for any s the map [0,1] -►	given
by ti-> Fr, is continuous in the norm topology. Consequently, ind(Fr) is
constant in t by Proposition 7.1.
Recall that two elliptic operators P0,Pi: Г(Е) -► F(F) over X are homo-
topic if they can be joined by a continuous family Pt9 0 t < 1, of elliptic
operators. Our remarks above can be restated as follows:
Corollary 7.8. The index of an elliptic operator on a compact manifold
depends only on its homotopy class.
An immediate consequence is this:
Corollary 7.9. The index of an elliptic operator on a compact manifold
depends only on its principal symbol.
Proof. If Р^Р/.ЦЕ) -► T(F) have the same principal symbol, then so
does each element in the family Pt = (1 — t)P0 + tPv 
This result suggests making deformations of the principal symbol. Re-
call that the principal symbol of an elliptic operator is a section ae
Г[(ОтТХ) ® Hom(E,F)] with the property that
: E -> F is an isomophism for all £ / 0.	(7.6)
We shall say that two such symbols are regularly homotopic if there
is a homotopy ot9 0 < t < 1, joining them such that at satisfies (7.6) for
all t.
Theorem 7.10. The index of an elliptic differential operator on a compact
manifold depends only on the regular homotopy class of its principal symbol.
Remark 7.11. Using Theorem 3.19 and the subsequent discussion there,
one can generalize this result to elliptic pseudodifferential operators.
Proof. In light of 7.8 and 7.9 it will suffice, given ar, to construct a family
of operators Pf with a(Pt) = ot. This is evidently possible locally given
coordinates on X and trivializations of E and F. Patching together with
a partition of unity over some good presentation of E and F does the job
globally. 
§8. INDEX OF A FAMILY
205
The question, manifest at this point, is whether the index of an elliptic
operator can be computed directly from its principal symbol. A procedure
for doing this was first given by Atiyah and Singer. It will be discussed
in detail in §13.
§8. The Index of a Family of Elliptic Operators
In this section we shall present the important concept, due to Atiyah and
Singer, of an index for families of elliptic operators. In examining the
topological invariance of this index we shall prove that the Fredholm
operators on Hilbert space constitute a classifying space for K-Theory.
Let 4 be a Hausdorff topological space, and let E and F be smooth
vector bundles over a compact manifold X. The definition given in §7 of
a continuous family РГ:Г(Е) -► T(F), t g [0,1], of elliptic operators, can
be extended directly by replacing [0,1] with 4. (We shall call this a product
family.) However, when 4 is homotopically non-trivial, it is important to
allow the manifold and the operators to twist globally over 4 (in the spirit
of the families of complex analytic objects considered by Kodaira and
Spencer).
For a smooth vector bundle E -> X, let Diff(£;X) be the set of diffeo-
morphisms of E which carry fibres to fibres linearly. Note the homo-
morphism p: Diff(£;X) -► Diff(X) onto the diffeomorphism group of X.
Let Of = Diff(£,F;X) be the subgroup of Diff(£ ф F;X) which maps £ to
£ and F to F. This group acts naturally on the space Opm(£,F) of all
differential operators Р:Г(£) -> T(F) of order m (by setting g(P) =
02 ° P 0 011 for 9 = (0i,02))-
Definition 8.1. A continuous family of smooth vector bundles over X
parameterized by the Hausdorff space 4 is a fibre bundle ► 4 whose
fibre is a smooth vector bundle £ over X and whose structure group is
Diff(£;X).
From the homomorphism /?:Diff(£;X) -> Diff(X) we get an associated
fibre bundle -> 4 whose fibre is X and whose structure group is Diff(X).
Note that 8 is just a vector bundle over the total space of ЗС which is
smooth on each fibre. Furthermore, this smooth structure is changing
continuously over 4.
Definition 8.2. By a continuous pair of vector bundles over X param-
eterized by 4 we mean a bundle S ф & -> 4 whose fibre is a split bundle
£ ф F over X and whose structure group is & = Diff(£,F;X).
Associated to such a pair is the family of differential operators of
order m from £ to F. This is the bundle Opm(<£,«^r) -► 4 associated
206
III. INDEX THEOREMS
to the principal ^-bundle of © S? by the action of 9) On Opw(E,F).
A continuous section P of such a bundle whose fibre is elliptic for all
a g A is called a family of elliptic operators parameterized by A.
Example 8.3. Let X be a compact manifold and consider its asso-
ciated Jacobian torus Jx = H1(X;R)/H1(X;Z). Recall the isomorphisms:
H\X;R) Hom(^xX,IR) and Hx(X;Z) = Hom^X;/). Let X denote the
universal covering of X. Then we define a flat complex line bundle L over
X x Jx by taking the quotient
L= X x H'tX-M) x C/nJX) x Hx(X;Z)	(8.1)
where the action of 7txX x Hl(X;Z) is given by
= (gx9v + h9e2niv(g)z).	(8.2)
We think of L as a family of flat line bundles on X parameterized by
Jx-
Suppose now that X is even-dimensional and oriented, and let S be
any Dirac bundle over X with associated Dirac operator D. (For example,
one could choose the Clifford bundle or the spinor bundle for some
riemannian metric on X.) Then S ® L = (S+ ® L) ® (S" ® L) is a con-
tinuous pair of vector bundles on X parameterized by Jx. Using the given
flat connection on L, the Dirac operator on S extends to a family of Dirac
operators
D; : T(S+ ® Lv)-----> r(S” ® Lv)	(8.3)
for v g Jx. For S — C£(X), this family was introduced by G. Lusztig [1]
is his proof of the Novikov Conjecture for = Zm.
Suppose now that P is a family of elliptic operators defined over a
compact Hausdorff space A. Following Atiyah and Singer [3] we shall
define an analytic index ind(P) in the group K(A). If the dimension of
ker Pa (and therefore also of coker Pa) were locally constant on 4, then
this index would simply be the formal difference of finite dimensional
vector bundles:
ind P = [ker P] — [coker P] g K(A).	(8.4)
In general these dimensions are not constant, and so we must stabilize
the picture. Let <£, S' and 9C be as above. For a g 4, let 8a (and ^) denote
the fibre of 8 (and S' respectively) at a. This is a smooth vector bundle
on X and we denote by Г(<£а) its space of smooth cross-sections.
Lemma 8.4. There exists a finite set of sections {wb ... ,wN} of S' over
9C such that for each ae A the map Pa:CN © F(<%) -► Г(^Ра) given by
P^tp • • •	= У, Ijwjo’a + surjective for all a. The vector spaces
§8. INDEX OF A FAMILY
207
ker Paform a vector bundle over 4, and the element
[ker P] - [CN] g K(A)	(8.5)
depends only on the operator P.
Proof. From the local triviality of fibrations each point aQ g A has a
neighborhood U in which Pa9 a g U, is a product family. For any s, this
family extends to a continuous map P: U -►	and we
can apply Lemma 7.2. For W = ker(PJ0) this lemma shows that the maps
Pa:W® L*(E) -► L2_m(F) given by Pfl(w,<p) = w + Pfl(<p) are all surjective
in a neighborhood U' of a0. Theorem 5.2(i) implies that W c T(F) and
that the restriction Pa : W ® Г(Е) -► T(F) is also surjective for all a g U'.
Corollary 7.4 shows that dim(ker Pfl) is locally constant on V.
We now globalize this construction. Let {wp ... ,wr} be a basis for
W. Each Wj can be considered as a (constant) section of & over 17', or
alternatively as a section of the vector bundle & over the open set
л-1(17') c (where -► 4 is the fibration). Clearly each w, can be
extended to a global section Wj of & over all of 2E. Taking a finite
covering of 4 by such neighborhoods, and taking {vvp... ,wN} to be the
union of the sections constructed from each neighborhood, we establish
the first statement of the lemma. The local constancy of dim(ker Pfl) follows
by repeating the argument above with W replaced by the direct sum of
W*s from each neighborhood.
It remains to prove that the class (8.5) is independent of the choice of
global sections .. ,wN. Let w'p ... ,w'N> be another such choice, and
consider the family of maps Pa :CN ® CN' ® Г(^а) -► Г(^а) given by
PJt/,^) = £	+ E	have a commutative diagram
ker Pa с С" © Г№)	W
n	n	II
ker Pa с CN © CN' © Г(£а) -Ь— Г(^а)
рг
С"' =	С*'
where the map pr denotes projection. Since the map Pa is surjective one
easily sees that the sequence
0----► ker Pa----► ker Pa----► C*'----► 0
is exact. It follows that [ker Pfl] — [CN+N ] = [ker Pfl] - [CN] in K(4).
The argument applies symmetrically to the operators P'a: Cv ф -►
Ц^д) to prove that [ker Pfl] — [CN+N ] = [ker Pj - [CN], and the
proof is complete. 
208
III. INDEX THEOREMS
We are now authorized to make the following definition:
Definition 8.5. The analytic index of a family of elliptic operators
P over a compact Hausdorff space A is the element ind(F) g K(A) defined
by (8.5).
As one might imagine, this index enjoys invariance properties analogous
to those of the ordinary index. It depends only on the principal symbol
of the family, and in fact only on the homotopy class of the principal
symbol (all taken in the obvious sense; see Atiyah-Singer [3] for details).
This invariance is related to a basic and interesting fact. Let Я be a
(infinite-dimensional) separable Hilbert space. Recall that any two such
spaces are isomorphic. Furthermore, there is Kuiper’s Theorem which
states that the group = <5?Х(Я,Я) of linear isomorphisms of H with
the norm topology is contractible. Consider a family of elliptic operators
P over A as above. This is a section of the bundle Opm(<?;^j with structure
group Q). We can fix s and complete each fibre in the Sobolev norms to
get a bundle with fibre Я s 8(b?(E),L2_w(F)). This is the bundle asso-
ciated to the principle ^-bundle by the homomorphism 9)
Since <2?Х(Я,Я) is contractible, this bundle is trivial as an <2?Х(Я,Я)-
bundle in a homotopically unique fashion. Under the trivialization, the
family P becomes a continuous map P :A -> H = ^Ht,H2) where Нг =
LS2(F) and H2 = L2_W(F).
Note that the set 5 = <5(Я,Я) of Fredholm operators on a Hilbert
space Я has a continuous associative semi-group structure given by the
composition 8 x 8	8- For any topological space A, this makes the
space [Л, 8], of homotopy classes of maps of A into 8, into an associative
semigroup.
Theorem 8.6. (cf. Atiyah [4]). For any compact Hausdorff space A there is
a natural isomorphism
ind:[4,8] -----► K(A)	(8.6)
It has the property that for any continuous map f: A' -> A between such
spaces,
ind of*=f*o ind	(8.7)
Consequently, 8 is a classifying space for K-Theory.
This theorem completely generalizes Proposition 7.1 (where A = [0,1])
and is the foundation for proving the homotopy invariance of the index
for families.
Proof. Let T: A 8 be a continuous map. By Lemma 7.2 and the
compactness of A we know that there is a closed subspace V cz Я of
§8. INDEX OF A FAMILY
209
finite codimension so that for all a e A
(i)	Fn(kerTfl)={0},
(ii)	TaV is closed and of finite codimension in H, and
(iii)	H/TV = lJaGX (H/TaV) is a vector bundle over A.
One then defines
ind(T) = [Я/F] - [Я/TV] g K(A),	(8.8)
where [Я/F] = [Л x (Я/F)] denotes the trivial bundle. We must show
that this element is independent of the choice of V. Let V be another
such choice. Since V n V is also a choice, we may assume V cz V. There
is then an exact sequences of vector bundles: 0 -> V/V H/TV
H/TV - 0, which shows that [Я/TV] - [Я/ГГ] = [И/Г] = [Я/F] -
[H'V']. Hence, (8.8) depends only on T.
Given T :A -► and a map f:Af -► A, the subspace V chosen for T
also works for T ° f and we evidently have ind(T ° /) = /*(ind T).
Suppose now that T: A x I -► g is a homotopy between To and Tr
where Tj = T ° ij, and ij: A -► A x I is the inclusion i/a) = (a, j). By the
above paragraph ind(Tj) = i*(ind T) for j = 0,1, and it is a basic fact that
i* = if in К-theory. This proves the homotopy invariance of ind.
It remains to show that (8.6) is an isomorphism. We show first that it
is a homomorphism. Let T:A -> Q and T': A -> be continuous maps
and choose V H for T as above. Note that T' is homotopic to prv о T'
where prv: Я -► V is orthogonal projection. Hence, we may assume
T'H V. Let V be a choice of subspace for T', and note that К is also
a choice for the composition T ° Tf. Therefore we have
ind(T о Г) = [Я/Г] - [Н/ТТТ]
ind(T) = [Я/F] - [Я/7Т]
ind(T') = [Я/Г] - [Я/ТГ].
From the exact sequence: 0 V/T'V' Л H/TT’V H/TV ->0 of vec-
tor bundles over Л, we see that [Я/TT'F'] = [H/TV] 4- [V/T'V'] =
[H/TV] 4- [H/T'V'] — [Н/V]. Plugging this into the equation above
shows that ind(T о T') = ind(T) 4- ind(T') as required.
To prove that ind is surjective we first recall from Chapter I, §9, that
every element in K(A) can be written as [C*] — [E] where E is a vector
bundle on A and C* denotes the trivial /с-plane bundle. For each integer
к we define an operator Sk e 5 of index к by fixing a complete orthonormal
basis {e;}J°=1 of Я and setting
w=
eJ-k
о
if j — к > 0
otherwise.
The constant map T = Sk on A has ind T = [C*] for к 0.
210
Ш. INDEX THEOREMS
Fix a vector bundle £ over A and recall from Chapter I, §9, that for
some N there is a continuous map f: £ -> CN which is a linear injection
on each fibre. Let pr„: CN -» CN denote orthogonal projection onto f(Ea)
and let prx = Id — pre denote projection onto the orthogonal comple-
ment. Then the map TE:A -»	CN® H) given by setting
T^a) = pr„ ® S-i + prx ® Id has the property that ind TE = — [£].
Choosing an isomorphism CN ® H = H yields an isomorphism
3(CN® H, CN® Я) = 3. Since ind(S* ° TE) = [C*] - [£], we have
proved that ind: [A, 3] -♦ K(A) is a suijective homomorphism.
Our next step is the following. Let 3 х denote the invertible elements
in®.
Lemma 8.7. If T: A -> 3 has index zero, then T is homotopic to a map
T':4->3X <=3.
Proof. Choose a subspace V <= H as above so that ind T =
[H/V] — [H/TV], The hypothesis ind T = 0 means that for some k, we
have A x [(Н/И) ф C*] = (H/TV) ф C*. This implies that if we replace
У by a closed subspace of codimension к in V, we have the bundle
isomorphism
A x (H/V) & H/TV.
It is elementary to verify that there is a continuous map H/TV -» H which
carries H/TaVisomorphically onto (ТаИ)х for each a e A. Combining with
the isomorphism above, we get a linear map T1: A -> £P(H/V,H) where
for each a, T„ is a linear injection of H/V onto (TeK)x. The direct sum
Txa = Tx ф Ta:(H/V) ф V------► H
defines a continuous map Tx: A -* 3х c 3- This map can be connected
to T in 3 by the homotopy T, = tTx ф T for 0 < t £ 1. This proves the
lemma. 
Using the theorem of Kuiper [1] that 3 х is contractible, we conclude
that any T: A -» 8 of index zero is homotopic to the constant map T = Id.
Hence ind is injective and Theorem 8.6 is proved. 
One of the important results of Atiyah and Singer is the establishment
of a topological formula for the index of a family P of elliptic operators.
We shall present this formula in §15.
As a final remark we point out that the arguments given above adapt
easily to prove the following real analogue of Theorem 8.6. Let 3r =
SrC^r) denote the space of (real) Fredholm operators acting on real
Hilbert space Hn.
§9. THE G-INDEX	211
Theorem 8.8. For any compact Hausdorff space A there is a natural
isomorphism
ind : [Л,5И] -^KO(A)	(8.9)
having the functorial property (8.7) above. Consequently, is a classifying
space for KO-theory.
An important consequence of 8.6 and 8.8 is the following:
Corollary 8.9. For all к 0, there are isomorphisms:
як($)K(Sk) = К ”*(pt)	(8.10)
KO(Sk) ее KO-fc(pt).	(8.11)
§9. The G-Index
In this section we shall study operators which are preserved by the action
of a compact Lie group G. To be specific we fix vector bundles E and F
over a compact manifold X together with an action of G on the triple
(X,E,F). This means a smooth action p: G x X X of G on X together
with smooth actions of G on both E and F which carry fibres to fibres
linearly and which project to p. In this context we have the following
notion:
Definition 9.1. A differential operator Р:Г(Е) -► T(F) is called a G-
operator if Р(ду) = gP(q>) for all g e G and all у e Г(£).
A good example is given by the isometry group Gx of X acting on
Ct(X) s A*X. This action preserves both splittings Ct(X) =
C€°(X) © CP(X) and C€(X) = C€+(X) © C€"(X) and commutes with
the Dirac operator. Therefore both D° and D+ are G^-operators.
Similarly, if X is spin, then either Gx or a two-fold covering of Gx acts
on the spinor bundle of X and commutes with the Atiyah-Singer operator.
The basic observation here is that if P is elliptic and a G-op er at or, then
ker P and coker P are finite-dimensional representation spaces for G. This
leads us to consider the representation ring (or the ring of virtual repre-
sentations) R(G) of G. This can be defined as the free abelian group gen-
erated by the equivalence classes of irreducible finite-dimensional complex
representations of G. Since every finite-dimensional representation of G
can be decomposed uniquely, up to equivalence, into a direct sum of ir-
reducible ones, R(G) can also be defined as the Grothendeick group of all
finite-dimensional representations. In other words, each element of R(G)
can be expressed as a formal difference [F] — [IF], where [F] and [IF]
212
Ш. INDEX THEOREMS
are equivalence classes of finite-dimensional representations of G, and
where [F] — [W] = [F'] — [W"] if and only if V ф W' is equivalent to
У ф W.
The tensor product of representations is distributive over the direct sum
operation and makes R(G) into a commutative ring.
Example 9.2. Let G = S1 and let tm stand for the 1-dimensional repre-
sentation: (pm((f)z = eimrz. Then R(Sl) is easily seen to be the ring of Lau-
rent polynomials in t.RlS1)
Definition 9.3. Let P be an elliptic G-operator on a compact manifold.
Then the G-index of P is the element
indG(P) = [ker P] — [coker P] g P(G).
Note that when G = {!}, R(G) s Z and we recover the usual index of P.
The G-index can be specialized to individual elements of G. Given a
complex representation p*. G GL(F) we define its associated character
to be the function %p(g) = trace(p(#)). This function completely determines
the representation up to equivalence, and it has the properties that
Xpi+P2 = XP, + Xp2> Xp, ®P2 = Xp,Xp2, etc. (see Adams [1] for details). For
this reason R(G) is alternatively called the character ring of G. Given a
G-operator P as above and given g e G, we can define
indg(P) = trace(g|ker P) - trace(g|coker P).	(9.1)
This is the difference of the characters of the two representations, ker P
and coker P, evaluated at g. Consequently indg is the specialization of
indG to g as claimed.
Note that for D = d 4- d* : Acven(X) -> Aodd(X), and for g e Isom(X), the
number indg(D) is just the classical Lefschetz number of g. The general
topological formulas for indG given by Atiyah, Singer, Bott and Segal
represent generalizations of fundamental work of Lefschetz and Hopf.
Our object at the moment is to establish some elementary stability
properties of indG. As before, the regularity theory of elliptic operators
implies that the spaces ker P and ker P* coker P remain unchanged if
we pass to any Sobolev completion P: Ц(Е) -► Lj_m(F). Since G is com-
pact, we may choose metrics on X, E and F which are G-invariant. The
actions of G on Г(Е) and T(F) then extend to unitary representations on
Ls2(£) and L2(F) which commute with P and P*. This leads us to examine
the following concept.
Let and H2 be separable complex Hilbert spaces equipped with
unitary representations G ->	= 1,2, of a compact Lie group G. Let
8g = 8g(^i,^2) denote the space of all G-equivariant Fredholm maps,
i.e., all Fredholm maps T:Ht —> H2 such that Tgv = gTv for all g gG
§9. THE G-INDEX
213
and all и g ЯР To each T g gG we associate the index
indG(T) == [ker T] — [coker T] g R(G)
Proposition 9.4. The map indG: <JG -► R(G) is constant on connected com-
ponents of 3G. Furthermore, if and H2 are G-isomorphic, then indG
induces an injection
indG: n0(3G) c_> K(G).	(9.2)
Note. The map (9.2) will be a bijection provided that each finite-dimen-
sional irreducible representation of G occurs with infinite multiplicity
in = H2. The map (9.2) is also an additive homomorphism since
indG(T ° S) = indG(T) 4- indG(S). This is proved exactly as above (see (7.5)).
Proof. The argument for the first statement is identical with the one given
for the first part of Proposition 7.1 above. It is only necessary to check
that all subspaces are G-invariant and that all maps (such as (7.2), (7.3)
and (7.4)) commute with G.
We now assume that = H2 = H (as G-spaces). To prove the second
statement we must show that any two elements T0,Ti g gG with the same
G-index must lie in the same connected component of gG. We begin by
observing that any T g <Jg can be deformed to one which is “relatively
prime,” that is, one for which no non-zero subspace of ker T is G-isomor-
phic to a subspace of coker T = (Im T)1. Indeed, if such a G-isomorphism
L exists, it can be extended to a map L: H -► H by defining it to be zero
on the complementary subspace. The family T 4- tL, t 0, gives the de-
sired deformation.
We may assume now that To and Tv satisfy this “relatively prime”
condition. Consequently, the hypothesis indGT0 = indGT1 implies that
there are G-isomorphisms: ker To = ker Tx and coker To = coker 7\. We
now observe that given two finite dimensional, G-invariant subspaces
V0,Vi cl H which are G-isomorphic, there exists a G-isomorphism
C:H -► H with C(F0) = V\. In fact, C may be taken to be the identity on
(Fo 4- Pi)1. The subspace Vo n Vt is G-invariant and its complements in
Ko and Vi respectively are G-isomorphic; so the existence of C is clear.
It now follows that there exists a G-isomorphism C:H -> H which
carries Im(T0) isomorphically onto ImfTj). We define a second G-isomor-
phism С: H -► H by taking the given isomorphism ker To A ker Tx and
extending by the map lCTQ: (ker TJ1 A (ker TO1. We find that
Ti = CT0(C')-1, and the existence of a homotopy from To to Tx is an
immediate consequence of the following:
Lemma 9.5. The set	of G-isomorphisms of H is connected.
214
III. INDEX THEOREMS
Proof. The argument here is identical to the one given for Lemma 7.7
above. One needs only to check that the entire construction is G-
equivariant. 
Arguing as in §7 we see that Proposition 9.4 has the following imme-
diate consequence. Let Р:Г(Е) -► T(F) be an elliptic G-operator over a
compact manifold X.
Corollary 9.6. The G-index of P depends only on the homotopy class of P
in the space of elliptic G-operators. In particular, indG(P) depends only on
the principal symbol of P.
Note that the action of G on X, E and F makes E = (OWTX) ® Hom(E,F)
into a G-bundle, i.e., a bundle with a smooth G-action which maps fibres
to fibres linearly (and isomorphically). The principal symbol of P is an
invariant section of E, i.e., o(P) e TG(E) where TG(E) = {a g T(E):^a = tr
for all g g G}. Two principal symbols of elliptic G-operators, o0 = a(P0)
and = a(Px), are said to be regularly G-homotopic if there is a regular
homotopy 0 < t < 1, joining them (see (7.5) forward), such that at e
Tg(E) for all t.
Theorem 9.7. The G-index of an elliptic G-operator on a compact manifold
depends only on the regular G-homotopy class of its principal symbol.
Proof. As in the proof of Theorem 7.10 we construct a family of elliptic
operators Pt with o(Pt) = trf for 0 < t < 1. Averaging over G, by integrating
Лф = ^G(gPtg~1<p)dg = |j*c з(Л)^|ф
with respect to Haar measure on G, produces a homotopy of G-operators
with o(Pt) = ot for all t (since ot is G-invariant). The result now follows
from Corollary 9.6. 
§10. The Clifford Index
In this section we shall discuss, in general terms, elliptic operators which
are Ctfc-linear for some k. Several important examples of such operators
have been introduced and discussed in detail in Chapter II, §7. Motivated
by these Ctk-Dirac operators, we make the following definitions.
Definition 10.7. By a C€fc-bundle on a space X we mean a bundle of
real, left Ctfc-modules. This is a real vector bundle E over X together with
a continuous map T :C€fc x £ -> E such that Тф( ) = Т(ф, ): E -> E is a
bundle endomorphism for all <p g Ctfc, and the restriction Ctk x EX^EX
makes the fibre into a Ctfc-module for each x g X.
§10. THE CLIFFORD INDEX
215
Note that -bundles and Ct2-bundles are simply complex and qua-
ternion bundles respectively. In general, a Ctk-bundle is thought of as
having the algebra Clk as “scalars.”
Remark 10.2. A Ctk-bundle E will be called riemannian if it carries a
bundle metric which is preserved under multiplication by each unit vector
e e Rk c Ctk. Starting with any metric and averaging over the Clifford
group, as in 1.5.16, makes any Ctk-bundle riemannian. Note that if E is
riemannian, then multiplication by any element w g Rk <= Ctk is fibre-wise
skew-adjoint. Thus
<w •	u2> 4- <ub w • u2> = 0	(10.1)
for all u19u2 g Г(Е).
Let us fix smooth Ctk-bundles E and F over a manifold X.
Definition 10.3. A differential operator P: Г(Е) -> T\F) is said to be
Ctk-Iinear (or simply a Cfk-operator) if cpP(u) = P((pu) for all cp e Ctk and
all и g T(E).
Tensoring with Ctk makes any differential operator trivially into one
which is Ctk-linear. More interesting examples, such as the Ctk-Dirac
operators of 11.7, occur when there exists a Z2-grading. We say that a
Ctk-bundle E is Z2-graded if there is a bundle decomposition E = E° ® E1
making each fibre into a Z2-graded Ctk-module. A Ctk-linear differential
operator P : Г(Е) -> Г(Е) on such a bundle is called Z2-graded (or simply
graded) if with respect to the decomposition E = E° ® E1, it has the form
Note that Р°:Г(Е°) -► ЦЕ1) is C£g C£k_ plinear. When E and X are
riemannian, the operator P is (formally) self-adjoint if and only if
P1 = (P0)*.
We would like to define the analytic index of an elliptic operator of this
type. For this we recall the groups
= KO’fc(pt)	(Ю.З)
where 9Wk and 9Mk denote respectively the Grothendieck groups of equiv-
alence classes of Ctk and Z2-graded Ctk modules. (For detailed discussions
of these, see 1.5.20, 1.9 and II.7.) Recall also that KO"k(pt) KO"k + 8(pt)
for all k9 and we have
к	1	2	3	4	5	6	7	8
KO-*	z2	z2	0	Z	0	0	0	Z
216
III. INDEX THEOREMS
Suppose now that P: Г(Е) -> Г(£) is an elliptic self-adjoint graded Co-
operator on a compact riemannian manifold X. Then ker P is a finite-
dimensional Z2-graded C£k-module, and we have the following:
Definition 10.4. The analytic Clifford index of P is the residue class
indkP [ker P] e 9Wk/i*9Wk+i KO"fc(pt). (10.4)
Under the isomorphism (10.3) this element corresponds to the residue class
indkP [ker P°] g 9Wk _ Ji*^	(10.5)
where P° is given in (10.2). (The equivalence of these definitions is discussed
in detail in Chapter II. See II.7.4.)
The category of self-adjoint graded C£k-linear operators is a natural
and important one as we have seen by examples in Chapter II. However,
there is a twin category which has some advantages when studying sta-
bility properties of the Clifford index. This is the category of graded op-
erators which are skew-adjoint and C£k-antilinear. A differential operator
P: Г(Е) -> T(F) between C£k-bundles is said to be C£k-antilinear if P(wu) =
— wP(u) for all weRkc C£k and all и e Г(Е). This is equivalent to the
requirement that
P((pu) = a[(p)P(u)	(10.6)
for cp g C£k and и g Г(Е), where a denotes the involution of C£k engendered
by w -► — w.
Suppose now that E = E° ® E1 is a riemannian Z2-graded C£k-bundle
over a compact riemannian manifold X, and let P: Г(Е) Г(Е) be a Z2-
graded elliptic differential operator which is C£k-antilinear and formally
skew-adjoint. With P written as in (10.2) above, this means that P1 =
—(P0)*. We define the analytic Clifford index of such an operator to be
indkP = [ker P] g	KO"fc(pt)	(10.7)
and note as before that this is equivalent to taking [ker P°] in 9Mk_ 1/i*2Kk.
Observation 10.5. There is a natural transformation between graded elliptic
differential operators which are formally self-adjoint and Clk-linear, and
those which are formally skew-adjoint and Clk-antilinear. It is given by
associating to
(0 РЛ
p~\p° o)
the operator
P-(Q -pl\
\p°	0 /
Evidently, indtP = indtP.
§10. THE CLIFFORD INDEX
217
We shall now explore the topological invariance of these indices. To
begin we recall from 5.9 above that the elliptic self-adjoint operator P
above can be diagonalized on L2(£) with finite-dimensional eigenspace Vj
and discrete eigenvalues with lim, = oo. Thus, P can be written as.
P = £ XjTtj where я/Ь2(Е) -► Vj denotes orthogonal projection. In these
terms the (essentially) positive operator P2 is written as P2 = £ 22Note
that P2 is Ctk-linear and preserves the factors of the splitting L2(E) =
L2(E°) ® L^E1). We shall now define an associated Ctk-linear operator
on L2(E) which preserves these factors by the formula
(l+p2)4 = S(1+12)-^
The operator
£
P = (1 + P2)~2P	(10.8)
is a Z2-graded Ctk-linear self-adjoint Fredholm operator on L2(E) =
L2(E°) © Ь2(Е1). We leave as an exercise the verification that this con-
struction associates to a continuous family Pt, 0 < t < 1, of such elliptic
operators, a continuous family Pf, 0 < t < 1, of Fredholm operators.
The reader will note that if P is tlw companion of P in the sense of
10.5, then the operator P = (1 + P2) 2P = (1 - P2) 2P is a Z2-graded
Fredholm operator which is Ctk-antilinear and skew-adjoint. These con-
siderations motivate the following definitions.
Let H = H° ® H1 be an infinite-dimensional separable real Hilbert
space which is a graded module for the algebra Ctk. Assume in addition
that for each unit vector e e <= Ctk, the corresponding map e :H -> H
is a skew-adjoint isometry. Such a space will be called a graded Hilbert
module for C£k. Examples are easily constructed by taking the tensor
product H = H' ® Ctk = (Hf ® Ctk) ф (H' ® Ctk)for some real Hilbert
space H'.
Let <Jk c: gR(H, H) denote the subset of those Fredholm operators
Т.Н which are Z2-graded (i.e., Г(Я°) £ Я1 and Т(ЯХ) £ Я0), Co-
linear and self-adjoint. Similarly, let gk <= 3К(Я, Я) denote the subset of
graded operators which are C£k-antilinear and skew-adjoint. (As should
be obvious, an operator T e 8к(Я, Я) is called Ctk-linear if T((pu) = (pT(u)
for all cp g C£k and ue H. It is called Cfk-antilinear if T((pu) = a(<p)T(u)
for all such cp and u.)
As in Observation 10.5 we have a natural homeomorphism 3k = 8*-
We define the Clifford index of an element T e 3k (or 3k) to be the residue
class indkT = [ker T] e 9Mk/i*9Mk +1 £ KO-k(pt). Our first main result is
218	III. INDEX THEOREMS
the following:
Proposition 10.6. The Clifford index
indk:gk(^5k)---------------------->KO-k(pt)
is constant on connected components of <Jk.
Proof. Fix T g gk and recall that since T is Fredholm, 0 is an isolated
point in the spectrum of T. Replacing T by an appropriate scalar multiple
we may assume that the non-zero spectrum of T lies outside [ — 2,2], i.e.,
that T2 > 2Id on (ker T)1. Choose a neighborhood U of T in gk so that
for all S g U, we have spectrum(S2) c= [0,|) u (l,oo), and \\T2 — S2|| < 1/2.
Fix S g U and let W a H be the range of the spectral projection of S2
onto [0, j]. We claim that orthogonal projection pr: H -> ker T (= ker T2)
restricts to give an isomorphism
pr:PV-^ker T.	(10.9)
To begin, suppose v g W n (ker T)1 and note that <(T2 — S2)v, v)
(2 — i)||u||2 IMI2. Since \\T2 — S2|| < 1/2 we must have v = 0 and so
(10.9) is injective. Similarly any v g РУ1 n ker T satisfies <(S2 — T2)v, v)
||d||2 and is therefore 0. Hence, (10.9) is surjective and the claim is proved.
Observe now that W is a Z2-graded C£k-submodule of Я, and fur-
thermore splits into graded submodules W = ker S ® V where V =
(ker S)1 n W is S-invariant. One easily sees that the projection (10.9)
preserves the graded module structure, and so we have a graded C£k-
module equivalence
ker S ® К = ker T.
It remains only to enhance the structure of V to that of a graded C£k+1-
module. This is done as follows. Let Sv: V -► V denote the restriction of
S to V. This is a symmetric, Z2-graded C£k-linear map, and so also is
J = [S£] 2SV. Note that J2 = Id. With respect to the decomposition
V = V° © K1 we can write J in the form
J = f°
v° ° /
The graded endomorphism of V given by
J=f°
V° 0 /
is Ctk-antilinear and satisfies J2 = —Id. It therefore makes V into a
graded Ctk+ 1-module as required. Hence indkT = indkS and the proof is
complete. 
§10. THE CLIFFORD INDEX
219
Consider now an elliptic differential operator P: Г(Е) -► Г(Е) which is
3tk-linear and graded. Note that its principal symbol is also Ctk-linear
ind graded. In this context a regular homotopy of symbols is required to
^reserve these additional properties. Arguing as in Theorem 7.10 and using
:he proposition above directly proves the following. Assume, as before,
hat P is defined over a compact manifold and is self-adjoint.
Theorem 10.7. The Clifford index of P depends only on the regular homo-
topy class of its principal symbol.
Proposition 10.6 implies that indk induces a map on the connected
components of gk. In previous cases this map was a bijection, and the
analogous result is almost true here. However, we need some preliminary
adjustments. To begin, we shall assume for each к that H = H° ® H1
is a graded Hilbert module for Ctk+2 and is considered to be a graded
C£k-module under restriction to C£kci Clk+2- (This algebra inclusion
s induced by the euclidean space inclusion Rkc[RH2 as the first fc-
coordinates.) This assumption assures us, for example, that when к =
0(mod 4), each of the two distinct irreducible graded Ctk-modules appear
with infinite multiplicity in H.
For each fc, we jiow define a new space gk as follows. If fc 0 — l(mod 4),
then 5k = gk % gk. If fc = — l(mod 4), then we consider for each T e Sk,
the associated operator w(T) = ex • • • ekT|Ho where e15... ,ek is a fixed
orthonormal basis of cz Ctk. Note that w(T): H° -> H° is a self-adjoint
Fredholm operator. We now decompose gk into three disjoint subsets
gk+, 8k~> and gk consisting respectively of those T’s such that w(T) is
essentially positive, essentially negative, or neither. (A self-adjoint operator
is essentially positive if it is positive on a closed invariant subspace of
finite codimension.) Each of these subsets is open in gk and hence is a
union of connected components of gk.
We now show that when fc = — l(mod 4), the space gk is not empty.
Recall that H is a module for C£k+2 => Ctk. Let e15... ,ek + 2 be an exten-
sion of our orthonormal basis above, and write the action of ek + 1 on
H = H° © Я1 as
_/0 -S*\
e*+1 \S 0 J
Then the map ek+1:H -> H given by
/0 S*\
e*+1 “ \S 0/
(10.10)
is Ctk-linear and self-adjoint. Furthermore w(ek+i) anticommutes with
ek+2£k+i, and so we have £k + 1 g Йк.
220
III. INDEX THEOREMS
The following striking result is due to Atiyah and Singer [5]:
Theorem 10.8. For each k, the Clifford index induces a bijection
indk!7i0(&)->KO’k(pt).	(10.11)
Proof, From Proposition 10.6 we know that the map (10.11) is well de-
fined. To show that it is surjective, let V = К0 ф V1 represent an element
of	= KO~k(pt), and let ek+1 be as in (10.10) above. Then the
map £k+i ®0 is an element of ф V) and has kernel V. Since
H ф V = H, this shows that the map (10.11) is surjective.
To prove injectivity, it is convenient to first divest ourselves of the
grading. Recall that the endomorphisms e? = eker, ... ,ek-! =ekek_x
make the subspace H° into an ungraded module over Ctk_x = Ctk. Let
gk denote the space of all skew-adjoint, Ctk_ x-ant ilinear Fredholm op-
erators on H°. There is a natural map
(•)°:&—(Ю.12)
defined by setting T° = ekT|Ho. It is easily checked that this map is a
homeomorphism. Passing from graded to ungraded modules as in (10.4)
and (10.5), we find that indkT= [ker T0].
Assume now that we are given two operators S,T e gk with indkS =
indkT. It will suffice to show that S° and T° are homotopic in gjj. Our
first step is to observe that we may assume S° to be a Hilbert space
isometry on (ker S0)1 (and similarly for T°). This is accomplished by
putting the operator in polar form and deforming away the “radial” part
as in the first step of the proof of Lemma 7.7. Properties of skew-adjoint-
ness and Ctk_x-antilinearity are preserved. The operator S° now satisfies
(S0)2 = — Id on (ker S0)1, and setting ek = S° makes (ker S°) into a
Ctk-module. The same remarks apply to T° of course.
Observe now that our hypothesis indkS = indkT implies that there
exist ungraded C€k-modules V and W together with a Ctk_x-module
isomorphism
ker S° ф V ker T° Ф W.	(10.13)
We now claim that the module V can be realized as an S°-invariant
subspace of (ker S0)1. This means simply that V is isomorphic to a C£k-
submodule of (ker S0)1 where e° acts by S° as above. When к й
— I(mod4), all irreducible Ctk-modules are equivalent, and our claim
is obvious. When fc == — l(mod 4), there are two equivalence classes of irre-
ducible Ctk-modules and we must show that each of them appears with in-
finite multiplicity in (ker S0)1. Recall that these two representations are
distinguished by whether the central element cu = e? • • • ek acts by 1
or —1. There is a splitting (ker S0)1 = M = M+ ф Af " where M1 =
§10. THE CLIFFORD INDEX
221
(1 ± co)M, and we want to know that dim M + = dim M" = oo. However,
о =	. eo = eo ... eQ_	where the sign depends only on fc,
and so the desired property follows from our assumption that w(S) is neither
essentially positive nor essentially negative.
The same argument, of course, shows that W may be realized as a T°-
invariant submodule of (ker T0)1. We now consider the two orthogonal
decompositions:
H° = (ker S°) ф V ф V' = (ker T°) ф W ф W'.
It is clear from the discussion above that V and W' are isomorphic as
Ctk-modules. This means simply that there is a Ctk-linear Hilbert space
isomorphism L:K' ^>W' such that S° = L-1T°L on V. By the obvi-
ous homotopy we may assume that S° = 0 on V and T° = 0 on W. Ex-
tending L by the isomorphism (10.13) gives an isomorphism L:H° -► H°
such that S° = L~iT°L.
The theorem now follows from the fact that the group of Ctk-linear
isometries of H° is connected. This fact is proved by suitable adaptation of
the argument given for Lemma 7.7 above. 
Remark 10.9. The mapping (10.12) above gives a natural transforma-
tion between graded and ungraded operators. All the previous discussion
of this section could be so transformed. Consequently we find that there
exists a parallel index theory for real elliptic operators which are skew-
adjoint and (ungraded!) C^k_ {-antilinear.
Let us examine some examples. Let P be the operator. If к = 1, we find
that indxP = dimR(ker P)(mod 2). (Thus, every real skew-adjoint elliptic
operator has a well-defined index in Z2!) If fc = 2, then ker P is a
C£x C module and we find that ind2P = dimjker P)(mod 2). If fc = 4,
then Cf3 = H ® H and the bundle E on which P is defined splits as
E = E+ ф E~ where E± = (1 ± co)E and co = е^вз. By antilinearity,
P(E±) с: Г(ЕТ) and we split P into P+ and P" = —(P+)* as before.
Note that ker P = ker P+ ф ker P" and that ind4P = 0 if and only
if dimM(kerP+) = dimM(ker P”). (This is because the ± spaces are the
two distinct Ct3-modules.) It follows easily that ind4P = dimM(P+) —
dimM(P") = the index of P+ considered as a quaternionic operator.
We conclude this section with a discussion of a nice result of Atiyah
and Singer which states that the spaces gk form a “spectrum” (in the sense
of homotopy theory). Together with the periodicity phenomena in Clifford
modules and Theorem 8.8, this will give a new proof of Bott periodicity.
To begin we notice that for each к 1 there is a natural inclusion
<5fc+1c—► <5fc
222
III. INDEX THEOREMS
and a distinguished element
£*+1^8*
given by (10.10). We extend this by convention to к = 0 as follows. Let
50 denote the space of all real Fredholm operators on Я0, and let fa	fa
be the map which associates to Те fa the operator T° = е^до. (This
identifies 3^ with the space of all skew-adjoint operators in g0.) Set
= Id g go-
Let Clfa denote the loop space of gk defined here to be the set of all
continuous paths у:[0,я] -► fa with y(0) — ^k+i and у(тс) = — fy + i- This
space carries the compact-open topology. It is homotopy equivalent to
the space of all paths which both begin and end at ek + 1. The following
is the main result of Atiyah-Singer [5] and is presented here without proof.
Theorem 10.10 (Atiyah and Singer). For each к 0, the map
fa+i—>nfa
which assigns to Те fa + i the path from ek+1 to — ek + 1 given by
(cos t)ek+ x 4- (sin t)T, 0 < t n9
is a homotopy equivalence.
This gives us a generalization of Theorem 8.8 above.
Theorem 10.11. For any compact Hausdorff space A and for any к there is
a natural isomorphism
indk:[4,8t]-----> K0~\A)
with the functorial property (8.7). Hence, 8* is a classifying space for the
junctor KO~k.
Proof. The case к = 0 is just Theorem 8.8 above. For higher к we have
148*] a [A. Q*8o] = [E48o] = K0&kA) = KO~k(A) where ZkA de-
notes the /с-fold suspension of A. 
Theorem 10.12 (Bott Periodicity). For each к 0 there is a homeomor-
phism 8* = 8*+e- Therefore, by 10.10 there is a homotopy equivalence
8*~я’8» which implies that
K0~k(A) s K0~k+6(A)
for any compact Hausdorjf space A.
Proof. Let H = H° ф Hl be a graded Hilbert module for Ctk as above,
and let V = V° ф V1 be an irreducible graded (real) module for C£8. The
graded tensor product H ® V is a module for Ctk <§) Ctg s Ctk+8.
§10. THE CLIFFORD INDEX
223
There is an isomorphism V £ R16 which yields an explicit identifica-
tion C£8 s R(16). The fundamental results of Chapter I, §4, give an isomor-
phism Ctk+8 Ctk ® R( 16) (ungraded tensor product), where elements of
the form 1 ® (p act by Id ® cp on H ® V.
Let &k(H) denote the Banach space of bounded operators on H which
are C£k-linear and graded. Define a map T:^fk(H) <£k+s(H ® V) by
setting T^T) = T ® Id. This map is continuous and injective. Further-
more, we claim that it is surjective. To see this, fix T g <tfk + s(H ® V), and
for each x g H consider the map Tx: V -► H ® V defined by Tx(v) =
T(x ® v). Smce tx((pv) = (ptx(v) for all cp g R(16), we conclude that
dim(Image Tx) is either 0 or 16. It follows that Image(fx) = £ ® V for a
unique 1-dimensional subspace £ с H. Consequently, T(x ® v) = T(x) ® v
for a unique element T(x) g H. The map T : H -► H is easily seen to be
bounded, Ctk-linear and graded. Clearly T(T) = f. We have shown that
is surjective and therefore by the Open Mapping Theorem that is
a homeomorphism. It is easy to check that 4х carries the self-adjoint
Fredholm operators in &k(H) onto those in £fk + s(H ® V). This gives
the desired homeomorphism gk s 8* + s- 
Kuiper’s results [1] show that the group of C£k-linear isometries of
Я0 is contractible. This shows that the construction of the homeomor-
phism above is canonical up to homotopy.
Note that for each к 0 we have isomorphisms
KO-*(pt)	я0(&) Wk/i*Wk+1
where a is given by 10.11 and where ft is the Clifford index defined in
10.4. It is shown in Atiyah-Singer [5] that the map a ° Д"1 coincides with
the isomorphism given by the Aityah-Bott-Shapiro construction (cf. 1.9).
There is a natural ring structure in KO~* where the multiplication is
induced by the tensor product of operators. This is defined as follows. Let
Hk and Нг be graded Hilbert modules for C£k and C£^ respectively. The
graded tensor product Hk ® H, is then a module for Ctk ® C£^ = C€k+,
(see 1.5). Representing Rk+/f by 5к+/Нк ® Hk ® ЯД we define a map
®:Rfcx	(10.14)
by requiring that for elements v g Hk and w g of pure degree with
respect to the grading,
(S ® T)(v ® vv) = (Sv) ® w + (- l)de8l’p ® (Tw),
where S' g gk and T g ft,. One checks easily that S' ® T is C£k-linear and
C£rlinear, and therefore Ctk+/-linear. The inner product on Hk ® H, is
given by ® w, v' ® vv'> =	and S' ® T is clearly self-
adjoint. Since S' and T interchange even and odd factors, so does S ® T,
224
III. INDEX THEOREMS
and one computes that
(S ® T)2 = S2 ® Id + Id ® T2.
From this one sees that S ® T is Fredholm and that there is an
identification
ker(S ® T) = (ker S) ® (ker T).	(10.15)
The graded tensor product of modules induces a multiplication in
ЭД* = фкЯЙк, which descends to the quotient KO " *(pt) ©(9Wk/i*9Wk+1).
Hence, (10.15) gives the following:
Proposition 10.13. For Sefik and T e one has that
indk+XS ® T) = indk(S) ® ind/T).
It is straightforward to check that the map (10.14) above preserves the
subsets 8k, that is, it restricts to give a continuous mapping
x 3г-----* 3*+г-
Applying the isomorphism K0~k(A) s [A, <5k] of 10.11, we get a
multiplication
K0~k(A) x KO~'(A)-------► K0~k'(A)
defined for any compact Hausdorff space A and for all k, £ 0. In fact,
for any pair of such spaces A and B, we have a transformation
K0~\A} x KO^B)----------► K0~k~'(A x B)
given by associating to f'-A—>fa and 0:	8^ the map (fg)(a,b) =
f(a) ® g(b). (When A = В the multiplication is obtained by restricting to
the diagonal.) These transformations coincide with the ones convention-
ally defined in KO-theory (see 1.9).
Remark 10.14. Recall that the homeomorphism (10.12) identifies fa
with the space gskcw of all skew-adjoint Fredholm operators on real
Hilbert space. Consequently, Theorem 10.11 shows that ftskew is a clas-
sifying space for the functor KO~l.
Similarly, the assignment T -► £7T|Ho associates to each element
T g g7, a C£6-linear map of the ungraded Hilbert module Я6 for C£6.
Note that C£6 £ R(8) and we can take H6 to be the product H6 = H ® R8
with R(8) acting on the right-hand factor. Any R(8)-linear map of H6 is
then of the form A ® Id. This identifies g7 with the space gsymm of
all self-adjoint Fredholm operators on real Hilbert space, and Theorem
10.11 shows that gsymm classifies the functor KO~\
§11. MULTIPLICATIVE SEQUENCES
225
Remark 10.15. The entire discussion of this section can be carried
through in the complex case. One considers complex Clifford algebras,
complex graded modules, complex Hilbert spaces, etc. All of the analogous
fundamental results remain true. One essentially recaptures the “classical”
index theory discussed in §§7 and 8. However, a new fact which emerges
is that the functor Kl is classified by the skew-adjoint Fredholm operators
on complex Hilbert space. The analogue of Theorem 10.10 (also proved
in Atiyah-Singer [5]) leads to a proof of Bott Periodicity for the unitary
group.
§11. Multiplicative Sequences and the Chern Character
In this section we present some fundamental constructions in K-theory
and the theory of characteristic classes. These constructions will be needed
later when we discuss the cohomological formula for the index of an
elliptic operator.
Throughout the section cohomology groups will be taken with rational
coefficients, although much of what we do carries over to more general
coefficient rings.
There is a principle underlying much of what we do here. Roughly
stated it asserts that for computational purposes every complex vector
bundle is a direct sum of line bundles. Moreover, if the bundle is the com-
plexification of a real bundle, the non-trivial line bundles occur in complex
conjugate pairs. To make this precise we need the following result, often
referred to as the “Splitting Principle”:
Proposition 11.1. Let Ebe a complex vector bundle over a manifold X. Then
there exists a manifold ifE and a smooth, proper fibration n\SfE^>X such
that
(i) The homomorphism n*: H*(X) -► H*(SPE) is injective.
(ii) The bundle n*E splits into a direct sum of complex line bundles:
ф<>	(11.1)
Proof. Let p: P(E) -> X denote the projectivization of E, i.e., the bundle
whose fibre at x is the projective space P(EX) of all complex lines in Ex.
The bundle p*E contains a line bundle defined tautologically at a line
t c Ex to be I itself. Using some fixed hermitian metric in E, this gives us
a tautological splitting
p*E = t ф Л
The homomorphism p*: H*(X; Z)	Я*(Р(Е); Z) is injective by the Leray-
Hirsch Theorem C.14 in Appendix C.
226	Ш. INDEX THEOREMS
We now repeat the process for the bundle and continue inductively
to complete the proof. 
There is a direct analogue of the proposition and its proof for the case
of real vector bundles. We shall not state this. However, we do want to
signal the following “hybrid” result.
Proposition 11.2. Let E be an oriented real vector bundle of dimension 2n
over a manifold X. Then there is a smooth proper fibration n: ifE X such
that 7i*: H*(X) -► H*(SfE) is injective and the bundle n*(E ® C) splits into
complex line bundles:
n*(E ® С) ф ф • • • <, ф	(11.2)
where tdenotes the “inverse” or “complex conjugate” bundle to (see below).
In fact, there is a splitting
ф En	(11.3)
into oriented real 2-plane bundles such that Ek ® С = £k ф Skfor each k.
Proof. Suppose, to begin, that dimK(E) = 2. Fix a metric in E and let
J: E -> E be the map which rotates each fibre by я/2 in the positive di-
rection. We let £fE = X and note that
E ® С = £ ф t
where at x e X
£x = {t? — idv : v g Ex} and ёх = {v + iJv: v g Ex}, (11.4)
are the + i and — i eigenspaces of J ® C. Note incidentally that as com-
plex bundles we have the bundle isomorphism
E^S.	(11.5)
For the general case we fix a metric in E and consider the bundle
p: G(E) —> X whose fibre at a point x consists of all oriented 2-dimen-
sional subspaces of Ex. Then there is a canonical splitting
p*E = Er ф Ef
where Er -► G(E) is the tautological oriented 2-plane bundle whose fibre
at P g G(E) is P itself. The argument given for Theorem C.14 adapts im-
mediately to prove that the homomorphism p*: H*(X; Z) -► H*(G(E); Z) is
injective. Repeating the process for the bundle Ef and proceeding in-
ductively, we construct the desired splitting bundle ZfE. 
Remark 11.3. An analogous result holds when dimR(E) = 2n + 1. One
must add a trivial line bundle onto the decompositions (11.2) and (11.3).
§11. MULTIPLICATIVE SEQUENCES
227
For many purposes it is permissible to add a trivial real line bundle to E
and work directly in the even-dimensional case.
Note 11.4 (conjugate bundles). Recall that if E is a complex vector
bundle, then its conjugate bundle Ё is obtained from E by redefining scalar
multiplication. The new scalar multiplication by t e C is the old scalar
multiplication by L For any given hermitian metric (•,•) in E, the map
-►(?„(•) = (•,!?) identifies Ё with E* = Homc(E,C). If E = Eo ® C is the
complexification of a real vector bundle Eo, then there is a complex bundle
isomorphism E = Ё.
Note 11.5 (line bundles). Theset ^(X)^ Яx(X; Sx)of equivalence classes
of complex line bundles on a manifold X has a natural commutative multi-
plication given by the tensor product. Since	trivial, the
multiplication is invertible, and ^f(X) is a group. The first Chern class
cr:^(X) -> H2(X;Z) is a group isomorphism (see Appendix A, Example
A.5).
The Splitting Principle above can be applied directly to characteristic
classes.
Observation 11.6 (splitting the Euler class). Let E be an oriented real
vector bundle of dimension 2n, and let x(E) g H2"(X;Q) denote its Euler
class. If E decomposes into a sum of oriented 2-plane bundles E =
Er ф • • • ф En, then since x(E ф E') = z(E)/(E') we can write
X(E) = Xi - -x„	(11.6)
where xk = x(Ek) for each fc. If we now complexify and write E ® C =
Л Ф А Ф • ’ • Ф Ф then we see from (11.5) that
xk — ctfj for each fc.
Using the Splitting Principle, formula (11.6) can, in fact, be used to define
the Euler class.
Observation 11.7 (splitting the total Chern class). Let E be a complex
vector bundle, and denote by
c(E) = 1 + C1(E) + ... + c„(E)
the total Chern class of E. Recall that c(E ф E') = c(E)c(E'). Hence, if E
decomposes as a sum of line bundles E = ф • • • ф £n, then
c(E)= П (1 + **)	(И.7)
k= 1
where xk = ct(^k) for fc = 1,... ,n. In particular, we find that
Cj(E) =	... ,x„), j = l,...,n	(11.8)
where Oj denotes the jth elementary symmetric function of xn ... ,x„.
228
III. INDEX THEOREMS
If E does not split as a sum of line bundles, we may lift it to the space
ZfE where it does. The map Я*(Х) H*(SfE) is injective and identifies
with а/хх,... ,xn) as in the previous case. Note that any symmetric
polynomial expression in the x/s can be rewritten as a polynomial in
(i.e., in cx,... ,cn). This will enable us to define characteristic
invariants in a particularly useful way.
As a simple application note that since cx(Z) = — сх(/), we have
с/Ё) = (-1)ед	(11.9)
for all j.
Observation 11.8 (splitting the total rational Pontrjagin class). Let E
be a real oriented vector bundle of dimension 2n, and recall that the
rational Pontrjagin classes of E are defined by p/E) = ( — 1ус2/Е ® C),
j = 1,... ,n. (Since E ® С = (E ® C), the Chern classes of odd degree are
zero by (П.9).) The total (rational) Pontrjagin class is defined to be
p(E) = 1 4- p^E) 4-... 4- pn(E).
It has the property that p(E ® F) = p(£)p(F). If £ ® C decomposes as a
direct sum, E ® C =	®	®	®	® then c(E ® C) =
fj c(/’fc)c(Zfc) = П (1 — xk), and we And that
p(E) = П (1 + x*2)	(11.10)
k = 1
where xk = cx(/k) as before. In particular, we have
p7(£) = ay(x2,...,x2)	(11.11)
for each j.
Let Q[[x]y denote the set of formal power series in x with rational
coefficients and with constant term 1. It is easily seen that Q[[x]]" is a
group under multiplication. Fix an element f(x) e Q[[x]], and for each
neZ+ consider the formal power series in n indeterminates given by
/(xi) ’ ’ ’ f(xn)- This is evidently symmetric in the x/s, and so it has an
expansion of the form
/(*1) • • /(*») = 1 + F 1(^1) + ^2^1. °1) + ^1,^2. ff3) + • • •
where
ak(x1?... ,xn) =	£ xh • • • xik for 1 < к < n
ii < • <ik
denotes the fcth elementary symmetric function in x1?... ,xn, and where
Fk is weighted homogeneous of degree k, i.e.,
Fk(tol,... ,Лгк) = tkFk(ol,... ,ak) for all t e Q.
§11. MULTIPLICATIVE SEQUENCES	229
Each of the polynomials Fk(ax,... ,ak) is well defined and independent of
the number of variables Xj. This is easily seen by adding more variables
and using the obvious fact that ак(х1?... ,хл,0,... ,0) = ак(х1?... ,хл) if
к n, and ak(xb ... ,хл,0,... ,0) = 0 if к > n.
The sequence of polynomials {Fk(ax,... ,ak)}£°= x is called the multi-
plicative sequence determined by the formal power series f(x). It has a
universal multiplicative property which we shall now describe.
Let В be a commutative algebra with unit over Q, and assume that В
has a direct sum decomposition В = B° ® В1 ® B2 ®  • • with the prop-
erty that Bk • В/ с вк+/ for all fc, / 0. For example, В could be H2*(X;Q)
or H4*(X;Q) for a space X. It could also be the polynomial ring Q[x]
with Bk = Qxk.
Given such an algebra B, let B" denote the set of all formal sums
1 + br + b2 + ... where bk e Bk for each fc. Note that the finite sums (those
with only a finite number of non-zero terms) actually belong to В and
form a set which is closed under multiplication. This extends to B" by
defining
(1 + br + Ь2 + . . .XI + c 1 + c2 + • • •) = 1 + (^1 +cl) + (^2+C2)+ • • • ,
that is, by defining the nth term of the product to be £ bkcn_k. Every
element of B" has a multiplicative inverse, and so B" is an abelian group.
As an example, note that if В = Q[x], then B" = Q[[*]]'.
Fix a multiplicative sequence {Fk(al9... ,ffk)}2°sl. Then to each alge-
bra В as above we associate a map F: B" -► B" by assigning to
b = 1 + br + b2 + ... g B" the element
F(b)=l + F1(b1) + F2(b1,b2) + ... .	(11.12)
Lemma 11.9. The map F:B" -> B~ is a group homomorphism, i.e.,
F(bc) = F(b)F(c)
for all b,c g B~.
Proof. In the polynomial algebra В = Q[x1?... ,хл] consider the element
a = (1 + xx) • • • (1 + хл) = 1 +	+ ... + сгл g В". By definition of {Fk}
we have that F(a) = f(xr) • • • f(xn) — 1 + F^aJ + F^g^gJ -I-... . We
now increase the number of variables. Let В = Q[x1?... ,хл+т], and
consider the subalgebras B' = Q[x15... ,хл] and В" = <0>[хл + 1,... ,хл+ш].
Let g,g',g" be the corresponding elementary products in each case. Then
we have
a = (1+ xj • • • (1 + хл+т) = o'g",
and
F(a)=/(x1)---/(xn+m) = F(a')F(H.
230
III. INDEX THEOREMS
The general result now follows easily from the algebraic independence of
the a/s. 
It is easy to see that the universal multiplicative property 11.9 char-
acterizes the sequence. The formal power series is recaptured by taking
F(1 -I- x) = f(x) in the algebra Q[x].
The concept of a multiplicative sequence is due to F. Hirzebruch, who
established their importance in the theory of characteristic classes.
Basic Construction 11.10 (multiplicative sequences of Chern classes).
Let {Fk} be a multiplicative sequence associated to the formal power series
/(x)g Q[[x]y. To each complex vector bundle E over a space X, we
associate the total F-class
Fc(E) = F(c(E))€H2*(X;Q)".
Since c(E Ф £') = c(£)c(£'), this class has the property that
FC(E ф £') = FC(£)FC(£'),	(11.13)
for any two complex vector bundles £ and £' over X. If we decompose
E = /’i ф • • • ф 4 according to the Splitting Principle, then
FJE) =/(xx) • • •/(xn)	(11.14)
where Xj = ct(^) for each j.
Example 11.11 (the total Todd class). Associated to the formal power
series
„ x .1	1 ,
tdw = l—1 + ix + Tix +•••
1 — e	2	12
is the multiplicative sequence {Tdm} called the Todd sequence. The total
Todd class is denoted by Tdc- Its first few terms are:
TdJcJ^C!
Td2(c j,c2) = — (c2 + cf)
TdjfcuC^Cj) = 24 c2ci-
If X is a compact complex manifold of dimension n, and if £ = TX, then
the number Td(X) = Td„(TX)[X] (where [X] denotes the fundamental
class of X in H2<(I;Q)) is called the Todd genus of X.
Basic Construction 11.12 (multiplicative sequences of Pontijagifi
classes). Let {Ft} be a multiplicative sequence associated to the formal
power series /(x). To each real vector bundle £ over a space X, we asso-
§11. MULTIPLICATIVE SEQUENCES	231
ciate the total F-class
F(E) = F(p(E))eH4*(X;Qr.
Given two such bundles E and E' over X, we have p(E ф E') = p(E)p(E')
and so
F(E © E') = F(E)F(E').	(11.15)
Assume E is oriented and of dimension 2n, and decompose E ® C as
Л © Л ® • • • © 4 ® 4 according to the Splitting Principle. Then
F(£)=/(x?) -/(x2)	(11.16)
where xk = cr(^k) for each fc.
Example 11.13 (the total Л-class). Associated to the formal power series
ч/х/2	.	1	7	2
a(x) =------=— = 1 — — x +	.,> , x2 + ...
sinh(^/x/2)	24	27 - 32 5
is a multiplicative sequence {Am} called the Л-sequence. The first few terms
of the sequence are
A(Pi)= ~^Pi
Л^РнРг) = 2=7 ,y2 .~5 ( —4Рг + 7Pi)
^(РиРг-Рз) = ~2ю . Зз . 5 у (>6Рз “ 44Рг/’1 + 3,Pi)-
Given a real bundle E, the total Л-class of E is the sum
A(E) = 1 + A^PiE) + Л2(р1Е,р2Е) + ...
If we write E (x) C = f v © 7X © • • • © 4 © 7n as above, then from (11.16)
we see that
-	" x 72
A(E)= П	0 1.17)
./= i sinh(x;/2)
where of course pjE = cr/x2, ... ,x2).
Closely related to the Л-sequence is the Л-sequence {Лш} determined
by the power series d(x) = a(16x). One easily sees that Am = 16ШЛШ for
each m.
The Todd class and the Л-class are intimately related.
Proposition 11.14. For any oriented real vector bundle E it is true that
Tdc(E (x) C) = A(E)2.
232	III. INDEX THEOREMS
Proof. We may assume dim E is even (see Remark 11.3) and we consider
a formal splitting E ® C =	© Л © • • • ф /n © ?n- Then by definition
Tdc(£0C)-n , / /г/
where	Multiplying by exj/2e Xj/2 in the denominator gives
Tdc(E®C)- n Уе-,1П
J = 1 l_e	e
x/2 ~12
sinh(Xy/2)
= [A(E)]2. 
= n
Example 11.15 (the total L-class). Associated to the formal power series
„ 4 Jx 4	1	1 9
Дх) =-------= = I + - x - — x2 + ...
tanh^/x 3	45
is a multiplicative sequence {Lm} called the Hirzebruch L-sequence. The
first few terms of the sequence are
Li(Pi) = з Pi
Ь1(Р1,Р2) = ± (7p2 - pj)
Ь3(Р1,Р2,Рз) = Зз .5.7 (62Рз - 13p,p2 + 2p?)
Given a real bundle E, the total L-class of E is the sum
L(E) s 1 + L'(piE) + L2(plE,p2E) 4-... .
If we write Е®С = /1ф Z, ф-"Ф4Ф4 according to the Splitting
Principle, then from (11.28) we see that
<1И8>
where PjE =	. ,x„).
Closely related to the L-sequence is the L-sequence {Lm} determined
by the power series ?(x) = f(x/4). One easily sees that Lm = 4mLm. For a
real oriented bundle E of dimension n, we have
- n x 72
L(E) = П —(11.18')
Л i tanh(Xj/2)
§11. MULTIPLICATIVE SEQUENCES
233
Suppose we fix a multiplicative sequence {Fk} as above. Then for each
lifferentiable manifold X we define the total F-class of X to be
F(X) = F(TX) e H4’(X;Q)
n particular, the examples above give us a total Л-class A(X) and a total
L-class L(X).
If we furthermore assume that X is compact, oriented and of dimension
i, then we define the F-genus of X to be the rational number F(X) obtained
эу evaluating F(X) on the fundamental homology class [X] e Hn(X; Q) of
the manifold. In other words:
KX) = F(X)[X] = |0	if „^o (mod 4)
(1119)
From (11.15) one easily deduces that the F-genus is multiplicative in the
sense that
F(X x X') = F(X)F(X')	(11.20)
for any pair of compact oriented manifolds X and X'.
Suppose now that У is a compact oriented manifold with boundary
cY = X. Note that TY\X = TX ® (trivial), and so p(Y)(x = p(X). Con-
sequently, Г(У)|Х = F(X) and, since X is homologous to zero in У, the
F-genus of X must be zero. This proves that the F-genus of a manifold
depends only on its oriented cobordism class. In fact the F-genus gives a
ring homomorphism
F:Q*°-----► Q	(11.21)
from the oriented cobordism ring into Q.
Two important examples here are the Л-genus and the L-genus. An
important result of F. Hirzebruch says that for any compact oriented
4/c-manifold X,
L(X) = sig(X).
This can be proved by direct verification on a set of generators for the
ring Q^° (x) Q (cf. Hirzebruch [1]). It also follows from the Atiyah-Singer
Index Theorem.
Note in particular that L(X) is always an integer. This is not true for
Л(Х). The formulas above show, for example, that Л(Р2(С)) =
-(1/8)L(P2(C)) = — 1/8. Nevertheless, it will follow from the Index Theo-
rem that Л(Х) is an integer when X is a spin manifold. It is a fact, inciden-
tally, that the Л-genus is always an integer.
We now discuss the Chern character. Let E be a complex vector bundle
of dimension n over a manifold X, and via the Splitting Principle express
234
III. INDEX THEOREMS
the total rational Chern class of E formally as
c(E) — 1 4- Cj 4- . . . 4- cn = П (1 4- xfc)
4=1
so that ck =	. ,x„). Consider the expression
ch(E) = eXx + ... 4- ex” = n + £ XjF + 1 £ xj + ...	(11.22)
j=i lj=i
The term of degree к in this expression is just the symmetric polynomial
ch‘£=^-.i^	(1L23)
which can be rewritten as a universal polynomial expression in the ele-
mentary symmetric functions ct.... ,cn. In particular, ch E = n 4- ch1^ 4-
ch2E + ... is a well-defined element of H2*(X;Q). It is called the Chern
character of E.
Note that if £ is a complex line bundle, then
ch(^) =	(11.24)
where сД/’) denotes the first Chern class of /.
The importance of the Chern character lies in the fact that it respects
the (semi) ring structure on the set of vector bundles.
Proposition 11.16. The Chern character has the following properties for any
pair of complex vector bundles E and E' over X:
(i) ch(E © E') = ch(E) + ch(E')
(ii) ch(E ® E') = ch(E)ch(E').
Proof. Consider formal splittings
c(E)= П (1+xt) c[E’) =	(1 + x})
k=l	j=l
where as above the Chern classes are the elementary symmetric functions
of the x’s. Then we have corresponding splittings
n	m
c(E®E’)= П (1 +**) П (!+*;•)
k=l	J=1
c(E ® E') = П fl U + xk + x’J
j = 1 4=1
The first is obvious. The second is a consequence of the basic fact that
for complex line bundles cff ® Г) = cff) 4- cf/'). (Compare Note 11.5
§11. MULTIPLICATIVE SEQUENCES
235
or (A.7) in App. A.) By definition we now have
ch(E ® E') = J eXk
k=l
ch(E ® E') = f f
j=lk=l
+ f ex> = ch(E) 4- ch(E')
J=i
eXk+x' = ( f ^)( f eXj) = ch(E)ch(E') 
\k=i / v=i /
Corollary 11.17. For any compact Hausdorff space X, the Chern character
descends to a ring homomorphism
ch:K(X)----► H2*(X;Q).
Remark 1148. It is a result of Atiyah and Hirzebruch [2] that if, say,
X is a finite complex, then the associated map ch: K(X) ® Q -► H2\X\ Q)
is an isomorphism. They show, moreover, that this map extends to a ring
isomorphism ch: K*(X) ® Q H*(X; Q) which carries K\X) ® Q onto
Hodd(X;Q)
We now examine some basic constructions in К-theory. To make cal-
culations we shall always assume our bundles to be a direct sum of line
bundles. This is justified by the Splitting Principle 11.1 and 11.2 provided
the answer is independent of the splitting. We assume throughout that X
is a manifold or a finite simplicial complex.
Construction 11.19 (the exterior power operations). Let £ be a com-
plex vector bundle of dimension n over X and for each k, 1 < к < и, con-
sider the bundle AkE. This operation on vector bundles has the property
Лк(Е®Е')= X (Л‘Е) ® (A'E').	(11.25)
i + j = к
To extend the operation to К-theory we consider the ring K(X)[[t]] of
formal power series with coefficients in K(X). Assigning to a vector bundle
E the element
ЦЕ) = £ [A*E]t*	(11.26)
k»0
gives a map with the property that
ЦЕ © F) = А/Е)Л(П	(11-27)
by (11.25). From the universal property of K(X) the map (11.24) extends
to a group homomorphism
X, -K(X)---> K(X)[[t]f.
The к-component of this map is called the fcth power operation on K(X).
236	III. INDEX THEOREMS
For vector bundles, Ar(E) is a polynomial and zm(E) is a well-defined
element of K(X) for all me Z. For example, Ar(/) = 1 + t[f] for any line
bundle For general elements of K(X) the above statement is false. Note,
for example, that Ar(-[/]) = (1 + t[/])' 1 = £ (-t)w[/]w.
Fix a vector bundle E and consider a formal splitting E = /\ ф • • • ф 4
with xk = CJ4). From (11.27) we have that zr(E) = f] 2f(4) = П (1 +
and so
сЦл,Е) = П (1 + teXk).	Щ.28)
k- 1
In particular, we conclude that
ch(z_ XE) = ch(AevenE - AoddE) = f] (1 - ex*).	(11.29)
k = 1
Construction 11.20 (the Adams operations). Closely related to the
power operations are a family of ring homomorphisms фк: K(X) -► K(X)
called the Adams operations. For a line bundle /, they are defined by
setting,
W*0 = Л	(11.30)
The extension is then determined by the Splitting Principle. Write E ==
Л ф • • • Ф 4 and define фк(Е) = /\ ф • • • ф tkn. We see that
фк(Е ф E') = фк(Е) ф WE'D
фк(Е (x) E') = фк(Е) ® фк(Е').)
The first is obvious. For the second, note that W(E 4) ® (E 4’)] =
WE < ® 6] = E (6 ® Ok = E ® (Ok = (E 4) ® (E Ф =
WE <•) ® WE 4)-
Note that ch фкЕ = J ekXj = pkch E where pk: H2*(X; Q) -► H2*(X; Q) is
defined by setting pk = km on Н2ш(Х;О).
The Adams operations transform naturally under the homomorphism
/*: K(X) -► K(Y) induced by a continuous map f: Y -► X.
It is an interesting exercise to show that Ek°=i ( — 0*W£) =
— t[(d/dt)Af(E)]/Af(E).
Construction 11.21 (the Clifford difference element). Let E be a real
oriented riemannian vector bundle of dimension 2n, and let co£ =
inex* - * егп be the oriented unit volume element in the complex Clifford
bundle Ct(E) = Ct(E) ® C. Since = 1, we have a splitting Ct(E) =
С£+(Е)ф Ct”(E) where Ct^E^fl ± co£)Cf(E). We then define the
Clifford difference element
<5(E) = [Ct+(E)] - [Cf (E)] g K(X).
(11.32)
§11. MULTIPLICATIVE SEQUENCES	237
Suppose E' is another such bundle. From the fact that C£(E ф E') =
C£(E) ® C£(E') and cu£e E> = one sees easily that
<5(E © E) = d(E)6(E').	(11.33)
The proof of the Splitting Principle 11.2 shows that we may consider E
to be a direct sum of 2-plane bundles E = Ex ф • • • ф En9 and E ® C =
Л © 6 © ” ’ ® 4 © 4 where Ej ® С = ф
Consider therefore the case where dim E = 2. We have E ® С = ф Z
for some complex line bundle In fact, if at a fixed point x e X we choose
an oriented orthonormal basis (e19e2) of Ex9 then lx = C • (er — ie2) and
4 = C • (ex + ie2). Clearly C£(E) = C- l®C-cu£®/’®7, and since
a>E = ieve2 we find that
C£+(E) = C • (1 + a>E) © 7 and C£"(E) = C • (1 — cu£) ф Л
Since the bundles C • (1 + co£) are trivial, we find that for dim E = 2,
3(E) = [Z] - [/].
Applying the Splitting Principle and (11.33), we conclude that for E =
Ei ф • • • ф EH,
<5(E)= П ([4] - [4D
k=l
and so
ch[<5(E)] = П (e~Xk “ e*k)	(11.34)
k=l
where xk = ct(4) for к = 1,... ,n. From (11.6) we know that /(E) =
• • • xn. Consequently, one verifies that
Л ip ______ pi*k \
ch[<5(E)] = z(E) П -------------
k=l \ Xk /
Л /________ 0'“Xk/2,\
= (- l)"x(E) j] I-------------)(eXfe/2 + e“Xk/2)
= (-2)"v(E) П (s*nh(x*/2) Y f_____ъЛ____\
( 2)х(£)ЦД	J^tanh(Xt/2)J
From (11.17) and (11.18') we conclude the following:
Proposition 11.22. For any oriented real vector bundle E of dimension 2n,
one has the relation
ch[<5(E)] = (-2)"z(E)L(E)A(E)-2.
238
III. INDEX THEOREMS
Construction 11.23 (the spinor difference element). Let E and a)E be
as in Construction 11.21 and suppose E carries a spin structure. Let SC(E)
be the complex spinor bundle for E and consider the decomposition
Sc(£) = S£(E) © Sc(E) where S£(E) = (1 ± (Oe)Sc(E). We define the
spinor difference element to be
s(E) = [5+(£)] - [5-(E)] ё K(X).	(11.35)
It follows easily from the material of Chapter I that if E is another such
bundle, then
s(E © £') = s(£)s(£').	(11.36)
As above we may now restrict attention to the case dim £ = 2 where
£ ® C = £ © ?. Recall from (11.5) that as oriented 2-plane bundles £ (.
The fact that £ is spin is equivalent to the fact that /(£) = ct(£) is even, i.e.,
that ( has a square root ^1/2. The spinor bundle is given by
Sc(£) = £112 Ф ?1/2
One verifies that S£(E) = t1'2 and Sq(E) - ^1/2, and so
S(£) = [?V2] _ ^1/2]
Passing to the general case £ = Ex ф • • • ф En via the Splitting Principle
as above, we then have
«(£)= П(И/23-1А1/2]),
k= 1
and so
ch[s(£)] = П ^~Xk/2 ~ eXkl2)	(1L37)
k=l
This proves the following:
Proposition 11.24. For any real spin vector bundle E of dimension 2n, one
has that
ch[s(E)] = (—1 )"/(£)&(£)-1
§12. Thom Isomorphisms and the Chem Character Defect
We present here some material relevant to understanding the general form
of the Index Theorem. It is not necessary, however, for understanding the
cohomological formula for the index in the basic cases.
Let X be an oriented n-dimensional manifold which is not necessarily
compact. Let H*pt(X) denote the cohomology of the complex of rational
§12. THOM ISOMORPHISMS
239
singular cochains with compact support on X. (A cochain c has compact
support if there is a compact subset К cz X such that c(a) = 0 for any chain
a which does not meet k.) Poincart duality states that there is a canonical
isomorphism
^:Я?Р,(Х)^ЯВ_Р(Х)	(12.1)
for p = 0,..., n (see de Rham [1]). If Y is another such manifold of di-
mension m, and if f: Y -> X is a continuous map, then for each p with
p m — n there is a linear map
/:H?pt(K)----->H&m~*XX)	(12.2)
called integration over the fibre (or the Gysin homomorphism). It is de-
fined by setting /(w) =	where Д is the usual map induced on
homology.
An important example is provided by the following. Let E be an oriented
vector bundle of fibre dimension к over X. Consider the maps
я: E----► X and i: X-----------► E
where я is the bundle projection and i denotes inclusion as the zero sec-
tion. Since these maps are homotopy equivalences, я* and i* induce iso-
morphisms on homology with я*1* = Id. Consequently, the maps
я,: H^k(£)----> H?pt(X) and f,: H?pt(X)--------> Hpptk(E)
are isomorphisms for all p. Since я,/, = ^х1п^е^е1^^х = И, we see
that
7c! = (f!)’1.	(12.3)
Definition 12.1. The map i,: H£pt(X) -> Hppik(E) is called the Thom iso-
morphism of E for compactly supported cohomology.
Note that if X is compact, then Hppi(X) = HP(X). Furthermore, if DE de-
notes the disk bundle of £, then by excision and Lefschetz duality we have
/ЗД Hn_p(E) Hn_p(DE)	Hk + p(DE,dDE).	(12.4)
We thereby recover the Thom isomorphism in its more conventional form
(cf. Milnor-Stasheff [1]). We also have the following basic result:
Lemma 12.2. If X is compact, then for all и g H*(X) = H*pt(X), one has
i*i£u) = %(E) • и	(12.5)
where /(E) is the Euler class of E.
Proof. We shall only outline an argument (for full details, see Milnor-
Stasheff [1]). Let Z = i*[X] = i^x(l) be the class of the zero-section in
240
III. INDEX THEOREMS
The equation
Z(E) = i*i!(l)=i*^£1Z
is one of the standard characterizations of the Euler class. To wit, if c e
Hk(X), then
(z(E),c) = (S^ZJ^c) = (l,Zn i#c),
that is to say that x is given by intersection in E with the zero section.
For a general class w g Hp(X), we fix c g Hk+p(X) and note that
(z(E) • w, c) = (z(E), &xu n c) = (l,Z n i*№xu r\ c))
= (1,(i^xu) i*c) = (Яё^хЪ^с)
= (ци, i*c) = (1*ии, c) 
In К-theory there is a group Kcpt(X) analogous to the group H*pt(X)
discussed above. It consists of homotopy classes of triples [E, F; rr] where
E and F are complex vector bundles over X and where a is a bundle
isomorphism from E to F defined outside some compact subset of X (see
1.9). If X is a compact manifold, then Kcpt(X) = K(X). If U c= X is an
open subset of any manifold, then there is a natural inclusion homomor-
phism Kcpt(U) Kcpt(X).
Suppose now that n :E -► X is a complex vector bundle of rank к over
a manifold, and let i: X -> E be the inclusion as the zero section. Then,
as proved in Appendix C, Theorem C.8, there is a natural Thom iso-
morphism I,: Kcpt(X) Kcpt(E) of the form
it(w) = A_j • tc*w
where A_ t = [я*ЛеспЕ,7i*AcdE;a] and where о is defined at each non-
zero vector e in E by setting
ae = e л — (e*)L.
(The element e* is the dual of e under some fixed hermitian metric.) If
X is compact, then A-i = ij(l) is a well-defined element of Kcpt(E). When
X is not compact, the product A_t • n*w can still be shown to be a well-
defined element of Kcpt(E) (cf. Karoubi [2]). Roughly speaking, n*w has
compact support in the “X-directions” and A_t has compact support in
“fibre-directions”. If we restrict the element Л-i to the zero section, we
recover the element A.^EJg K*(X). This gives the following.
Lemma 12.3. If X is compact, then for all £ g K(X) = Kcpi(X). one has
/%Ю = Я_1(Е)-е	(12.6)
where JL^E) = [AevenE] - [AoddE] g K(X).
§12. THOM ISOMORPHISMS
241
Assume now that X and У are smooth manifolds and that f: X Y
is a smooth proper embedding. Assume furthermore that the normal
bundle N to f(X) is equipped with a complex structure. (Hence, dim Y —
dim X is even.) Under these circumstances we can define a natural mapping
/:Kcpt(X) -+ Kcpt(T)
by taking the Thom isomorphism i,: Kcpt(X) -> Kcpt(N) followed by the
map Kcpt(N) -► Kcpt(y) obtained by identifying N with a regular neigh-
borhood of X in У.
Observe now that for any proper embedding f:X У of manifolds,
the normal bundle to the associated (proper) embedding f*:TX TY
has a canonical complex structure. This normal bundle is just the pull-
back to TX of N ® N where N is the normal bundle to X. The first
factor is thought of as lying in “manifold-directions,” the second in “fibre-
directions.” The complex structure is given by
fO “Id\
T =	I
\Id 0 J
Consequently, for any proper embedding of manifolds f: X c-> У, there is
an associated map
/:Kcpt(TX)-----► Kcpt(Ty)	(12.7)
which is of fundamental importance in defining the topological index of
an elliptic operator.
In the last section we defined the Chern character ch: K(X) Heven(X).
This homomorphism has a direct extension
ch:Kcpt(X)----
to the case of compact supports. For any given complex vector bundle
7t: E —► X, we have defined Thom isomorphisms:
Kcpt(X) Kcpt(E) and Hcpt(X) Hcpt(£),
and it is natural to ask whether Z,ch = ch i,. This is not true in general
and the resulting “correction term” is of basic importance.
We assume from this point on that the manifold X is compact. Then
to each complex vector bundle я: E -► X we associate the class
I(£) = 7CjCh if(l)	(12.8)
Note that for any £ e K(X) we have Tc.ch i,£ = TtjChG^l) • 7r*{) =
7i,[ch i,(l) • ch 7i*c] = [7c,ch i,(l)]ch £, and so, since я, = (z,)"1 on H*pt(£),
G!)-1ch(i!O = 3:(£)cht	(12.9)
That is, T(£) is just the “commutativity defect” mentioned above.
242	HI. INDEX THEOREMS
Now it is not difficult to check that X(£) is natural, i.e., /*I(E) =
X(/*E) for any continuous map between manifolds. Consequently 1(E)
is a characteristic which we shall compute. Note that iiX(£) = ch it(l) =
ch A_p Applying i* and Propositions 12.2 and 12.3, we find that
X(EJZ(E) = i*iX(E) = i*ch A_t = ch /♦A-i = ch A.^E). This compu-
tation can be carried out for the universal bundle E over the classifying
space BU„ whose cohomology ring is a polynomial ring generated by the
Chern classes c19... ,c„. Here we are authorized to write the equation
If we split c(E) = П (1 + xj) formally as in §11, we find from equations
(11.6) and (11.29) that
From (11.11) and (11.9) we can rewrite this as
1(E) = (—l)"Tdc(E)-1	(12.11)
Assume now that £ is a real oriented riemannian vector bundle of even
dimension over X. Then one can define the basic element
6(E) = [я*СС+(£),я*С€‘(£);д] g Kcpt(E) (12.12)
where де = е- denotes Clifford multiplication by e. It is evident that
i*6(E) = 6(E), the difference element considered in 11.21. Arguing as above
we see that: x(£)n,ch 6(E) = f*i!7i1ch 6(E) = i*ch 6(E) = ch 6(E). Applying
the calculations of Proposition 11.22 then proves the following.
Propositon 12.4. For any oriented real vector bundle E of dimension 2n on
X9 one has
rc.ch 6(E) = (-2)"L(E)A(£)"2
If we now assume that £ has a spin structure, we can construct the
element
<£) = [7i*sa (£),	(£); д] в кер.(£)	<1213)
where again = e- denotes Clifford multiplication by e. Clearly, i*s(£) =
s(£) = the element discussed in Construction 11.23. Arguing as above and
applying the calculation of Proposition 11.24, we find the following:
Proposition 12.5. For any real spin vector bundle E of dimension 2n on X9
one has
Ttich s(£) = (—1)"A(E)“1
§13. THE ATIYAH-SINGER INDEX THEOREM	243
These last two propositions extend easily to the case of coefficients.
Given any element и e K(X), one immeidately verifies that
K{ch[5(E) • n*u] = (- 2)"ch и • L(E)A(E) ’ 2	(12.14)
^ch[s(E)n*u] = (-irchuA(£)-1	(12.15)
Indeed, note that n!ch(5(E)7t*u) = n1[ch 5(E)ch n*u] = n,[ch 5(E)n*ch u] =
[n,ch 5(E)]ch u. If и = [£'] is the class corresponding to a complex vector
bundle Ef over X, then
5(E)-я*и [я*СГ(Е)®Е', 7t*Cf"(E) ® Е';д],
s(£)-n*u S [n*S£(£)®£', n*S^(£)®£'; Д].
These elements S(E) and s(E) are fundamental. Using them, one can define
Thom isomorphisms in К-theory for E as follows:
Proposition 12.6. Let n:E -► X be an oriented real vector bundle of dimen-
sion 2n on X. Then the map
i*: K(X) ® Q----► Kcpl(E) ® Q given by if(u) = S(E) • я*и
is an (additive) isomorphism. If E is spin, then the map
i?: K(X)---► Kcpt(E) given by $(и) = s(E) • л*и
is an (additive) isomorphism.
For a proof of this proposition and a discussion of related results, the
reader is referred to Appendix C.
§13. The Atiyah-Singer Index Theorem
We present here the topological formulas of Atiyah and Singer for the
index of an elliptic operator on a compact manifold. We begin with the
general К-theoretic formula for which we give a detailed proof. Then,
using material derived above, we shall rewrite the formula in cohomo-
logical terms and work out the details in some important special cases.
Let X be a compact differentiable manifold of dimension n and consider
an elliptic operator Р:Г(Е) -► T(F) where E and F are smooth complex
vector bundles over X. Recall from §1 that the principal symbol a(P) of
P defines a class
<r(P) = [л*£,л*Р;а(Р)] e Kepl(TX)	(13.1)
where л: TX X is the tangent bundle of X. (We have identified Kcpl(TX)
with K(DX,dDX) since DX/dDX is naturally homeomorphic to the one
point compactification of TX.) Choose now a smooth embedding
244
III. INDEX THEOREMS
f: X	R5 into some euclidean space, and consider the induced map
f,: Kcpt(TX)---► Kcpt(TRw)	(13.2)
defined in (12.7). Follow this by the homomorphism
^:Кср1(ТГ)----->K(pt)^Z	(13.3)
where q: TRN -> pt is the canonical “scrunch” map taking TRN to a point.
Note that TRN = R5 ® RN = CN and q: CN -* pt can be considered as a
vector bundle. Viewed in this way, the map qt is just the inverse of the
Thom isomorphism i,: K(pt) Kcpt(CN). (It is also just the Bott periodic-
ity map. For an alternative view, consider the embedding TRN = R2JV ->
S2N = R2JV u {00} and the induced map Kcpt(TRN) -► K(S2JV) K(pt).)
Applying scrunch-shriek to f gives the index.
Definition 13.1. The topological index of P is the integer
top-ind(P) = qJ]a(P).	(13.4)
One must verify that this definition is independent of the choice of f.
To begin consider f =j°f where y:RN RN+N' is a linear inclusion.
The induced map f: Kcpt(TRN) -► Kcpt(TRN+N ) is just the Thom isomor-
phism for the bundle QN+N' CN, and one easily checks that qj\ = qj\
where q: TRN+N/ pt. If we are given two embeddings f0: X	RNo and
fi'.X RNl, then the embeddings jaf0:X RjVo+N1 and j\ °fY:X
RJVo+N|, defined as above, are isotopic. That is, Ft = tjY °	+ (1 —
tj/o 0 /о, 0 < t < 1, is a smooth family of embeddings. Applying the
homotopy invariance of Kcpt completes the proof that (13.4) is independent
of the choice off. One of the basic results in mathematics is the following:
Theorem 13.2 (The Atiyah-Singer Index Theorem [1]). For any elliptic
operator P over a compact manifold, one has
ind(P) = top-ind(P),
that is, the topological and analytic indices of P coincide.
Proof. For the purposes of the proof we introduce a special class of
operators. Let E and F be (smooth) com lex vector bundles over a com-
pact riemannian manifold X. An operator P g WDO^E, F) is called clas-
sical if its principal symbol is homogeneous of degree m in £ outside of
some compact subset of T*X, that is, P is classical if there is a constant c
so that ar<s(P) = tmo^(P) for all { g T*X with ||£|| c and for all t 1. If X
is not compact, we define the classical operators to be those which have
this property over every compact subdomain of X. The set of all such
operators will be denoted ^СОт(Е, F).
§13. THE ATIYAH-SINGER INDEX THEOREM	245
Given an operator P e ЧСОЛЕ, F) we can consider its asymptotic prin-
cipal symbol
r-> oo 1
defined for all £ in dDX = {£ g T*X: ||f|| = 1}. This gives us an exact
sequence
0----► '¥DOm_l(E,F)------► ^C0^E,F) T(Hom(n*E,n*F))---------------► 0
(13.5)
where я: dDX -> X is the bundle projection. The surjectivity of 6 is seen
as follows. Given a section s g Г(Нот(я*Е,я*Р)), extend s smoothly to all
of T*X so that it is homogeneous of degree m in £ for ||f ||	1. Given local
trivializations of E and F over a coordinate chart U on X, one easily con-
structs an operator P in 17 with principal symbol s. Let {Uj} be a finite
covering of X by such charts and let {%j} be a partition of unity subor-
dinate to this covering. Then the operator P = £ Xj^Xj € 'УСОЛЕ, F) has
principal symbol cr(P) = s, and the surjectivity is proved.
Our first main step in the proof will be to show that the analytic index
makes sense at the symbolic level and, in fact, gives a well-defined homo-
morphism ind:Kcpt(T*X) Z. We begin with a technical lemma which
will be useful later on. For this lemma, X is assumed to be a manifold
which is not necessarily compact but is of finite topological type.
Lemma 133. Let я. В -► X be a smooth, real vector bundle over X. Then
every element in Kcpt(B) can be represented by a triple of the form
(я*Е,я*Р;а) g <SPi(B)cpt where E and F are vector bundles on X which are
trivial outside a compact set, and where a:n*E я*Р is homogeneous of
degree 0 on the fibres of В (wherever it is defined).
Note that outside a compact subset of X, о is defined everywhere on
the fibres. At such points the homogeneity implies that a is in fact con-
stant on the fibres.
Proof. We know from Chapter I, §9 that any element in Kcpt(B) can be
represented by a triple (E0,F0;a0) where rr0:£0 -► Fo is a bundle equiv-
alence defined outside a compact subset К c= B. There exists a bundle
Eq on В so that the sum Eo © Eq is trivial, and we can replace (Eo, Fo; a0)
with the equivalent triple (E,F;a) = (Eo © Eq, Fo © Eq; a © Id). Then
there exist trivializations
tE-:E|(B_K) (В - K) x Cm and	(В - K) x Cm
246
III. INDEX THEOREMS
so that 5 = Tp 1 ° Tg. (Let tf be the assumed trivialization, and set tf =
тё off-1.)
Choose now a compact domain £2 c= X so that К c= B|n. Set E = i*E
and F = i*F where i : X В is the zero-section, and let te and tf denote
the restrictions of the trivializations tf and tf to E and F respectively.
We claim that over В there exist bundle isomorphisms
fE:E n*E and /f:F n*F (13.6)
which are compatible with the given trivializations over B|(X_n), i.e., which
have the property that
/e = te1otf and /f = tf1otf.
at points of B|(X_n). These isomorphisms are constructed as follows. Let
h: В x [0,1] -► В be the homotopy defined by h(bj) = th, and set £ = h*E
and 2F — h*F. Note that
^|вх{о} ~ ft* E, *^|bx{o} = ft*E
^|в X {1} = Д *^|вх{1} = F
Introduce connections on S and & which extend the canonical flat con-
nections (compatible with the trivializations) over B|(X_n). Parallel trans-
port along the curves b x [0,1] gives the desired maps (13.6). The bundle
map a = fF ° J ° fE 1: tc*£ -► tc*F is an isomorphism which is defined on
В — К and constant on the fibres of В — л-1(£2). Fix r > 0 so that К cz
{b g B|n: ||b||	r}. We now redefine в in the set where \b r so that it is
homogeneous of degree zero (by setting	for |£ = r and t 1).
This gives the desired triple and completes the proof of the lemma. 
Suppose now that X is a compact manifold and choose an element
и g Xcpt(T*X). Represent и by an element (я*Е, n*F; a) as in Lemma 13.3.
Fix an integer m. From the discussion above we know that we can choose
an (elliptic) operator P g ТСОт(£, F) whose asymptotic principal symbol
is exactly a, and so in particular a(P) = u. We now set
ind u = ind P	(13.7)
and show that this definition is independent of all the choices involved.
We know from §7 that ind P depends only on the homotopy class of its
principal symbol. Now if P' e УСО^Е, F) satisfies a(P') = <r(P), then a(P')
and c^P) are homotopic (rel oo). Therefore, ind P is independent of the
choice of P with a given asymptotic principal symbol. It is also indepen-
dent of the homotopy class of the representative (rc*E,rc*F;a) of u. To see
this suppose that (л*£', rc*F';a') is another such representative and that
there exists an element a = (E, F;a) € St\(T*X x [0,l])cpt whose restric-
§13. THE ATIYAH-SINGER INDEX THEOREM
247
tion to T*X x {k} is (л*Е(к), n*F(k); <т(к)) for к = 0,1. The argument used for
Lemma 13.3 applies here to prove that a can be replaced with an element
of the form (л*Ё,л*Г;(7) where Ё and F are bundles on X x [0,1] and
where d is homogeneous of degree zero outside a compact set. Given mth
order operators P and P' associated as above to these representatives, one
can easily use the element (л*Ё, 7t*F;(7) to construct a homotopy between
them and thereby show that ind P = ind P' as claimed.
Suppose now that we have two distinct representatives uk =
(7t*Ekin*Fk;<Tk) for our class u, and that we have chosen associated zero-
order operators Pk where к = 0,1. By the definition of the equivalence
defining L^T^X)^ s X(T*X)cpt this means that there exist elementary
complexes ek = (7t*Gk,7t*Gk;Id), к = 0,1, so that a0 ф e0 and ax ф ex are
homotopic. We can choose associated operators Pk = Pk ф Id for ak ф ek,
and one easily sees that ind P'k = ind Pk for each k. Hence, we have
ind PQ = ind P1? and so ind u is well defined at least if we choose operators
of order m = 0.
The definition is also independent of the choice of the order m. To see
this, suppose we are given an elliptic operator P e ^PCO^E, F). Choose a
metric and a unitary connection on F, and let V*V be the associated
laplacian on E (see II.8.3). Then for any integer S, we consider the com-
position Pg = P ° (1 4- V*V//2 g '¥COm+t(E,F). It is easily seen that
6(Pg) = o(P) and since (1 4- V*V//2 is invertible, that ind Pg = ind P. It fol-
lows that ind и is well defined using operators of any order.
We have now shown that for any compact manifold X, the analytic
index (13.7) gives a well-defined homomorphism:
ind:Xcpt(T*X)----->Z	(13.8)
Our task now is to prove that this coincides with the homomorphism
top-ind defined above. This will be accomplished if we can establish the
following two properties:
Property 1. In the special case where X = T*X = pt, the homomorphism
ind: X(pt) -> Z is the identity.
Property 2. If X and Y are compact manifolds and f'.X^Yisa smooth
embedding, then
ind(u) = ind(/tu)
for all и g Kcpt(T*X), where f is the homomorphism (12.7).
To see that these properties suffice to prove the theorem, we first choose
an embedding f:X<^SN and let j:pt SN denote the inclusion of a
248
III. INDEX THEOREMS
point. By Property 2 we have ind(u) = ind(/u) = ind(y," ^u), and by Prop-
erty 1 we know that ind ° у,"1 = y,"1 = q,. We conclude that ind(u) =
q<Mu) = top-ind(u).
Property 1 is easily established. Each element in K(pt) can be repre-
sented in the form [Cr] — [Cs] where (Cr,Cs) g =2\(pt) is a pair of vector
spaces. An elliptic operator on this pair is just a linear map P:Cr -► Cs,
and we see that ind P = r — s.
Property 2 is more difficult to establish. Following Atiyah and Singer
[1], we split this up into more easily established properties. The first one
shows that ind is well defined over open manifolds of finite topological
type.
The Excision Property 13.4. Let О be an open manifold, and let
and	f':(9c—► X'
be two open embeddings into compact manifolds X and Xf. Then ind ° f =
indo f\ on Kcpt(T*0).
Proof. Fix и g Kcpt(T*&). By Lemma 13.3 we know that и can be repre-
sented by a triple (я*Е, n*F; o) where E and F are bundles over О which
are trivial outside a compact subset of О and where a is homogeneous
of degree 0 outside a compact subset of T*@. In particular, outside a
compact set Q c= О there are trivializations
т£:£|(г-а)-^^-«)хСи and rF:%_n)- Q) x C"1
(13.9)
with respect to which ахЛ = ox = (tf)~ 1 ° (t£)x at all points (x,{) g
T*(6? — Q). This means that over T*(0 — Q) the morphism a comes from
a bundle map o0:E -> F over the base. Moreover, with respect to the
trivializations (13.9), oQ becomes the identity mapping, i.e., а0(гх,... ,zm) =
(z1?... ,zm) at all points x g О — Q. Recall that a bundle map cr0 g
T(Hom(E,F)) is just a differential operator of order zero.
We now choose a zero-order elliptic operator P g WCOo(E, F) which
has symbol o(P) = a outside a compact set in T*0 and which is the
operator oQ = Id in О — Q. Such an operator clearly exists.
Suppose now that we are given an open embedding f:O^X. Using
(13.9), we extend the bundles E and F trivially over X — f(O), and we
extend the operator P to be the identity there. This defines an elliptic
operator fP on X with the property that
MZP)]=Z[a(P)]=/;u.	(13.10)
Clearly any element in ker(/IP) has support in Q and hence belongs to
the subspace ker P (under the natural embedding ker P cz ker f P given
§13. THE ATIYAH-SINGER INDEX THEOREM
249
by extending by zero). Hence, dim(ker fP) = dim(ker P). The same re-
marks apply to the adjoint (fP)*. Assuming that X is compact, we con-
clude from this and from (13.10) that
ind(/ju) = ind(/jP) = dim(ker P) — dim(ker P*).
Since the right hand side is independent of f, our assertion 13.4 is proved.
The Multiplicative Property 133. Let X and У be compact manifolds. Then
for all elements и e KepfT*X) and v e Kevi(T*Y) we have that
ind(u  v) = (ind u)(ind t>)
(Ш1)
Proof. Naively the argument goes as follows. We represent u and v as
above by first-order elliptic operators
Р:Г(£)—>Г(Р)	and Q.T(£')—► T(F)
over X and У respectively. We introduce metrics and define a “graded
tensor product”
Р:Г((£®£') ф (F®F)) —► T\(F®E') ф (£®F)
by
О = Г01
\i®e
-i®e*\
p*®i)
(13.12)
Note that £®£', etc., here denotes the exterior tensor product over
X x У. The operators P® 1, etc. are uniquely determined by requiring
that for q> e Г(£) and t/r e Г(£') we have (P® 1)(<р(х)® ф(у)) = (Pcp(x)) ®
ф(у). Using the fact that P® 1 and 1 ® Q commute, one easily computes
that
p*p®i + i®e*e
о
n_. /pp*®i + i®e*e
DD -(	0
0
PP*® 1 + 1®QQ
0
P*P® 1 + 1®QQ
(13.13)
Note that: D*Dcp = 0 => (D*D<p,<p) = (D(p,D(p) = 0 => D(p = 0. Hence,
ker D*D = ker D. Furthermore, since is diagonal, it suffices to com-
pute ker D*D separately on each summand, E ® E' and F ® F'. Given
(ре Г(Е ® E'), we see that: D*D(p = 0 => (Р*Рф,ф) 4- (Q*Q<p,<p) = 0 =>
||Рф||2 + ||6ф||2 = 0 => Рф = Qcp = 0 (where R denotes R ® 1 or 1 ® R,
whichever is appropriate). Note that ker P n ker Q s ker P® ker Q.
250	HI. INDEX THEOREMS
Continuing in this fashion we deduce that
ker D = ker D*D (ker P ® ker Q) ф (ker P* ® ker Q*)
coker D ker D* = ker DD* s (ker P* ® ker Q) ф (ker P ® ker Q*)
are therefore in К (pt)
[ker D] - [coker D] = ([ker P] - [coker P])([ker Q] - [coker g]).
In particular we have
ind D = (ind P)(ind g).	(13.14)
Now the principal symbol of D is exactly the (outer) tensor product
of the symbols of P and Q. Therefore, naively we have established (13.11).
However, there is one technical flaw in the argument. This is the fact that
for P e ФС0 i(E,P), the operator P ® 1 g WDO^E ® E',F ® £') does not
in general belong to the class lPCO1(E ® £', F ® E') because its principal
symbol is not homogeneous outside a compact set in T*(X x У). It is only
homogeneous outside a uniform neighborhood of the “T* У-axes” in
T*(X x У).
This flaw is repaired as follows. We shall construct a continuous fam-
ily of operators (P ® l)c g ЧСО^Е ® E', F ® £') for £ > 0 such that
lim£S0 (P ® 1)£ = P ® 1, where this limit is taken in the space of bounded
linear maps from Lf(£ ® £') to L%(F ® £'). Applying the construction
to each entry in (13.12) will give us a family of elliptic operators Ds in
TCO1 such that lim£40 Ds = D as bounded (Fredholm) maps between
Sobolev spaces. From the local constancy of the index (see 7.3.) we have
ind D£ = ind D for all £ > 0.
On the other hand it will be evident from the construction that given any
compact subset К c= T*(X x У), there exists a constant eK > 0 so that
ct(D£) = o-(D) on К for all £ £K. It follows by excision that [a(DJ] =
[ct(D)] — и • v for all £ sufficiently small. Hence, ind(u • u) = ind(DJ =
ind(D) = (ind P)(ind Q) = (ind u)(ind u), and the property will be
established.
It remains to construct the operator (P ® 1)£. This is done by multi-
plying the symbol of P ® 1 by a function iA£(|{|,|^|) of the cotangent
variables (£,fj) g T*X x T*Y. This function is constructed as follows. Fix
a C°° function ф:	-> [0, 1] such that 0(t) = 0 for t 1 and 0(t) = 1 for
t 2. Then for £ > 0 and for r,s 0, set ^£(r,s) = 1 — (fay/r2 4- s2)0(£s/r).
In multiplying the symbol of P ® 1 by ^C(|£|,M), one can use a good co-
ordinate presentation or some symbol calculus. The choice of method is
not critical. It is a straightforward and worthwhile exercise to check that
the resulting family (P ® 1)E has the properties claimed above. This com-
pletes the proof of 13.5. 
§13. THE ATIYAH-SINGER INDEX THEOREM
251
We shall actually need the multiplicative property in the more general
context of twisted products, i.e., fibre bundles. However, we only need to
consider the special case of sphere bundles which arise from vector bundles
by adding a section at infinity. More specifically, let я: P -♦ X be a prin-
cipal OB-bundle over a compact manifold X and consider the associated
bundles
F = PxOrR" Z = PxOnS"	(13.15)
where O„ acts on R" by the standard representation and acts on S" by
extending this representation to the one-point compactification on R".
(that is, O„ acts on S" by trivially extending the standard representation
to R"+1 = R" x R1 and then restricting to the unit sphere.) We define a
product
K^(T*X) ® KOn(T*S\„---------► Xcpt(T*Z) (13.16)
as follows. Choosing a metric in Z we get a splitting T*Z = n*T*X ф
T(Z/X) where T(Z/X) = T*Z/n*T*X denotes the tangent spaces along
the fibres of the projection n: Z -» X. This splitting gives us a multipli-
cation
Kcpt(T*X) ® KcptT(Z/X)-------► Kcpt(T *Z).
(Given a direct sum of vector bundles E ф E' on X, the map on
Kcpt(£) ® Kcvl(E') is defined by first taking the outer tensor product on
252
III. INDEX THEOREMS
E x E' over X x X and then restricting to the diagonal.) Combining this
with the composition
KOn(T*S")cpt - KOn(P x T*S")cpt - Kcpt(P xOn T*S") = Kcpt(T(Z/X))
gives the desired multiplication (13.16).
The associated bundle construction, which associates to a linear repre-
sentation p: Ол -> ON the vector bundle Vp = P x p extends naturally
to a homomorphism
аР:К(Ой)------(13.17)
The ring Kcpt(T*X) is naturally a K(X)-module. Therefore, via (13.17) it
becomes an K(OJ-module. We are now in a position to state the main
property.
The Multiplicative Property for Sphere Bundles 13.6. Let Z be an Sn-bundle
defined as above over a compact manifold X. Then
ind(u • u) = ind(u • indOnu)
for all и e Kcpt(T*X) and v e KOn(T*S")cpt.
Proof. The proof of this fact follows very much the argument given for
13.5. We represent и and v by first-order elliptic operators P and Q re-
spectively. (Q is an O„-equivariant operator on O„-bundles.) Using local
trivializations of the bundle, we cover P by a finite number of product
neighborhoods {Uj x Ол}*=1. We lift the operator P back over each pro-
duct and glue together with a partition of unity with respect to {Uj} on
X, to get an 0,,-invariant operator P on P. We now consider the tensor
product operator D on P x S" defined as in (13.12) (with P replaced by
P). This is an 0„-operator and can be pushed down to give an operator
D on the quotient Z. Notice that “pushing down” is equivalent to restricting
D to the subspace of sections coming from the base. It is easily checked
that a(D) represents the class u • v where the multiplication is that defined
in (13.16) above.
It remains to compute the analytic index of D in terms of ind P and
indOn Q. We shall work upstairs with the operator D restricted to sections
coming from the base. Using (13.13) and the arguments which follow it,
we see that:
ker D = ker 5*5
= (ker(P ® 1) n ker(l ® Q)) ф (ker(P* ® 1) n ker(l ® Q*))
with an analogous statement for ker 5* = coker 5. Since the operators
P ® 1 and 1 ® Q commute, we can carry out the computation in steps,
that is, we first pass to the kernel of 1 ® Q (or 1 ® Q* whichever is rele-
§13. THE ATIYAH-SINGER INDEX THEOREM	253
vant) and consider the operators P ® 1 and P* ® 1 acting there. For
example, on Г(£ ® £') the space ker(l ® Q) consists of those sections ф
which when restricted to each factor {p} x S" с P x S", lie in the space
£„(p) ® ker Q. To say that ф comes from a section on the base means that
ф satisfies the transformation law: ф(рд~ Sjjx) = рдф(р,х) for g e O„, where
p is the natural representation of O„ on ker Q. This means precisely that
ф corresponds to a section <p over X of the bundle £ ® ker Q where ker Q
is the vector bundle on X associated via the principal bundle P to the
representation p. Therefore, in passing to ker(l ® Q) and ker(l ® Q*)
the operator D descends to an operator on X on the form Pk + P*. where
Pk: Г(£ ® ker Q) ----* T(F ® ker Q), and
Ри: Г(£ ® ker Q*)-----> T(F ® ker Q»).
It follows that ind D = ind Pk — ind P£> = ind[P ® (ker Q — coker Q)] =
ind[u • indOn v] as claimed. 
From the properties we have established, matters can be easily reduced
to computing some simple cases. However, it is unavoidable that one must
compute the index of some operator at some point. We do this now.
Lemma 13.7. Consider the n-sphere S" to be an On-manifold under the re-
liction of the standard representation on R" ф R => S" (i.e., by rotations
about an axis). Let i: pt S" denote the inclusion of one of the two fixed-
points of the action. Then
indoJiJ^lefqO.)
froof. Consider the operator Z>°:CC° -* Cf1 with respect to the stan-
dard metric on S*. Recall that D° is just the de Rham-Hodge operator
d + d*: Aeve” -» Aodd. This is an ©„-operator and from Hodge Theory (II.5)
we see easily that indOii(£>°)= [H°] +(-l)"[H"]. The action of O„ on
H° = {constant functions} is always trivial. The action on H" = R{the
volume п-form} is trivial if and only if n is even. Therefore, we have
„	(2 if n is even
indoD° = <	„•
"	(1 — £ if n is odd
where i represents the non-trivial 1-dimensional representation on ©„•
We leave the details of the computation of the symbol class of D° to the
reader. One finds that in Ko„(S")
"is even
'J ((1-^(1) if n is odd.
Combined with the above, this completes the proof of the lemma. 
254
III. INDEX THEOREMS
We now complete the proof of the Index Theorem. We have shown
that it will suffice to establish Property 2, namely that ind = ind о /, for
embeddings f: X Y. By the Excision Property we may replace the com-
pact manifold Y with a tubular neighborhood of X in У, which is diffeo-
morphic to the normal bundle of X in Y. Consequently it will suffice to
prove that
ind u = ind(/ju) for all u g Kcpt(T*X)
where V is a vector bundle over X and f: X И is the inclusion of the
zero-section. Again by the Excision Property we may compactify V by
passing to the associated sphere bundle as in (13.15). We then apply the
Multiplicative Property for Sphere Bundles 13.6, with v = irl. Using
Lemma 13.7 we find that: ind(u • i,l) = ind(u • indo„(U)) = ind(u). How-
ever, by definition fu = и • i, 1, and the proof is complete. 
The remainder of this section will be devoted to deriving certain co-
homological formulas for the topological index. The most general one is
the following:
Recall that for any manifold X, the tangent bundle TX is canonically
an almost complex manifold since T(TX) = я*ТХ ф я*ТХ n*TX ®
C. This gives TX a canonical orientation as a manifold, A positively
oriented basis of T(TX) is of the form (e1,Je1,e2,Je2,... 9en9Jen) where
ei9... 9en is a basis of л*ТХ and J carries the “horizontal” to the “vertical”
factor. With this orientation we can evaluate any element ug H^TX)
on the fundamental class [TX] of the manifold. The result is denoted by
u[TX],
Theorem 13.8. Let P be an elliptic operator over a compact manifold X of
dimension n. Then
ind P = (- l)"{ch a(P) • A(X)2}[TX]	(13.18)
where A(X) denotes the total А-class of X pulled back to TX.
Proof. We consider first the scrunch map q: TRN = ф = CN -► pt.
Consider this as a complex bundle and let i: pt CN be the inclusion
as the origin. Fix an element и g Kcpt(CN) and apply the defect formula
(12.9) with и = i£. Recalling that = (i,)-1, that 3(CN) = 1, and that
ch: K(pt) -> H°(pt) is an isomorphism, we find that: g,ch и = q,u. Since q,
on H*pt is just integration over the fibre, we find that
q{u = ch u[71Rn].	(13.19)
Consider now a real vector bundle p: v -► X and let i: X -► v denote
the inclusion as the zero-section. Taking derivatives gives the bundle
§13. THE ATIYAH-SINGER INDEX THEOREM	255
p: Tv -» TX with zero-section i: TX -> Tv. One sees easily that the bundle
p:Tv -» TX is equivalent to n*v ф n*v = n*v ® C. While TX is not
compact, the map i is proper and equation (12.9) can be shown to hold
for elements of Kcpl(TX). Thus for any a e КсрЛ(ТХ) we have
Pjch = 3(v ® C)ch <7.
Evaluating on the fundamental class and recalling that p, is integration
over the fibre give the formula:
(ch i,tf)[Tv] = {3(v ® C)ch <r}[TX].	(13.20)
Consider now an embedding f: X R1* with normal bundle v. We
identify v with an open tubular neighborhood of f(X) in RN. Similarly we
have an open embedding
Tv <= TRN	(13.21)
as a tubular neighborhood of f(TX). Given any a 6 Kcpt(TX), the class
i,e has compact support in Tv and naturally extends to TRN under the
open inclusion (13.21). This extended class is, by definition, the element
/jff e Kcpt(TRN) given in (13.2). In particular we have
ch(i(ff)[Tv] = ch(»[TRN].	(13.22)
Combining (13.19)-(13.21) gives
ind P = {3(v ® C)ch ff(P)}[TX].
It remains to identify 3(v ® C). For this we note that since v is the normal
bundle to X, we have TX ф v = (trivial). Since 3 ‘s multiplicative, this
means that 3(v ® С) = 3(TX ® C)-1. Applying formula (12.11) we see
that
3(TX ® С)"1 = (- l)"Tdc(TX ® C).	(13.23)
We have used here that TX ® C is self-conjugate. For the final step we
invoke Proposition 11.14. 
Integrating over the fibre immediately gives the following.
Theorem 13.	8 (the cohomological formula for the index; Atiyah-Singer
[2]). Let P be an elliptic operator on a compact oriented n-manifold X and
let в = <т(Р)е Kcpl(TX) denote the symbol class of P. Then
n(n 4- 1)
ind P = (-1) 2	{л,ch a) • A(X)2}[X]
where it;TX -> X is the bundle projection.
(13.24)
256
III. INDEX THEOREMS
л(и+ 1)
Note. The factor (-1) 2 compensates for the difference between the
orientation on TX induced by the one on X, and the canonical orientation
described above.
We now consider two important special cases.
Theorem 13.	9. Let X be a compact oriented manifold of dimension n = 2m
and consider the signature operator D+ : Г(С£+(Х)) -► Г(С£“(Х)). Then
indD+ = L(X) = sig(X)
More generally, if E is any complex vector bundle over X, then the index of
De : Г(С£ + (Х) ® E) Г(СГ(Х) ® E) is given by
ind(D+) = {ch2E • L(X)}[X]	(13.25)
where by definition ch2E = £ 2kchkE.
к
Proof. Clearly we have that <r(D+) = 3(TX) where S is defined in (12.12).
By Proposition 12.4, n,ch^(TX) = ( —2)mL(X)A(X)-2. Applying the for-
mula (13.24) above, we find that
• j	fL(X) ifmiseven
ind D = 2L X) =	.c
(0	if m is odd.
(A direct identification of ind D + with the signature of X was carried out
in II.6.2). For the more general case we apply the formula (12.14) and
(12.16). We conclude similarly that
ind(D^) = 2w{ch E • L(X)}[X].
Writing this out and using the fact that LZ = 2-2ZLZ, we find that
2m{ch E-L(X)}[X] = Z {2«chkE-LX^)}m = Z {2m~2'chkE-L/X)}[X]
= £ {2kchkE • ЬДХ)}[Х] since the sum is over (k/) with 2Z + к = m. This
proves (13.25). 
Similar arguments give the following in the spin case:
Theorem 13.1	0. Let X be a compact spin manifold of dimension n = 2m and
consider the Atiyah-Singer operator 0 + : Г($с(X))	Г($ё(С)). Then
ind = Л(Х).
More generally, if E is any complex vector bundle over X, then the index
of К :W(X) ® E) -+ Г(^(Х) ® E) is given by
ind(^) = {ch E • A(X)}[X].	(13.26)
§13. THE ATIYAH-SINGER INDEX THEOREM	257
Proof. Note that a(D+) = s(TX) where s is defined in (12.13). By Proposi-
tion 12.5, T^chsfrY) = (— l)mA(X)-1. Applying formula (13.24) above, we
find that ind p+ = A(X)[X] = A(X).
For the more general case we apply formulas (12.16) and (12.15) to con-
clude that «(Ch^TX) ® E] = ( — l)mch E • A(X)“ 4 Therefore ind(Dg) =
{ch E • A(X)}[X] as claimed. 
Remark 13.11. For a compact spin 2m-manifold X, formula (13.26) is
equivalent to the full Index Theorem. This is seen as follows. Any elliptic
zero,P0 s (1 + P*P)~ 1/2P, which has the same index and the same symbol
class в e Kept(TX). By the Thom isomorphism 12.6, о can be written in
the form e = s(TX) • ?r*u for some u e K(X). (We use here that X is spin.)
Writing u = [E] — [F] for vector bundles E and F on X, we see that
в = [я*$с ® E, ® E‘> д] —	® F, ® F; д]
i.e., at the level of the principal symbol, P is equivalent to the difference
of two Atiyah-Singer operators with coefficients. This means essentially
that P © is homotopic to Therefore, ind P = ind — ind f)p =
{(ch E - ch F) • A(X)}[X] = {ch и • A(X)}[X], However, (12.15) gives
я,® = ?t|(s(Ar)rt*u) = (— l)mch и • А(У)”', and ind P = {np • А(У)2}[Х] as
claimed.
For non-spin manifolds, one can argue similarly by using the signature
operator with coefficients.
One interesting corollary of the index formulas above is the following:
Theorem 13.12. On an odd-dimensional compact manifold, the index of
every elliptic differential operator is zero.
Note that this result does not remain true for pseudodifferential operators.
Proof. Consider the diffeomorphism c: TX -* TX given by c(v) s — v and
note that if dim X is odd, then сф[ТХ] = — [ТХ]. Let P be an elliptic
differential operator of degree m with principal symbol a(P). Since a _^(P) =
(— lytf^P), we see that c*o(P) = (— 1)"<t(P). Since a(P) and —a(P) are
regularly homotopic (by <r(t,P) = e*u0(P), 0 <; t £ 1), they define the same
elements in К-theory, and we conclude that c*a(P) = <r(P). Applying for-
mula (13.18) now gives
ind P = - {ch a(P)  A(X)2}[TX]
= —c*{ch <r(P) • А(Х)2}сф[ТХ]
= -{ch c*o(P) • А(Х)2}сф[ТХ]
= — {ch o(P) • A(X)2}( —[TX])
= -ind P. 
258
III. INDEX THEOREMS
Many important elliptic operators on a manifold X arise in the follow-
ing way. Suppose the structure group of X can be reduced to a compact,
connected subgroup G SO2m (where dim X = 2m), and that P: T(E) -►
F(F) is an elliptic operator where E and F are vector bundles associated
to unitary representations, pE and pF respectively, of G. To apply the index
formula (13.24) to P we must compute Ttjch a(P) where л: TX -► X is the
bundle projection.
To do this we pass to the universal case. Let тг: f -► BG denote the
universal 2m-plane bundle associated to the inclusion G c= SO2m, and let
E,F be the complex bundles over BG associated to the representations pE
and pF respectively. Let a = [я*Ё, tt*F; a] g Kcpt(T) be any elliptic symbol
from Ё to F. Then from (12.5) and the fact that ttj = (i,)-1, we have that
X(T)7TjCh a = i*f!7r!ch a = i*ch a = ch Ё — ch F.
The algebra H*(BG;Q) always embeds in the polynomial algebra H*(BT;Q)
where T c= G is a maximal torus. Therefore, if x(T) / 0, we can write:
TtiCh a = (ch Ё — ch F)/x(f). Pulling back to X by the classifying map for
TX (with its G-structure) gives the following corollary to Theorem 13.8.
Theorem 13.13. Let X be a compact 2m-manifold with structure group
G cz SO2m as above. Let P: Г(Е) T(F) be an elliptic operator where E and
F are associated to unitary representations of G. Suppose that the image of
the Euler class x under the map H2m(BSO2m) H2m(BG) is not zero. Then
the characteristic class (ch E — ch F)/x(TX) g H*(X; Q) is well defined, and
ind P = (- l)4-ch E-~^ F • A(X)4[X],
I ZUл)	)
Example 13.14 (The Riemann-Roch-Hirzebruch Formula). Let X be a
compact complex manifold with a hermitian metric and let £ be a holo-
morphic hermitian bundle over X. The Dolbeault complex
Л0,0® Е-^ЛОЛ ®Е-^->------------^-»A0,m®E	(13.27)
converts to an elliptic operator
Л°.е«п Q E	A0.odd 0 E
where d* denotes the adjoint of d. Theorem 13.13 can be applied with
G = Um c SO2m and with P = d 4- d*. If we consider T = TX as an m-
dimensional complex vector bundle, then A0’* A*T and we see that
ch A°’even - ch A0 odd = ch A.JT). From (12.10) and (12.11) we see that
сЬ2_1(Г)/х(Г) = (-1)тТ(1с(Т)-\ From Proposition 11.14 we have
A(X)2 = Tdc([T]R ® C) = Tdc(T© T) = Tdc(T)Tdc(T). Plugging into
§14. FIXED POINT FORMULAS
259
13.13 immediately gives the following:
ind(3 4- 3*) = {ch E • Tdc(X)}[X]
where TdJX) = Tdc(T) is the total Todd class of X.
Observe that ker P = ker P*P = ker(dd* + d*d) = {<p e A0,even ® E:
дер =	= 0}, and similarly coker P = ker P* = {cp e A0,odd ® E:
dq> = a*<p = 0}. Applying the Hodge Decomposition Theorem (II.5.6)
gives the following:
Theorem 13.15. Let H*(X\E) denote the kth cohomology group of the
Dolbeault complex (13.27) over the compact complex manifold X. Then
Y (~ l)kdim Hk(X;E) = {ch E • Tdc(X)}[X].
Example 13.16. Let X be a compact oriented riemannian manifold of
dimension 2m, and consider the operator Р°:Г(СЕ°Х) -► ЦСРХ) given
in II.6.1. We leave as an exercise to the reader the verification that
ind D° = Z(X).
§14. Fixed-Point Formulas for Elliptic Operators
The proof of the Index Theorem outlined above carries over, almost
without change, to the cases of G-operators, Ctk-linear operators, families
of operators, etc. The main point is always to find the right K-theoretic
setting in which to work.
In this section we consider the case of G-operators and the associated
G-index Theorem. We assume throughout that X is a compact G-manifold,
i.e., a manifold equipped with a given smooth action p:X x G -> X of a
compact Lie group G. By a G-bundle on X we shall mean a complex vector
bundle E -► X with a G-action which carries fibres to fibres linearly and
projects to д. The Grothendieck group of equivalence classes of such G-
bundles (cf. 1.9) is called the equivariant К-theory of X and is denoted
KG(X).
Equivariant К-theory has the same properties as ordinary К-theory if
one restricts to the category of G-spaces and G-equivariant maps. Hence,
given a G-operator P on X, one can pass through the same constructions
as above (using a G-equivariant embedding X and the Thom iso-
morphism) to define the topological G-index top-ind^P) in KG(pt) s P(G).
Theorem 14.1 (Atiyah and Singer [1]). For any elliptic G-operator P on a
compact G-manifold^ one has that
indG(P) = top-indG(P)
260
III. INDEX THEOREMS
The proof of this “G-index Theorem” follows precisely the arguments
outlined above for the basic case where G = {e}.
In the case that G acts trivially on X, there is a cohomological formula
for the index which is deduced in analogy with the basic case. An impor-
tant fact is that when G acts trivially on X, there is a natural isomorphism
Kg(X)^K(X)®K(G)	(14.1)
determined as follows. Every finite-dimensional representation V of G can
be written in the form HomG(I<,K) ® Ц where the direct sum ranges
over the set {P<} of equivalence classes of irreducible representations of G.
Similarly, if G acts trivially on X, then any G-bundle E can be written as
E=@HomG(Ei,E)®Ei	(14.2)
where Et denotes the trivial bundle Ef = X x Ц. This association induces
the isomorphism (14.1) (see Segal [1]). Composing this isomorphism with
ch ® Id gives a homomorphism
chG: KG(X) —> H*(X; Q) ® K(G).
For non-compact spaces, such as TX, this extends to К-theory and coho-
mology with compact supports.
For each geG there is a homomorphism %g:R(G) -► C determined
by setting %g(p) = trace(p(#)) for each finite dimensional representation
p:G -> Hom(KF). Composing with chG gives a homomorphism
c\:KG(X)----->Я*(Х;С)	(14.3)
which, for an element u = X ® r« e ® E(G) is written chffu =
X(ch The isomorphism (14.1) together with the arguments for the
basic case given in §13 show the following:
Proposition 14.2. Let X be a compact n-manifold on which G acts trivially,
and let P be an elliptic G-operator on X with symbol class a = a(P) e
KG cpt(TX). Then one has that
indG(P) = (- l)"{chc<F  A(X)2}[TX],	(14.4)
In particular, for each g e G,
ind9(P) = (- l)”{ch9ff  A(X)2}[TX],	(14.5)
In the case that G acts non-trivially on X, the formula (14.5) can be
replaced by one which involves the data of the action in a neighborhood
of the fixed-point set. The result is a grand generalization of the classical
Lefschetz Fixed-Point Formula for maps of finite order. The key to the
computation is a certain “localization” theorem of Atiyah and Segal [1].
§14. FIXED POINT FORMULAS
261
An excellent summary of the arguments involved and a large collection
of illuminating examples are given in the book of Shanahan [1]. This
exposition, together with the highly readable original literature, are recom-
mended to the reader interested in full details. We shall present the result
here in concise form.
To begin, we recall that for each geG, the fixed-point set of g is defined
to be the set
Fg = {x g X : gx = x}.
Since G is compact, we know by averaging that G acts as isometries for
some riemannian metric on X. An elementary argument using the expo-
nential map then shows that for each geG, the set Fg is a smooth closed
submanifold of X (see Kobayashi [1]). In general, however, Fg is not con-
nected and the dimensions of the components can vary.
We wish to derive a cohomological formula for indfl(P), where P is an
elliptic G-operator. For this purpose, we may replace G by the closure of
the cyclic subgroup generated by g. This is a compact abelian Lie group
for which g is a “topological generator.” With this assumption we see that
Fg = {x e X : g'x = x for all д' e G} = FG, the fixed-point set for the entire
group. This is a trivial G-space; however, the normal bundle N of Fg is a
non-trivial real G-bundle. The normal bundle to the induced embedding
i:TFg<—>TX
is the complex G-bundle n*N ® C (where я: TFg -► Fg is the bundle pro-
jection). The element	r(n*N ® C) is the Thom class for this
bundle and is used, as in (12.7), to define a homomorphism
*: KG.cpt( TFg)-► KG.cpt(TX)	(14.6)
The fundamental result of Atiyah and Segal [1] is that after localizing at
g, i.e., after introducing formal inverses for all elements r e R(G) with
Xg(r) 0, the map (14.6) becomes an isomorphism with inverse given by
rl=_2L_
This leads to the following.
Theorem 143 (The Atiyah*Segal-Singer Fixed-Point Formula). Let X be a
compact G-manifold, where G is a compact Lie group, and let P be an elliptic
G-operator on X with symbol class a = e(P) e KGc-t(TX). For each element
geG, the "Lefschetz number” ind^P) s tracef#^,>) - tracetolcokerp) is
given by the formula
tow-'-4s^3)ilp')I}[rF'] ,14-71
262	III. INDEX THEOREMS
where Fg denotes the fixed-point set of g, where	denotes the Thom
class of the complexified normal bundle of Fg, and where d denotes the di-
mension of Fg (an integer-valued function which varies from component to
component).
Note that G is not assumed to be connected; in particular, Theorem
14.3 applies when G is a finite group. Some of the most important appli-
cations of the result come from this case. This theorem also gives rise to
a number of intriguing relations between topology and elementary num-
ber theory (see Hirzebruch-Zagier [1]).
The most basic example of a G-operator comes from the de Rham com-
plex. Let G act by isometries on a compact riemannian manifold X, and
consider the Dirac operator D°: C£°(X) -> Cf^X). (Recall that this is ex-
actly the operator d -I- <5:Acen(X) -► Acdd(X)). This operator is G-equiv-
ariant and for each geG, the number indfl(D°) is just the classical
Lefschetz number of g9 i.e., indfl(Z>°) = Цд) = traced Jfcven) trace(^|Hodd).
The symbol class of D° is just the class <r(D°) = z_1(n*TX ® C). Re-
stricting to TFg we have n*TX ® C = (it*TFg ® С) © (л*Х ® C)
Tc © and so ГМ00) = X(TC © Nc) =	It follows
that chff(i*tf(D0)) = ch^Z-^TJJch^A-^Nc)), and formula (14.7) becomes
Ш) = X(^).
In particular, if g has isolated fixed points, then Цд) — card(F J. This result
uses strongly the fact that g is contained in a compact group of diffeomor-
phisms. The general Lefschetz formula applies to any diffeomorphism f
with isolated non-degenerate fixed points, i.e., points where det(/ — df)
0. In this case points are added with the weight factor sign[det(/ — df)~f so
that the Lefschetz number becomes equal to the algebraic sum of the fixed
points.
One might ask whether the fixed-point formula (14.7) can be extended
to cover geometric automorphisms of an elliptic operator which do not lie
in a compact group. Such a formula was given by Atiyah and Bott [3,4]
for the case of automorphisms with isolated non-degeneratc fixed points.
To give the reader some feeling for the fixed-point formula, we shall
work out the details for the signature operator D+: T(Cf +(X)) ->
Г(СС(Х)) and the Atiyah-Singer operator 0": Г($ + (Х)) -> Г($"(Х))
(where X is spin).
Let us suppose that X is a compact oriented riemannian manifold and
that g: X -> X is an orientation preserving isometry. Let N denote the
normal bundle of the fixed-point set Fg, and note that the differential dg
of g gives a bundle isometry
dg: N-----> N.
(14.8)
§14. FIXED POINT FORMULAS	263
Fix x g Fg and note that since g is an isometry, we have ^(expxu) =
expx(d0-u) for all ve Nx. It follows that dgv / v for each v / 0, since
otherwise exp^tu) would lie in Fg for all t e R. Since g lies in a compact
abelian Lie group, we know from elementary representation theory that
there is an orthogonal decomposition Nx =	® @0<e<jr Nx(0),
where dgx\Nx(n} ~ — Id, and where the space Nx(0) splits into 2-dimensional
subspaces in which dgx rotates every vector by 0. This decomposition is
“constant” on each component of Fg. This follows from (14.2) or by simply
observing that parallel translation along any curve joining x to у in Fg
gives a ^-equivariant isometry of Nx(6) with Ny(0). (This follows without
difficulty from the fact that g is an isometry.) Consequently, we conclude
that the bundle N admits a decomposition
N = N(7t)© © N(0)	(14.9)
0 <6<it
where dg acts on by multiplication by —1, and where for each 0,
0 < 0 < я, N(6) is a complex bundle in which dg acts by multiplication
by eie.
Remark 14.4. Note that the bundle N(n) may not be orientable but it
carries the same orientation class as Fg, i.e., for every loop у cz Fg, the
orientation of Fg changes along у if and only if the orientation of N(ti)
also changes along y. (This is because X is orientable and each N(0),
0 < 0 < я, is complex and hence also orientable.)
For any real vector bundle E, we introduce the characteristic class
Ъп(Е) = х(Е)Ъ~1(Е)	(14.10)
where L(E) is the total L-class of Hirzebruch and where %(E) is the Euler
class of E. If E is not orientable, then %(E) lives in cohomology with twisted
coefficients.
For any complex vector bundle E, and any 0,0 < 0 < 7t, we define Ъв(Е)
to be the total class associated to the multiplicative sequence of Chern
classes with formal power series coth(x + |i0). Therefore, if E =	© ...
ф is a formal splitting into complex line bundles with сД^) = x;, then
Ц£) = П coth(x7 + |i0)	(14.11)
With these definitions we can state the following.
We recall now that if X is oriented and of even-dimension, then there
is a canonical splitting C£(X) = C£+(X) © C£~(X) and the Dirac con-
struction gives the “signature operator” D+ :T(C£+(X)) -► Г(С£"(%)).
This operator is preserved by the group G of isometries of X. For each
g e G we denote
sig(X0) = ind,(D+)
264
III. INDEX THEOREMS
to emphasize that this invariant depends only on X and g. If dim X = 4k
and g = 1, then sig(X,#) = sig(X).
Theorem 14.5. (The G-signature Theorem). Let g:X -+ X be an orienta-
tion-preserving isometry of a compact oriented 2m-dimensional manifold X.
Then
sig(X,0) = | П WW)) L(F9)llA] (14.12)
(O<0£ir	J
where N = ®N(6) is the decomposition of the normal bundle to Fg given in
(14.9).
Note. The cohomology class in (14.12) factors into a product of an ordi-
nary cohomology class with x(N(n)). As observed in Remark 14.4, N(n)
and Fg have the same orientation type. Hence, the pairing in (14.12) is
well defined.
Proof. We begin with a remark on dimensions. Since g is orientation
preserving, we see that dim[M(N(7i)) is even. Hence we may write
dim Fg — 2d, dim N(ti) = 2r, dimR/V(0) = 2s(0)
where d,r and s(0) are integers which vary from component to component
on Fg.
Recall that <r(Z> + ) = S(TX), and therefore by the multiplicativity (11.33)
of 6 we have i*d(TX) = 8(TFg ® N) = d(TFg)d(N). By pushing forward
to Fg and using Proposition 12.4, the Fixed-point Formula (14.7) becomes
in^D’,=2‘kr^^	(14J3)
It remains to evaluate the term chq(d(N)7_ fN ® C)-1). Since every-
thing is multiplicative, we can consider the factors N(0) separately. Via
the Splitting Principle it suffices to compute in the case where v is an
oriented real 2-plane bundle on which g acts by rotation by 0. We write
v ® C = 7 ф 7 where 7 is a complex line bundle, and we set x =
cf7) = x(v). Then we have
5(v) = 7—7 and j(v ® C) = (1 - 7)(1 - 7).
From the fact that ch^) = ch^e10 = ex + ie, we find that
p~x~ie _ pX + i'0
che(^(v))A <(v ® Cp1 =-------	-—-
\ И IV чу / (j _ ex + i6^ | _ e~x-i0)
= coth |(x -I- iO).
§14. FIXED POINT FORMULAS	265
This leads us to introduce the multiplicative sequence of Chern classes
be associated to the formal power series coth |(x 4- id). The above
computation shows that
ch^^M.^MO) ® С)’1) = I-0(N(0)).
This computation is equally valid when в = я provided that N(n) is
orientable. In fact, under this assumption we take a formal splitting
N(n) = ¥!©•••©¥,. into oriented 2-plane bundles, and with Xj = x(v;)
we find that
® С)’1) = П coth |(xj + in)
J= 1
= П tanh(x/2)
J=1
= fi - п Г Xj/2 1
Л i 2 М |_tanh(Xj/2)_
=
where x(N(rc)) s 2~rx(N(n)). Assembling these calculations, we find that
sig(X,0) = 24 П
where we have set L* = x • L"1. Multiplying the appropriate homoge-
neous component of this cohomology class by 2d allows us to remove the
hats from the L’s and gives us the desired formula (14.12). 
Remark 14.6. This formula undergoes only minor modification if one
takes coefficients in a complex G-bundle E. Specifically, if :
Г(С£+ ® E) -► T(Ct” ® E) is the twisted signature operator, then
setting sig(X,E,g) = indg(D£), we have
sig(X,E, g) = {ch E •	(14.14)
where & is the cohomological expression appearing in formula (14.12).
The applications of Theorem 14.5 are numerous and varied (see Atiyah-
Singer [2], Shanahan [1], and Hirzebruch-Zagier [1] for example). We
mention here just a few corollaries. Note to begin, that the expression
ЬЯ = Х L ’1 involves the Euler class. Consequently we have the following.
Corollary 14.7. Let X, D+, g, etc., be as in Theorem 14.5. If dim N(n) >
dim Fe, then sig(X,g) = 0.
Note that when g is an involution, (i.e, g2 = Id), we have N = N(n). In
this case the corollary states that dim Fe < |dim X => sig(X,g) = 0. This
statement has a pretty generalization.
266
III. INDEX THEOREMS
Corollary 14.8 (Atiyah-Singer [2]). Let X be a compact oriented 4m-
mainfold, and let g\X X be an orientation preserving involution with
fixed-point set Fg. Let FgFg denote the closed oriented manifold obtained
by intersecting Fg with (a generic displacement of) itself Then we have
sig(X,g) = sig(Fg- Fg).
The right hand side of the formula is independent of the transversal
displacement of Ft used to define Fg • Fg. It depends, of course, only on
the oriented cobordism class of Fg • Fg.
Proof. Since g2 = Id, we have N — N(n). One can verify easily that
{L(F,)L- WN)}[f J -	• Fj = {HF, • F,)}[F, • F,]
= sig(F, • Fg). (We use here that is the normal bundle to Fg • Fg
in Fg.) Applying (14.12) completes the proof. 
The proof of the following corollary is left as an exercise for the reader.
Corollary 14.9 (Atiyah-Singer [2]). Let X be a compact connected oriented
4-manifold, and suppose that gtX -* X is a diffeomorphism of odd order
(necessarily orientation preserving). Suppose Fg consists of oriented embedded
surfaces SU...,S„ and that for each k, dg rotates the oriented normal bundle
to Sk through an angle 0k. Then
sig(g,X) = ft яп-2(Ш$»Л
*«i
where Sk-Sk denotes the homological self-intersection number of Sk.
Our second basic set of examples will come from considering the
Atiyah-Singer operator 0+: Г($с) -> Г($ё) acting on spinors over a com-
pact even-dimensional spin manifold X. We shall assume that G is a
compact Lie group acting by orientation preserving isometries of X. Note
that there is an induced action of G on the bundle PsoW of oriented
orthonormal tangent frames. Recall that the spin structure is a 2-fold
covering Papin(X) -» Pso(X) which is non-trivial on the fibres. It corre-
sponds to an element и e Hi(PstfX);Z2).
Definition 14.10. The action of G preserves the spin structure of X if it
lifts to an action on the bundle PSpin(X). An individual isometry g: X -» X
is said to preserve the spin structure of X if the closed subgroup G <=
Isom(X) generated by g preserves this structure.
Note that if g preserves the spin structure, then g*u = u, where ue
Я1(Р8О(Л'); Z2) is the element corresponding to the structure.
If G is connected, then it follows from elementary covering space theory
that either G or a 2-fold covering group of G preserves the spin structure.
In particular, if t^G = 0, then G preserves every spin structure on X.
§14. FIXED POINT FORMULAS	267
Whenever G preserves the spin structure, it acts on the bundle of spinors
and commutes with the Atiyah-Singer operator l/>+ above. For each geG
one defines
Spinet?) = indfl(0+),
and the Fixed-Point Theorem gives us a cohomological formula for this
invariant. Its expression involves the sequences of Chern classes Ae asso-
ciated to the formal power series [2 sinh |(x + 0)]”1 for 0 < в < я. If E
is a complex vector bundle with formal splitting E = 4 ф ... ф 4 into
line bundles with c^j) = Xj, then
к	1	k	^(xj + iO)
= 2 * Д sinh L(X' + j0) = Д 1
In the special case where в = я we define a characteristic class АДЕ)
for any oriented real 2/c-dimensional bundle E as follows. Let E =
Ег ф ... ф Ek be a formal splitting into oriented 2-plane bundles, and set
Xj = x(£j). Then
- k 1 k 1
=2-1Д !inhtlit,+,g) - <2'T* n coshlx ,2|
Theorem 14.11 (The G-spin Theorem). Let g: X -> X be a spin structure
preserving isometry of a compact even-dimensional spin manifold X. Then
Spin(X,0) = (-l)4 П AeW)) A(Fe)l[Fj
where N = фЛГ(0) is the decomposition of the normal bundle to Fg given in
(14.9), and where a : Fg -► {0,1} is a locally constant function which depends
on the action of g on the spin structure.
Remark 14.12. In general the calculation of the sign function cr is a
complicated and subtle affair. We refer the interested reader to Atiyah-
Bott [3],[4] and Atiyah-Hirzebruch [3] for details.
Proof. We proceed much as in the proof of the G-signature Theorem. Re-
call that <r(D+) = s(TX) and therefore by the multiplicativity (11.36) of s
we have i*a(D+) = s(TFfl ф N) = s(TFJs(N). Pushing forward to Fg and
using Proposition 12.5 converts the Fixed-Point Formula to
inde(D+) =	0 C)) •
Computing as before, we consider the case of an oriented 2-plane bun-
dle v with v ® C = /© A Here we have that s(v) = ?1/2 — ^1/2 and
268	III. INDEX THEOREMS
/LJv ® C) = (1 - <0(1 - ?), and therefore that ch^sfvJA.Jv ® C)"1) =
+ [ет(*+»в) _ e-|(x+iO)j- i, jf g acts on v by rotation by в. It follows directly
that
C)’1)= ±Ae(N(0))
for each 0, 0 < в <, n. This completes the proof. 
The G-Spin Theorem can be enhanced by taking coefficients in a G-
bundle E. The formula changes exactly as in the G-signature case (cf.
Remark 14.6).
§15. The Index Theorem for Families
Consider now a family of elliptic operators P on a compact manifold X
parameterized by a compact Hausdorff space A. In §8 it was shown that
this family has a well-defined analytic index ind P e K(A). On the other
hand the constructions of §13 can be generalized to define a topological
index in K(A). The main result of this section asserts that these two indices
coincide.
The topological index of the family P is defined as follows. Let n:	-► A
denote the underlying family of manifolds (recall that this is a bundle
with fibre X and structure group Diff(X)), and let TW -► A denote the
associated family of tangent bundles (so that (T&)a = T(3Ea) where =
тг-1(а)). It is not difficult to see that for sufficiently large m we can find
a map f: 9C -► A x which restricts to a smooth embedding fa: 3Ea
{a} x for each a e A. This induces a map A x TRW which for
each a e Л, restricts to an embedding T3Ea {a} x TRM with normal
bundle Na Ф Na = Na ® C. Here Na denotes the normal bundle of
faffia) <= Rw pulled back to T^a. We now define a map
/:Kcpl(T^)------>KcptM x CM)
by taking the composition
Kcpt(T.^)----► Kcpt(N ® C)----► Kcpt(4 x Cw)
where the first map is the Thom isomorphism, and where the second map
is induced via an embedding N ® С <-> A x Тйш s A x which is con-
structed fibrewise by identifying Na ® C with a tubular neighborhood of
/Л(ТЛ) in (л) x as before.
The projection q: A x Сш -> A induces a Thom isomorphism (or Bott
periodicity map)
q,:Kcp{(A
§15. INDEX THEOREM FOR FAMILIES	269
The composition	° J\ : Kcpt( ТЯГ) -► K(A) is independent of the choice of
the embedding f.
Definition 15.1. Let P be a family of elliptic operators on a compact
manifold X parameterized by a compact Hausdorff space A as in §8. Let
<r(P) g Kcpt( ТЯГ) be the class determined by the principal symbol of the
family. Then the topological index of P is the element
top-ind(P) = qJ^P))eK(A).
Straightforward generalization of the arguments given in §13 proves the
following:
Theorem 15.2 (The Atiyah-Singer Index Theorem for Families [3]). For
any P as above one has that
ind(P) = top-ind(P).
Applying the Chern character and arguing as in §13 establishes the fol-
lowing cohomological form of this result.
Theorem 15.3 (Atiyah-Singer [3]). If P is as above, then
ch(ind P) = (-1/^ 7t,{ch <r(P) • А(ТЗГ)2}
where n = dim X and where л:ТР£ -> A is the natural projection.
Suppose now that л: PF -► A is a family of riemannian manifolds, i.e., a
family of manifolds as above together with a family of riemannian metrics
introduced continuously in the fibres.
Corollary 15.4. Let л: PF -y Abe a family of compact oriented riemannian
manifolds of dimension 2m, and let&+: T(Ct+(ТЯГ)) -► Г(С€“(ТЯГ)) be the
associated family of signature operators. Then
ind(g>+) = 2w7i!{L(T3T)}.
More generally, if we take coefficients in a family of hermitian vector bundles
S with connection, then the associated operator :Г(С£+(ТЯГ) ®	-►
Г(С£”(ТЯГ) ® (?) has index given by
ind(S£) - 2mnI{ch S • ЦТЯГ)].
Proof. The symbol of is <r(0+) = 5(ТЯГ). Using 12.4 to push forward
over the projection я0: ТЯГ -► ЯГ, we find that (rc0)j{ch +) • А(ТЯГ)2} =
(-2)шЬ(ТЯГ)А(ТЯГ)-2А(ТЯГ)2 = (-2)жЬ(ТЯГ). Projecting on to A then
gives the first formula. The second formula follows similarly from the fact
that = 5(ТЯГ) • л*&). 
270
III. INDEX THEOREMS
It is an interesting exercise to apply this formula to the Lusztig family
(8.3).
Carrying through the computations above with 3 replaced by s gives
the following.
Corollary 15.5. Let rt\9C -> A be a family of compact oriented spin mani-
folds of dimension 2m, and let : Г(,99 + ® <?) -► Г(^~ ® <?) be the family
of Atiyah-Singer operators twisted by a family of coefficient bundles S,
Then
ind(^) = 7r,{ch <? • А(ПГ)}.
All of these results go through for families of G-operators. One replaces
К by KG in this case.
§16. Families of Real Operators and the Cfk-Index Theorem
In discussing the Index Theorem thus far we have considered only com-
plex differential operators. Given a real operator, say P, defined using real
vector bundles, one can always complexify and apply the index theorem
in the form above. Since dimR(ker P) = dimc(ker P ® C) (and similarly
for P*), we see that the ordinary real and complex indices of p coincide.
Thus, in the basic case no information is lost under complexification.
This is not true, however, if one passes to the index theorem for families.
The index of a family of real operators takes its value in the group K0(A),
and the complexification map K0(A) -> К(Л) is not always injective. Note
for example that KO(Sn) f°r n = l(mod 8), but K(Sn) = {0} in these
dimensions.
For this reason Atiyah and Singer established a separate index theorem
for families of real operators. It is a more subtle and profound result than
one might naively expect. The constructions and arguments outlined above
for complex families go through essentially unchanged in the real case,
provided one employs the appropriate “К-theory.” It is here that matters
become interesting, for the appropriate theory is not KO-theory, the
straightforward theory of real bundles. It is the more general KR-theory
which is defined on any space with involution and reduces to KO-theory
when the involution is trivial. Recall (cf. I. 10) that if X is a space with
involution f:X -> X, then KR(X) is the Grothendieck group of pairs
(EJe) where E is a complex vector bundle over X and where fE :E -* E
is an involution which covers f and is C-antilinear on the fibres.
The introduction of this theory is motivated by the simple fact that the
principal symbol of a real operator is in general not real. Consider, for
example, the operator д/дв*. CXJ(S1) -> C0C(S'1) whose symbol is = i{.
§16. THE Cfj-INDEX THEOREM
271
Recall also that for any real Dirac operator D, one has that = i£-
(cf. Example 1.5). The appearance here of the complex number i cannot
be ignored. It is essential in the calculus of pseudo-differential operators
and must be retained if the proofs discussed above are to carry over.
Suppose now that P: Г(Е) -> T(F) is a real differential operator between
real bundles E and F over a compact manifold X. In local terms, we have
that P = £ Aa(x)dMldxa plus lower order terms, where the A* are real
matrix-valued functions. Consequently for any tangent vector £, we have
a{(P) = £ A“(x)(i^)“, from which it follows that
a-^P) = ^Pj.	(16.1)
This is the key to defining the symbol class of a real operator.
Definition 16.1. Given a compact manifold X, consider the tangent
bundle я: TX -> X to be equipped with the canonical involution f: TX ->
TX defined by /(<*) = — £ Given any real bundle E -» X, consider
я*(£ 0 C) to be equipped with the antilinear involution defined by com-
plex conjugation. Then for any real elliptic operator Р:Г(£) -» T(F) the
Real symbol class of P is defined to be the element
[я*(£ ® С), яф(Г 0 C); o(P)] g KRcpl(TX). (16.2)
Here я*(£ 0 C) and n*(F 0 C) are Real bundles on the Real space TX,
and (16.1) says that o(P) is an isomorphism of real bundles outside the
zero section of TX. Hence, (16.2) naturally defines an element in KRcpl(TX.)
In general if one wants to remember that a given bundle is the com-
plexification of a real bundle, one carries along the associated complex
conjugation map. The new point here is that when the bundle is pulled
back over TX we think of this conjugation as covering the involution
I—* — £. With this little refinement everything works as one hopes.
To define the topological index of a real elliptic operator P we first
choose an embedding/:X «-♦ Rm. The associated embedding TX TRm
is compatible with the involutions i.e., is a mapping of Real spaces. If W
is the normal bundle to X in Rm, then ф s n*N 0 C is the
normal bundle to TX in TR". We consider this to be a Real vector bundle
on TX as in Definition 16.1 above. For such bundles, the Thom isomor-
phism holds for KR-theory. (It is defined exactly as it was for X-theory
in §12. We need only note that the de Rham element A_ t of a Real bundle
is itself Real. See Atiyah [2].) We can now define a map
/:KRcpt(TX)------► KRcpt(TR-)
as before by composing the Thom isomorphism with the map induced by
the inclusion of the normal bundle as a tubular neighborhood of TX in
TRm. This inclusion can be easily chosen to be compatible with the involu-
tions.
272
III. INDEX THEOREMS
If -► A is a family of smooth manifolds over the compact space Л,
this construction extends, by using local triviality of the family, to give
a map
/:KRcpt(T^)-------► KKcpt(4 x TV).
We now identify 71RW	Cw by associating (х,£) to x + i£.
Clearly the involution on TRW becomes complex conjugation on Cw. The
fundamental (l,l)-Periodicity Theorem (1,10.3), says that there is a natural
isomorphism
q,:KRcpt(A x Cw)------>KR(A)
for any compact (Real) space A. As before, the composition qx ° /J can be
shown to be independent of the choices involved in the construction.
Definition 16.2. Let P be a family of real elliptic differential operators
on a compact manifold X parameterized by a compact Hausdorff space
A. Let <r(P) g KRcpt(TdC} be the symbol class of the family (where, as
before, 9C -► A is the underlying family of manifolds). The topological
index of the family P is defined to be the element
top-ind(P) = qd\o(P) g KR(A) * K0(A).
(Note that A carries the trivial involution.)
With this definition, the arguments for the Index Theorem discussed
above go through easily to prove the following.
Theorem 16.3 (Atiyah and Singer [3], [4]). Let P be a family of real
elliptic operators on a compact manifold parameterized by a compact Haus-
dorff space A. Let ind(P)G K0(A) be the analytic index of the family de-
fined as in 8.5 by replacing complex objects with real ones. Then
ind(P) = top-ind(P).
It should be remarked that the index theorem for families is a useful
tool, much more powerful than the standard index theorem. A number
of applications are given in the next chapter.
One important consequence of this result is the derivation of a topo-
logical formula for the Clifford index discussed in §10. No such formula
appears in the current literature even though the derivation was known
to Atiyah and Singer. We shall present the details here.
Assume from this point on that E = E° © E1 is a real, Z2-graded Clk-
bundle over a compact riemannian manifold X. Assume E carries a bundle
metric for which Clifford multiplication by unit vectors in R* is orthogonal.
Let P: Г(Е) -► Г(Е) be an elliptic self-adjoint operator and assume that
P is C£k-linear and Z2-graded. In 10.4 we defined an analytic index
§16. THE C£k-INDEX THEOREM	273
indtP e KO“*(pt) for P in terms of the C£k-module ker P. We shall now
give a topological formula for this index.
To do this we construct the following family SP of elliptic operators
parameterized by R*. We assume that P has degree zero by replacing P
with (1 + P*P)~ l/2P. We recall from 10.2 that with respect to the splitting
E = E° ф E1 we can write P as
P=(Q pl\
\P° 0 J
where P1 = (P0)*. The product family & on Rk x X is now constructed
by assigning to each v e Rk the operator
^ЦЕ^ЦЕ1)	(163)
defined by the restriction to £° of the operator
0>v = v + P	(16.4)
where “v” denotes Clifford multiplication by v. Note that there is a “con-
jugate family” of operators defined by = v — P. Since P com-
mutes with Clifford multiplication, we have that
= 4II< + P2}-	(16.5)
In particular, is invertible for all v # 0. Since the invertible operators
on Hilbert space form a contractible set, we could easily pass to a family
parameterized by Sk. However, the calculations will be more transparent
if we treat as a family with “compact support” in R*, whose index lies
in KOcpt(Rk) KO"k(pt). The main result is the following:
Theorem 16.4. Let P be an elliptic self-adjoint graded Clk-operator on a
compact manifold. Then
indk(P) = top-ind(^°)
where & is the family over Rk defined by (16.4).
Proof. Set K° = ker P° c= Г(Е°) and Kl = ker P1 coker P° cz ЦЕ1).
By Theorem 5.5 there are L2-orthogonal direct sum decompositions
Г(Е°)=У°®К° and Г(Е1)=И1©К1	(16.6)
where P°: V° V1 is an isomorphism. Since P commutes which Clifford
multiplication, we have that v • K° = K1 for all v / 0 in Rk. Furthermore,
since multiplication by v/||v|| is L2-orthogonal, we have v • V° = V1 for all
such v. It follows that the family = v 4- Р°:Г(Е°) -► Г(Е!) decom-
poses, with respect to (16.6), as a direct sum of two operators. By (16.5)
274
HI. INDEX THEOREMS
the first summand
j/0	Ki
is an isomorphism for all v e R*, and thus can be ignored for the purposes
of computing the index. The second summand is just
This operator is independent of variables on X. Its index equals the index
of and is given by the element
[K0,*1;»] e KOcpt(R*) * KO~k(pt).
Under the Atiyah-Bott-Shapiro isomorphism KO"k(pt)®lk/i*SBlk+1,
this corresponds exactly to the element represented by the Z2-graded C£k-
module ker P = К0 ф Kl, i.e., it corresponds exactly to indkP. 
Theorem 16.4 can be applied to give topological formulas for the index
of any of the graded C£k Dirac operators discussed in Chapter II, §7.
In particular this includes the Atiyah-Singer operator whose index is a
basic invariant which we shall now discuss in detail.
Let X be a compact spin manifold of dimension n, and recall that X
carries a canonical graded С£л Dirac bundle
whose associated Dirac operator Ф is called the Atiyah-Singer operator (see
equation (II.7.1) forward). This operator has an index ind„($) 6 KO-"(pt)
which coincides, by 16.4, with the index of the family J# on R" x X defined
by setting
Ф>,х = V +
To compute the topological index of this family we must understand its
symbol class <r(^) e KRcpt(R" x TX). For this we note first that n:R" x
TX -> X is a Real bundle over X whose fibre at x e X is R" x TxX with
involution (r,<J) I—* (t’, — £). The fibre of the bundle ф(X) at x is the Clifford
algebra Ct(TxX) = C£„. Vectors £ e TxX act by left Clifford multiplica-
tion as usual, and vectors v e R" act by right multiplication. The principal
symbol of $ is the map o{$): Г(л*Фс) “* Пл*Фс) defined by
o^) = v + i^ (= Rv + iL(),
where я*$с а ® C is treated as a Real bundle on R" x TX with in-
volution given by conjugation. The symbol class. [я*Ф£,	<r(^)] when
restricted to any fibre R" x s R” x R"s C", becomes exactly the
element,
§16. THE C£k-INDEX THEOREM	275
which, as we have seen in 1.10.12 is the generator of the group KRcpt(C")
Z. (It corresponds exactly to the deRham element A_t under the usual
isomorphism C£* ~ Л*.) Consequently the symbol class
*($) =	л*^;<т($)] g XRcpt(R" x TX)
is a Thom class or “orientation class” for the bundle n: R" x TX -► X in
XR-theory. Under the Thom isomorphism i1: KR(X) -► XRcpt(IR" x TX)
we have
<№) = Ф)	(16.7)
Choose now a smooth embedding f:X Rn+8' and let N denote its
normal bundle. This induces a smooth embedding f: TX <-► TR" + 8' with
normal bundle ti*N © ?r*N tt*N ® C, and we get the following com-
mutative diagram of bundle maps:
R" x TX R" x (n.*N ® C)
X<---------~P------- N
The vertical arrows are Real bundles. Taking zero-sections and embedding
the normal bundles as tubular neighborhoods give the following commu-
tative diagram of embeddings:
R" x TX c R" x (7t*N ® C) c R" x C" + 8'
iU	iU	iU	(16.8)
X = N <=	R"+8'
J	fc
We now recall from II. 1.15 that N has a canonically induced spin
structure. Since dimRN = 8f there is a Thom isomorphism j,:XO(X) -►
KOcpt(N) defined by setting
jM = (P*w) * s(N)
where s(N) = [S+(N), S " (N); д] is defined as in (12.13) using the real spinor
bundle of N (see Appendix C). We could view this as an isomorphism in
XR-theory where the spaces X and N carry the trivial involution. There
is a completely analogous Thom isomorphism jj:XRcpt([R" x TX) -►
XRcpt(R" x (n*N ® C)) defined using the spin structure on n*N and the
Real bundle n*Sc(N) = n*S(N) ® C. One easily checks that
and
where k, and k, denote the natural maps induced by the open inclusions
276
HI. INDEX THEOREMS
к and k. Consequently, (16.8) leads to a commutative diagram
KKcpl(R" x TX) KKcpl(R" x C"+80
4» ff, KR~\pt) (16.9)
KR(X) --------:----> KKcpt(R"+8')
where / = kj’,, / = kj,9 and where q, and qf denote the Bott periodicity
isomorphisms.
The topological index of the family is defined to be the element
Applying (16.7) and (16.9) we conclude that
top-ind(^) = qj^l) = qf^V).	(16.10)
Definition 16.5. For a compact spin manifold X of dimension n, the
Atiyah-Milnor-Singer invariant is the element
з/(Х)^ф/(1)бХО-п(р0
where f : X <~+ [R"+8zr is a smooth embedding for some and where ql
and / are defined as above.
This definition is easily seen to agree with the one given in П.3.22. In
particular, we see that л/(Х) depends only on the spin cobordism class of
X. Combining (16.10) with Theorem 16.4 gives the following main result:
Theorem 16.6. Suppose X is a compact spin manifold of dimension n, and let
$>: Г($) -► Г($) he the canonical Cln-linear Atiyah-Singer operator. Then
ind„($) = J(X).
From Theorem II.7.11 and the spin-cobordism invariance of .c/(X), we
see that ec/ :Q*pin -► K0*(pt) is a graded ring homomorphism.
Suppose now that E is any real vector bundle on X, Furnish E with
any orthogonal connection and introduce the tensor product connection
on ® E. This is again a graded C£„-Dirac bundle and has an associated
Dirac operator Let = v + be its associated family. One easily
checks that	= i,([E]). Passing through the arguments above one
finds that top-ind((7£) = ^;/!([Е]). This proves the following result:
Definition 16.7. Let X be a compact spin manifold of dimension n.
For each real vector bundle E over X, the associated Atiyah-Milnor-Singer
invariant is the clement
^(X;E)^qJ^E])eKO~n(pl)
where q, f -. K0(X) -► KO "(pt) is defined as above.
Theorem 16.8. Suppose X is a compact spin manifold of dimension n, and
let E he a real vector bundle over X. Let TbE • ПФ ® E) -► Г($ (x) E) be the
§17. HEAT AHD SUPERSYMMETRY	277
Ct „-linear Atiyah-Singer operator with coefficients in E (defined using any
orthogonal connection on E.) Then
indB($>E) = з/(Х;Е).
§17. Remarks on Heat and Supersymmetry
Since the basic work of Gilkey and Patodi it has been known that index
formulas for classical elliptic operators (such as the Atiyah-Singer opera-
tor with coefficients in a bundle) can be derived from the asymptotics of
the heat kernel. As explained at the end of §6, it is necessary to make
detailed computations of the density p„ occurring in the asymptotic ex-
pansion trace K,(x,x) ~	as t \ 0. Expositions of this can
be found in Gilkey [2] and Atiyah-Bott-Patodi [1].
In 1979 W. Allard showed us a simple and elementary calculation of
these densities obtained by working with certain canonical operators on
the underlying principal bundle and using the natural filtration of the
Clifford algebra. (His resulting proof of the classical Index Theorem is
completely elementary.)
In 1982, E. Witten found a different approach to these formulas through
considerations of symplectic geometry and supersymmetry. There is a
fixed-point formula for S1 -actions on (finite dimensional) symplectic mani-
folds due to Duistermaat and Heckman [1]. Witten considered the anal-
ogue of this formula for the canonical Sx-action on the free loop space
of a manifold. Using this together with some ideas from supersymmetry
he outlined a proof of the index theorem for the Atiyah-Singer operator
(see Atiyah [13]). His ideas have engendered a series of interesting papers
on the subject, notably by L. Alvarez-Gaume, E. Getzler, N. Berline and
M. Vergne, and J. M. Bismut.
It should be pointed out however that none of these methods applies
to prove the index theorem for families or the C£t-index Theorem (in their
strong forms). These theorems in general involve torsion elements in K-
theory which are not detectable by cohomological means. Moreover, it
is known that the Z2-invariants appearing in the C£t-index Theorem are
not computable from local densities, for this would imply a certain multi-
phcativity under coverings which examples show not to be true.
CHAPTER IV
Applications in Geometry and Topology
In this chapter we shall use the results of our previous discussion to derive
a series of consequences in differential topology and geometry. Some of
the theorems we shall derive have other, completely different proofs, and
some (to date) can only be proved by these means. Nevertheless we hope
to make clear that spin geometry and the Index Theorem give a unified
approach to a wide range of geometric problems.
The applications presented here fall into several categories: integrality
and divisibility theorems for characteristic numbers, immersions of mani-
folds and the vector field problem, group actions on manifolds with posi-
tive scalar curvature, Kahler geometry, pure spinors and basic twistor
geometry, the theory of calibrations, and the study of riemannian metrics
with reduced holonomy. A few of these results are already mentioned in
previous chapters in order to add spice to the exposition there. On the
other hand there are applications given in previous chapters which do not
appear here. Notable among them are the results of §§7 and 8 in Chapter
I, the new curvature identities 5.15 - 16 of Chapter II and their application
to the Homology Sphere Theorem (II.7.6) of Gallot and Meyer.
The first part of this chapter is concerned with applications of the Index
Theorem to differential topology. Recall that the ordinary index of an
elliptic is an integer. However, the Atiyah-Singer formula computes this
index in terms of topological invariants which are in general just rational
numbers. The fact that these rational numbers are always integers under
certain hypotheses, plays a significant role in topology. Its importance
stems from the underlying elliptic operators. Two general problems to
which such integrality theorems often apply are: finding the smallest codi-
mension for immersing a manifold into euclidean space, and finding the
largest number of pointwise linearly independent vector fields on a mani-
fold. We shall show how to prove results in these areas by directly con-
structing the relevant operators via Clifford multiplication.
Using more sophisticated forms of the Index Theorem, one can prove
interesting results about smooth compact group actions on spin mani-
folds. For example, applying the G-Index Theorem to the Atiyah-Singer
operator shows that on spin manifolds admitting an Sanction the A-
APPLICATIONS
279
genus is zero. Applying the C£k-Index Theorem to the C£k-linear Atiyah-
Singer operator and invoking some basic riemannian geometry yields a
more refined result about S3-actions. All of this is done in §3.
Sections 4, 5 and 6 are devoted to the study of riemannian manifolds
of positive scalar curvature. For compact simply-connected spin mani-
folds, the KO-index of the Atiyah-Singer operator gives a nearly complete
set of invariants for deciding whether or not there exists a metric of posi-
tive scalar curvature. This is discussed in §4. In §5 the analogous question
is discussed for manifolds with non-trivial fundamental group. Here there
is an interesting interaction between spin geometry and the fundamental
group which is mediated by an appropriately twisted Atiyah-Singer
operator. The ideas developed here carry over to give results about the
existence of complete metrics with positive scalar curvature on non-
compact spin manifolds. For this it will be necessary to develop a “rela-
tive version” of the Index Theorem over open manifolds. This is discussed
in §6.
The techniques of spin geometry can be applied to say something about
the topology of the space of all riemannian metrics with positive scalar
curvature on a given manifold. This is done in §7.
On an even-dimensional riemannian manifold there is a concept, due to
Ё. Cartan, of pure spinors. These are related directly to almost complex
structures on the manifold and to the Penrose twistor construction. This
together with theorems on integrability are examined in §9.
A Kahler manifold is a riemannian manifold with a parallel almost
complex structure. (Such a structure is always integrable, i.e., the under-
lying manifold is always complex.) When X is a Kahler manifold, the
Clifford bundle C£(X) has a very pretty decomposition. There are dif-
ferential operators Qi and Q such that Qi2 = Q2 = 0 and Qi + Qi = D (the
Dirac operator on C£(X)). There are also О-order operators «5? and &
such that <«S?,^, [«S?,^]> s sI2(C) and «5? + & = L (cf. (II.5.20)). From
these one can define certain Clifford cohomology groups and establish
within them all the classical identities of Kahler geometry (see §8). One
can also decompose holomorphic Dirac bundles and define spinor coho-
mology groups. A wide variety of vanishing theorems are then proved
using Clifford formalism in §11. These are applied to prove the existence
of compact manifolds with Spm-holonomy.
There is an intimate relationship between riemannian manifolds with
reduced holonomy, calibrations, and spin geometry. This is presented in
detail in §10 with special emphasis on the exceptional cases of G2 and
Spin7 manifolds.
In the last section we shall examine the role played by spinors in a
proof of the Positive Mass Conjecture in general relativity.
280
IV. APPLICATIONS
§1. Integrality Theorems
The point of this section is to use the Index Theorem to prove the
integrality and divisibility of certain rational characteristic numbers.
These results were known before the Index Theorem and, in fact, they
provided part of the motivation and insight for its proof. We begin with
the following classical result of Atiyah and Hirzebruch [1].
Theorem 1.1. Let X be a compact spin manifold of dimension 4k. Then
A(X) is an integer. Furthermore, if dim(JV) = 4(mod 8), then Л(Х) is an even
integer.
Proof. The first statement follows from the fact that on a spin manifold,
the Л-genus is the index of the Atiyah-Singer operator. The second state-
ment follows similarly from Theorem 6.16 of Chapter II but can be
deduced directly as follows. Notice from the classification of Clifford al-
gebras (Theorem 1.4.3), that in dimensions 4 (mod 8), the complex spinor
representations are actually quaternionic. The connection in the spinor
bundle is always such that multiplication by quaternion scalars is paral-
lel. Hence, the kernel and cokernel of the Dirac operator are quaternion
vector spaces, and their complex dimension is even. 
For manifolds which are not spin, the Л-genus is very often not an
integer.
Recall that for a 4-manifold the signature is always 8 times the Л-genus
(see II.6.17). Thus, Theorem 1.1 generalizes the following result of Rochlin:
Corollary 1.2. The signature of a compact smooth spin 4-manifold is a mul-
tiple of 16.
It was pointed out in Chapter II (see 2.12ff) that on a spin 4-manifold
X, the intersection form is even. That is, x о x = 0(mod 2) for all x e
H2(X;Z). This implies easily that the signature of X is a multiple of 8
(see Milnor-Husemoller [1], p. 242). The importance of smoothness in
Rochlin’s Theorem was recently underlined by Mike Freedman’s profound
results [1] which proved, among other things, the existence of a topo-
logical spin 4-manifold of index 8.
Another integrality result comes from our discussion of Spinc-manifolds.
Theorem 1.3. Let X be a compact orientable manifold and suppose c e
H2(X;Z) is a class such that c = w2(X) (mod 2). (Hence, X is a Spirf-
manifold.) Then the rational number
is an integer.
pA(X)}[*]
§2. IMMERSIONS AND VECTOR FIELDS	281
Proof. This number is the index of an elliptic operator (see App. D). 
Another immediate consequence is the following classical theorem of
Bott [1], [2]:
Theorem 1.4. Let E be a complex vector bundle over the 2n-sphere S2n.
Then the top Chern class of E is divisible by (и — 1)!, that is,
Proof. Consider the twisted signature operator
D+ :(Ct+ ® E)-----► Г(СГ ® E)
over S2n. Then the index of this operator is
ind(D+) = {ch E • L(S2"))[S2"]
= (ch E)[S2"]
= ch„(E)[S2"]
= (^T)!c-(£>P2-1
since all the Pontrjagin classes of S2n are zero and since H2k(S2n) = 0 for
к Ф 0,n. Of course, the index is an integer. 
This result is useful for many things, among them the study of immer-
sions of manifolds into R". Here, however, there is a direct approach
using spin geometry.
§2. Immersions of Manifolds and the Vector Field Problem
In this section we shall present a unified approach to the following two
problems. Let Xn be a compact manifold of dimension n and without
boundary.
I. Find the smallest integer q such that there exists an immersion
X” ВГЧ
II. Find the largest integer q such that there exist q pointwise linearly
independent vector fields on X
In the first case we always have that q n — 1 by the classical work of
Whitney. Furthermore, using Stiefel-Whitney classes and the product for-
mula w(T)w(N) = 1, one can sometimes find very good lower bounds for
q (see Арр. B, for a discussion of the product formula). Nevertheless, for
better results it is sometimes useful to use real characteristic classes or
KO-theory. This is the approach we shall take.
282
IV. APPLICATIONS
The techniques presented here generalize fundamental ideas of Atiyah
[9]. They appeared in Lawson-Michelsohn [1]. A second approach, also
using spin geometry, can be found in Mayer [1].
We shall assume throughout that X is a compact oriented manifold
of dimension n. For simplicity we let T denote the tangent bundle of X,
We begin with the first problem. Suppose that X admits an immersion
X R"+<?. Denote by N the normal bundle to this immersion, and note
that
T®N = t	(2.1)
where т denotes the trivial (n 4- g)-plane bundle. Let <•,•> and V be the
metric and connection on т induced from euclidean space. Using the de-
composition (2.1) we introduce on т a new riemannian connection V, called
the projected connection, as follows. For a section cp = (K,v) g Г(Т ф /V),
we set
V<p = ((VF)r,(Vvf)	(2.2)
where ( )r,( )N denote pointwise orthogonal projection onto T and N
respectively. The connection thus induced on T is the canonical riemann-
ian one for the induced metric. The difference V — V on т is the second
fundamental form of the immersion.
We now consider the bundle C£(t) as a bundle of left-modules over
C£(T) under the obvious inclusion C£(T) c= C£(t). This makes C£(t), with
the connection induced from V on t, into a Dirac bundle over X (cf. II.4.8).
If n = 4k, then we can split the bundle C£(t) via the parallel volume form
co =	• • en, and get the elliptic operator
D+:Г(С£+(т))------> Г(С£'(т)).	(2.3)
We shall prove below (Lemma 2.5) that
ind(D+) = 2M(X)	(2.4)
where A(X) = 2nA(X) is the Л-genus of X. It is a characteristic number
which is always an integer.
Notice now that while C£(t) is not in general trivial as a bundle of left
C£(T)-modules, it is trivial as an abstract vector bundle. In fact, we can
find n 4- q pointwise orthonormal sections q,... ,8„+g of t. These sections
generate a finite multiplicative subgroup G = {1} u {eh • • • «< }(,< ...<Ip in
the algebra of sections Г(С£(т)). Evidently, the group algebra R • G is just
the Clifford algebra C£n+(?.
We now consider the group G acting on C£(t) by right multiplication
K/tp) = (p ' g. This action does not quite commute with the Dirac opera-
tor, so we take the average
(25)
§2. IMMERSIONS AND VECTOR FIELDS
283
and note that
Ь о Rg = Rg о D for all g e G.	(2.6)
One can easily see that each commutator D о Rg — Rg о D is a differ-
ential operator of order zero. Hence, the averaged operator D has the
same principal symbol as D.
Of course right multiplication commutes with left multiplication, so the
bundles
C£±(t) = (1 ±co)C£(t)
are clearly G-invariant. It follows that D is also of the form
with respect to the splitting С£(т) = С£+(т) ф С£"(т), and the operator
D+ has the same symbol as D + . From the topological invariance of the
index we conclude that
ind(D+) = ind(D + )
The main step now is to observe that since D + commutes with G, the
kernel and cokernel of D + are RG C£n+fl-modules, and so the index of
D+ is divisible by a large power of 2.
There is one further observation which sharpens the result. Recall that
C£(t) carries a Z2-grading C£(r) = С£°(т) ф С£х(т) determined as the
+1 -eigenbundles of the parallel automorphism a (which extends v i—► — v
on t). This grading clearly carries over to C£+(r) and C£"(t), and makes
each of these a bundle of Z2-graded modules under right Clifford
multiplication. Furthermore, since a.(D(p) = — D(a(p), for any (p e Г(С€(т)),
we see that kerD + and kerD" are each Z2-graded under right multi-
plication by RG = C£„ + (?. This proves the following:
Theorem 2.1. Let X be a compact oriented manifold of dimension n = 4k.
If there exists an immersion X Rw + * then
2"~lA(X) = 0(mod a„+q)
where 2an+q is the dimension of an irreducible real ^-graded C£M+fl-module.
See the table below for values of anJrq. Note that by Proposition 1.5.20,
an+q is the dimension of an irreducible, real ungraded module over
cen+Q-r
As a quick illustration we note that Л(Р2(С)) = 2 and that 2^ = 0
(mod a4+q) only for q 3. Hence, P2(C) immerses only in codimension
^3, the range guaranteed by the Whitney Immersion Theorem. Moreover,
since the Л-genus is an oriented cobordism invariant, the conclusion is
much more general. In dimension four, A(X) = 2 sig(X) and we have the
284
IV. APPLICATIONS
following:
Corollary 2.2. If a compact oriented 4-manifold X can be immersed into IR6,
then the signature of X is even.
These results can be somewhat strengthened by two tricks. The first
trick allows us to deal with embeddings. Note that С£(т) = C£(T ф N) =
C£(T) ® C£(N) where ® denotes the graded tensor product of
Clifford bundles. Clearly we have CC^t) = [(1 ± co)C£(T)] ® C£(N) =
C£±(T)® C£(N).
Suppose now that the dimension of the normal bundle q = 0 (mod 4),
and let coN =	• •  eq be the normal volume element. Then <oN is parallel
in the projected connection and commutes with all tangent vectors. Hence,
oN commutes with the Dirac operator D. We can define bundles
C£±:t(T) = C£±(T) ® C£±(N) = (1 ± co)C£(T) ® (1 ± con)CI(N) (2.7)
and we get Dirac operators
D+ : Г(С£ + '(т))--> Г(С£- (т))	(2.8)
where can be either 4- or -. Evidently, D+ = D++ ф D+“, and so
ind(D+) = ind(D + +) + ind(D+).	(2.9)
Furthermore, we shall prove below the following fact:
Lemma 2.3. If the Euler class %(N) of the normal bundle is zero, then
ind(D + +) = ind(D+~).	(2.10)
Carrying out the averaging process above, we may replace all D’s by
D’s. (Note that each of Ct11^) is invariant under right-multiplication
by C£(t), and in particular by G.) It is an elementary fact that for an
embedded submanifold, = 0 (see Milnor-Stasheff [1]). Consequently,
we have the following:
Theorem 2.4. Let X be as in Theorem 2.1 and suppose q = 0 (mod4). If
there exists an embedding X c-*Rn+q, then
29~2A(X) = 0 (mod an+q).
The same conclusion holds for any immersion X	Rn+qfor which the Euler
class of the normal bundle is zero.
Our second trick is to simply take coefficients in a vector bundle over
X. This is technically quite easy and has the effect of introducing all of
the rational К-theory of the space into the problem. We get conditions
which are not simply in terms of characteristic numbers, but essentially
involve the entire cohomology ring. We proceed as follows.
§2. IMMERSIONS AND VECTOR FIELDS
285
Let E be a vector bundle over X. Here E may be real, complex or
quaternionic, and we assume it is equipped with an appropriate inner
product and connection. Then we take т = T ф N as above and consider
the bundle
C£(t) (x) E
with the tensor product connection. This remains a bundle of right and
left modules over X. It is easy to check that all of the constructions above
can be carried out with C£(t) replaced by С£(т) (x) E.
We need only compute the index of D + . We shall state the result here,
and give the proof below.
Lemma 2.5. The index of D+:T(C£+(t) ® E) -► Г(С£“(г) ® E) is given
by the formula
ind(D + ) = 2<{ch2£ • A(X)}[X]	(2.10)
where A(X) is the total А-class of X, and where ch2£ is defined as follows.
Let ch E = 1 4- ch!£ 4- ch2E + ... e H2*(X;Q), denote the Chern charac-
ter of E. Then for any t e R,
chtE = £ (chk£)?	(2.11)
kio
In the case that /(/V) = 0,
ind(D+") = | ind(D + )	(2.12)
where ind(D + ) is given by formula (2.10).
We note that when £ is real, we set ch £ = ch(£ ® C), and when £ is
quaternionic, we set ch £ = ch([£]c) where [ ]c denotes restriction of
scalars from 0-0 to C.
To state our results we shall need the following table of numbers which
is easily computed from the data given in Chapter I, §5. Let 2ak, 2bk and
ck denote the dimension over К of an irreducible Z2-graded K-module
for C£k where К = R, C and H respectively. Note that by Proposition
1.5.20, the numbers ak, bk and correspond to the dimensions of
irreducible ungraded modules for C(k _ p For all values of к we have that
ak + s =	bk + 8 — Ibbk'	Ck + 8 = 16q.
The values for к 8 are given in the following table.
n	1	2	3	4	5	6	7	8
								
<*n	1	2	4	4	8	8	8	8
ьл	1	1	2	2	4	4	8	8
ca	2	2	2	2	4	8	16	16
286
IV. APPLICATIONS
The method given above proves the following general result:
Theorem 2.6. Let X be a compact orientable manifold of dimension n, and
suppose there exists an immersion X Q* R"+g. Then for any complex bundle
E over X,
!{ch2E • A(X)}[X] = 0 (mod bn+q).	(2.13)
Furthermore, if q is even and the normal Euler class /(N) = 0, then
2q~2{ch2E • A(X)} = 0 (mod bn+q).	(2.14)
When n = 0 (mod 4), there are the following refinements: If E = ER ® C
for a real bundle ER, or if E = [EH]C for a quaternionic bundle EH, then
(2.13) holds with bn+q replaced by a„+q and c„+q respectively. If, furthermore,
q = 0 (mod 4) and = 0, the analogous improvements can be made in
(2.14).
Proof. Again the argument goes as follows. Let т = T ф N and consider
the Dirac bundle C£(t) (g) E. Average the Dirac operator over the group
in Г(С£(т)) generated by the global orthonormal sections ... ,cn+q of
t. Then the kernel and cokernel of the operator are Z2-graded modules
over the algebra С£и+(?, generated by .. ,cn+q. (The Z2-grading comes
from the even-odd grading of C£(t). In the complex case there are volume
forms in every even dimension: coc = imer • • • e2m). Applying the formula
from Lemma 2.5 gives the first part of the theorem. Splitting the operator
with the normal volume form and applying (2.12) gives the second part
of the theorem. 
We point out that the improvements in this method obtained by intro-
ducing coefficients are substantial. For example, they give results in every
even dimension.
The congruences (2.13) and (2.14) give a relatively simple machine for
calculating lower bounds on embedding and immersing dimensions.
For the complex and quaternionic projective spaces. P"(C) and IP”(IH1),
these methods give the best results of this kind for every n (see Lawson-
Michelsohn [1] for details). Similar non-immersion theorems were found
by К. H. Mayer [1]. There were earlier, slightly weaker results of this
type due to Atiyah and Hirzebruch [1], whose methods did not involve
elliptic operators.
Proof of Lemma 2.5. We note that the operator in question is just the
twisted signature operator (cf. II.6). More precisely, since С£±(Тф N) =
Ct^T) ® Ct(N), we see that we can write D+ as
D+:T\Ct+(T)®E)--------► Г(С£-(Т)® Ё)
§2. IMMERSIONS AND VECTOR FIELDS	287
where Ё = E® C£(N). It follows that
ind(D+) = {ch2E • L(X)}[X],	(2.15)
from the basic formula for the signature complex (see III. 13.9). Now
ch2E = ch2(E ® C£(N)) = ch2(E)ch2(C£(N)), and since T ф N = r =
Rn+fl, we see that C£(T)C£(N) = Cln + fl = R2" + 4 (trivial bundles). It follows
that
ch(C£(N)) = г^сЬ^Т))”1.	(2.16)
To compute this last expression, we apply the Splitting Principle and
express the total Chern class of T ® C formally as
л/2
c(T®C)= П (1 + xk)(l -xk)
k = 1
so that the Pontrjagin classes are
Pk(T) = ak(xl,... ,x*/2)
where ok denotes the /cth elementary symmetric function. In this language,
we have
Л/2 Xi
Recall that C£(T) ® С = Л*(Т ® C) and that therefore
ch(Ct(T)) = fl (1 +
k = 1
= 2" fl cosh2(xk/2).
k = l
Consequently,
ch(Cf.(N)) = 2’ fl cosh-2(xk/2),
k- 1
and for any t g R,
ch,(Cf(N)) = 2’ fl cosh"2(xkt/2).
k= 1
It follows that
n/2 x.
ch2(C£(N))L(X) = 2’ П •	'
k = i sinh(xk)cosh(xk)
2« П -------‘—
ЛА sinh(2xk)
= 2ЧЛ(Х),
288
IV. APPLICATIONS
since the Л-class is the multiplicative series given by the formal power
series 2x/sinh(2x). Formula (2.10) now follows immediately from (2.15).
For the second half of the lemma we make the following observation.
Let £ be a complex vector bundle with dimcE = 2m. Write c(E C) =
(1 — x^) so that pk(E) = <rk(xi,... ,x£) as above. Then we have
2m
ch(C£+E) - ch(C£-£) = J] (eXk - e’**)
*=i
= 2mX(E)a(E)
where a is a multiplicative sequence. It follows that
ind(£>+ +) - ind(D+ ") = {(ch2C£+N - ch2Cf N) • ch2E  L(X)}[X]
= {Z(N)-C}[X]
for some class C g Heven(X;Q). Hence, if %(N) = 0, we have ind(D+ +) =
ind(D + “) = |ind(D+) (cf. (2.9)). This completes the proof. 
Using the techniques developed above, we shall now analyze the second
problem mentioned at the outset of this section. Suppose our manifold
X carries q pointwise linearly independent vector fields ... ,eq. We
may assume e1?... ,eq to be pointwise orthonormal with respect to a
riemannian metric on X. Let us suppose for the moment that X is of
dimension n = 4k (and is orientable). Then we have the signature operator
D+:F(Ct+(X) -> Г(ССХ). As above, we may average D over the finite
group G c= Г(С£ X) generated by ... ,eq. That is, we set
|U| geG
where Rg denotes right Clifford multiplication by g. This multiplication
clearly preserves the sub-bundles Cf^T) = (1 ± co)C£(T), and we achieve
an operator D+: T(C£+(T) -► T(C£~(T)) which is G-equivariant and has
the same symbol as D+. As above we find that the kernel and cokemel
of D+ are Z2-graded modules over the Clifford algebra Ctq RG. Since
ind(D+) = ind(D+) = sig(X), we have the following:
Theorem 2.7 (Atiyah [9] Frank [1]). Let X be a compact oriented manifold
of dimension 4k. If X carries q pointwise linearly independent vector fields,
then
sig(X) = 0 (mod 2aq)
If, furthermore, q = 0 (mod 4), then
sig(X) = 0 (mod 4aq).
§2. IMMERSIONS AND VECTOR FIELDS
289
As an example, note that if X carries 3 such vector fields, then sig(X) =
0 (mod 8). If it carries 5, then sig(X) = 0 (mod 16). In general, if X carries
2m such vector fields, then sig(X) = 0 (mod 2W).
Proof of Theorem 2.7. The first statement follows from the argument
above. For the second statement, note that there is an orthogonal split-
ting T(X) = Т0®Тг where Tx = span(ej,... ,eq). Let w be the oriented
volume element of To. Since dim To = 0 (mod 4), we have that w2 = 1 and
w commutes with e1?... ,eq. By averaging ® as above, over the group Z2
generated by Rw, we may assume D commutes with Rw.
Since K2 = 1, we see that Rw has + 1 eigenbundles on each of C£+ and
СГ. They are given by C£ + ± = C£+(l ± w) and C£-± = C£"(l ± w).
Each bundle C£± ± is G-invariant, and we obtain two G-equivariant ellip-
tic operators
D+ + :T(C£ + +)--► Г(С£"+) and D+" :Г(С£ + ")-----------► Г(СГ’),
with the property that
D+ = D++ © D+~.
We know, therefore, that ind(D + ) = ind(D+ +) + ind(D+ “). A straight-
forward computation, in the spirit of that given for the second part of
Lemma 2.5, shows that ind(D+ +) — ind(D + “) = 2^/2x(T0)[X] = 0. Thus,
ind(D + +) = ind(D+) = | ind(D+)
and since the kernel and cokernel of D+ + are Z2-graded Ct^-modules, the
result follows as before. 
When Atiyah first presented these arguments (in [9]), he mentioned
that they could not achieve the additional power of 2 (when q = 0(4))
obtainable by other means. However, the last argument given above suc-
ceeds in getting the final 2 and gives the best known results of this type.
Theorem 2.7 is very pleasant because |sig(X)| is a homotopy invariant.
It is also an oriented cobordism invariant. However, for any given mani-
fold which carries a q-frame field, there is much more information avail-
able. For this we simply need to take coefficients in a vector bundle E.
Applying the arguments given above directly to C£(X) ® E proves the
following result (cf. Lawson-Michelsohn [1]).
Theorem 2.8. Let X be a compact oriented manifold of even dimension, and
suppose that X carries q pointwise linearly independent vector fields. Then
for every complex vector bundle E over X,
{ch2E • L(X)}[X] = 0 (mod 2bq)	(2.16)
290
IV. APPLICATIONS
If, furthermore, q is even and chq/2E = 0, then
{ch2E • L(X)}[X] = 0 (mod 4bq)	(2.17)
When dim(X) = 4k, there are the following refinements. If E = ER®C
for a real bundle ER, orifE = [Ен]с/Ог a quaternion bundle EH, then (2.16)
holds with bq replaced by aq or cq respectively. If furthermore, q = 0 (mod 4)
and chq/2E = 0, the analogous improvement can be made in (2.17).
Proof. The only point which needs attention here is the proof that
ind(D+ +) = ind(D+ ") when ch*/2E = 0. This follows from the equation
ind(D+ +) - ind(D+ ") = 2*/2{ch E • x(T0)}[X],
which can be verified by straightforward calculation. 
It is an interesting observation of Atiyah [9] that all of the above carries
over without change to prove the following generalization of Theorems
2.7 and 2.8.
Theorem 2.9. Let X be a compact oriented manifold whose tangent bundle
can be decomposed into oriented subbundles
T(X) = To © T. © T2 © • • • © Tq
where
dimR( Tk) = 1 (mod 4)	for к = 1,... ,q.
Then all the conclusions of Theorems 2.7 and 2.8 hold for X.
Proof. Let ek denote the oriented volume element for the plane field Tk,
к = 1,... ,q. Since these planes are mutually orthogonal and each has
dimension one (mod 4), the elements et,... ,eqe Г(С£(Х)) generate a
finite group G with R • G = C£q. The arguments now proceed exactly as
before. 
As a final application of these methods we consider the refined Clifford
index for C£k-Dirac bundles introduced in §7 of Chapter II. Let X be a
compact oriented manifold of dimension 4k 4- 1. Recall that the Kervaire
semi-characteristic of X is then defined to be the residue class
2k
<r(X) = Z MX) (mod 2)
j=i
= dim Heven(X) (mod 2)
= |x(X)(mod 2).
Theorem 2.10 (Atiyah [9]). Let X be a compact oriented (4k 4- i)-manifold.
If X carries an oriented tangent p-plane field for some p = 2 or 3 (mod 4),
then o(X) = 0.
§3. GROUP ACTIONS
291
Proof. By assumption there exists a splitting T(X) = To © where
dim(T0) = 2 (mod 4), and where To and Tx are oriented. This decomposi-
tion can be assumed orthogonal for some riemannian metric on X. Let
coo be the oriented volume form for To, and note that = — 1.
We consider now the Euler characteristic operator D°: Г(С£°(Х)) ->
ЦСС^Х)). Since dim(T0) is even, each C£k(X) is invariant under right
multiplication by coo. Averaging the Dirac operator over the group
generated by RWo makes ker DQ an RWo-invariant space, i.e., ker D° be-
comes aCtj = C-module. In particular, ker(D°) = Heven = H2k(X;W)
has even dimension (over R), and so a(X) = 0. 
§3. Group Actions on Manifolds
This section is concerned with smooth group actions. Recall that a Lie
group G is said to act effectively on a manifold X if there is a differentiable
map Ф: G x X -► X so that the corresponding map (p(g) = Ф(д9) gives
a continuous and injective homomorphism (p : G Diff(X). Here Diff(X)
denotes the group of Cx-diffeomorphisms of X in the standard C°°
topology.
There are two basic groups we shall consider here, namely,
S^zgC:^ 1}	(3.1)
S3 s e H : |</| = 1}	(3.2)
where multiplication is induced from multiplication in the field. There are
isomorphisms S1 = SO2 = Ux and S3 = Spin3 s SU2 = SpP These two
groups are basic for the following reason. We say that a Lie group G has
an Sk subgroup (for к = 1 or 3) if G contains a Lie subgroup isomorphic
to a finite quotient of Sk. (The only possibilities are S1 when к = 1 and
S3 or S3/%2 when к = 3.)
Proposition 3.1. Let G be a compact connected Lie group (of dimension >0).
Then G contains an S1-subgroup. In fact the Sl-subgroups are dense in G.
Moreover, if G is non-abelian, then it contains an S3-subgroup.
Proof. These statements follow directly from the elementary theory of
compact Lie groups (see Adams [1]). To prove the first, one notes that
every element of G is contained in a maximal torus, where ^-subgroups
are dense. To prove the last statement one takes the subgroup correspond-
ing to the subalgebra spanned by a root space and a root vector. 
Definition 3.2. We say that a manifold X admits an S*-action (k = 1 or
3) if there exists an effective action of a finite quotient of Sk on X.
292
IV. APPLICATIONS
Two main results concerning connected group actions on spin mani-
folds are the following:
Theorem 3.3 (Atiyah and Hirzebruch [3]). Let X be a compact spin mani-
fold which admits an Section. Then A(X) = 0.
Theorem 3.4 (Lawson and Yau [1]). Let X be a compact spin manifold
which admits an S3-action. Then j/(X) = 0.
Recall that j/ is the generalization of A to KO-theory given in (II.3.22).
In dimensions 4k, the two invariants are essentially the same. However, in
dimensions 1 and 2 (mod 8), the invariant j/ takes values in Z2 and can
be non-trivial while, of course, A = 0.
Remark 3.5. The stronger hypothesis of Theorem 3.4 is required for its
stronger conclusion. We can see this as follows. Let X8 be a compact spin
8-manifold with Л(Х8) = 1 (see the paragraph before II.7.12). Equip S1
with the interesting spin structure (cf. II.7.8), and take the product Y =
X x S1 with the product spin structure. Then Y admits an S^action, but
j/(Y) # 0.
The two theorems above could be combined into the following state-
ment by using Proposition 3.1.
Theorem 3.6 Let X be a compact spin n-manifold with j/(X) / 0. Ifn = 1
or 2 (mod 8), then the only compact, connected Lie transformation groups
of X are tori. If n = 0 (mod 4), then the only such group is the trivial one.
As we have noted before (cf. II.2.18), the spin manifolds with non-zero
j/ constitute half of the exotic spheres in dimensions 1 and 2 (mod 8). This
gives the following:
Corollary 3.6. Let E" be an exotic n-sphere which does not bound a spin
manifold (n = 1 or 2 (mod 8)). Then the only compact connected Lie sym-
metry groups of E" are tori of dimension [|(n -I- 1)].
Proof. The kernel of the surjective homomorphism : 0„ -► Z2, for n =
1,2 (mod 8), is the subgroup of homotopy spheres which do not bound
spin manifolds. This follows from the work of Milnor [7] and Adams [3].
If n is even, any toral transformation group Tk must have a fixed point
set, and the induced linear action on the normal spaces must be effective.
Therefore, 2k n. If n is odd and the fixed-point set is non-empty, a
similar argument applies. For the general case, see Ku [1]. 
Using other techniques, people have investigated the non-existence of
torus actions on exotic spheres. For example, using results of R. Schultz
§3. GROUP ACTIONS
293
[1] it can be shown that there are three exotic 10-spheres whose largest
connected transformation group is S1 or S1 x S1. These spheres are indeed
very unsymmetric!
It is interesting to note that this asymmetry is contagious. The
invariant is additive with respect to the connected sum operation Thus,
if X is a spin n-manifold with = 0 and if E is an exotic n-sphere as
above (n = 1 or 2 (mod 8)), then
rf(X К E) = з/(Е) / 0
and X # E only admits symmetry groups of toral type. This is particu-
larly striking when X = G/H is homogeneous. The manifold (G/H) # E is
homeomorphic to G/H, but has almost no symmetries. A good example
is the complex projective space X = IP4* + 1(C).
Another interesting phenomenon takes place under coverings. Fix an
exotic sphere E" as above, and let d be its order in the group 0n of
homotopy spheres. Consider the homogeneous space X = (S3f£d) x S"-3.
The manifold Y = X ft E" is quite inhomogeneous, however, its universal
(d-fold) covering space is
Y=(S3 x 5И~3)^(ЕЯ^;-^'ЕИ) = 53 x Sn~3,
d times
a symmetric space. This instability under finite coverings is not true of
the Л-genus.
Proof of Theorem 3.3. The argument is based on the Atiyah-Segal-Singer
Fixed-Point Theorem or, more specifically, the G-Spin Theorem (III. 14.11).
It proceeds as follows. Let X be an even-dimensional compact spin mani-
fold with an S'^action. We may assume by averaging that the action pre-
serves the riemannian metric. By passing, if necessary, to a 2-fold covering
group, we may assume that the S1 -action preserves the spin structure, i.e.,
that it lifts to an action on the bundle PSpin(X). This produces an ^-action
on the canonical complex spinor bundle $^, which projects to the given
one on X. This action commutes with the Atiyah-Singer operator 0,
i.e., g ° l/> ° g-1 = 0 for all g e S1. The volume form is S^invariant, and
so the splitting $c =	© $c is preserved. Thus the operator l/)+:
Г(#с) П#с) becomes an ^-operator, and we have defined an S^index:
indsi(0+) = [ker 0+] - [ker 0"] g ^(S1)	(3.3)
where K(S*) denotes the representation ring of S1. There is a natural
isomorphism of RiS1) with the Laurent polynomial ring
tri/^S1)----► Z^t"1]	(3.4)
obtained by taking the trace of the representation. Here t = eie corre-
sponds to the complex-valued function, or “character,” on S1 given by
294
IV. APPLICATIONS
the natural inclusion S1 <= C. Under the isomorphism (3.4) the index
defined in (3.3) becomes a Laurent polynomial which we denote by
4>*(0 = E
k= -N
Note that for any g e S1, we have
'₽>(») = t^|ker) - tr(0|ker₽-).	(3.5)
Now the G-Spin Theorem (III.14.il) gives a formula for ip+(g) of the
form
where Fg is the fixed-point set of g. For almost all g e S1, in fact, for all
non-torsion elements, the subgroup generated by g is dense. In this case
Fg = F = the fixed-point set of S1, and the expression &g varies only with
“normal rotation angles” as follows. The normal bundle to F has an S1-
invariant decomposition N = @otgZ Nm where each Nm carries the struc-
ture of a complex bundle and where S1 acts on Nm by scalar multiplication
by gm. For any given k-dimensional complex vector bundle E we define
the characteristic function
k i i	ii
V(E,t) = П
J= I
where t is taken to be an indeterminate and where, as usual, the x'/s arc
formal roots of the total Chern class of E. Because of the square roots, the
expression (3.6) is defined only up to sign.
Now for g e Sl с C a non-torsion element as above, the G-Spin Theo-
rem states that
<₽>(«) = (- 1Г (n V<</") • A(F)> [F]
f m	J
where the signs on each component of F are determined by the action of
S1 on the spin structure. It follows that we have an equality of rational
functions
<p• (0 = (-1 H П	tm) • A(F) J [F]
§3. GROUP ACTIONS
295
From (3.6) it is evident that the expression of the right is zero for
t = 0 and t = oo. Hence, the Laurent polynomial must vanish
identically. In particular, from (3.5) we have that (1) = dim(ker 0+) —
dim(ker 0~) = A(X) = 0. This completes the proof. 
The above argument actually proves the following stronger result.
Theorem 3.7 (Atiyah-Hirzebruch [3]). Let X be a compact spin manifold
of even dimension. For every non-trivial S1-action which preserves the spin
structure on X, the index
indsl(0+) = O in R(S1).
This result on S1 -actions has an interesting refinement observed by P.
Landweber, R. Stong and S. Weinberger. To state it we must differentiate
between actions of even and odd type. An S1-action on a spin manifold X
is said to be of even type if it lifts to an action on PSpin(Ar). Otherwise it
is said to be of odd type. (In the latter case the action of the 2-fold covering
group lifts to PSpinW ) These notions can be reformulated by considering
a lifting of the element -1 gS1 с C to PSpinW- The action is of even
type if and only if — 1 has order 2 on PSpinW- The action is of odd type
if and only if — 1 has order 4 on PSpin(X).
Suppose that the element — 1 acts freely bn X. Then one sees easily
that the S'1-action is even if and only if the quotient X/Z2 is a spin manifold.
If the fixed-point set XZ2 of — 1 is not empty, then
.. x (0 (m°d 4) if the action is even,
eodim(X•) = {2;шя)4;	if lhe action is odd. °”
We see this as follows. Recall that — 1 acts on its normal bundle N by
scalar multiplication in the fibres. Since the action of — 1 is orientation-
preserving (it is connected to the identity in Diff(X)), the rank of N is
even. Furthermore, there is a Z2-equivariant diffeomorphism of N with a
tubular neighborhood of X*1 in X. From this we see immediately that
the action on X is even if and only if the action on the normal sphere
bundle to XZ1 (with the induced spin structure) is even. Here the Z2-action
is free and by' the paragraph above the question comes down to whether
the quotient of the normal sphere is a spin manifold. By Remark II.2.4
we know that real projective (n — l)-space is spin if and only if n =
0 (mod 4). This proves (3.7). 
Theorem 3.8 (Landweber and Stong [1]). Let X be a compact spin mani-
fold of dimension 4k with an Section. If the action is of odd type, then
sig(X) = 0
296
IV. APPLICATIONS
Proof. A result of Hattori and Taniguchi [1] states that for any S1 -action
on an oriented manifold,
sig(X) = sig(Xs‘)
where Xs' is the fixed-point set of the action with an appropriately chosen
orientation. Consider now the inclusions: X5’ cz XZ2 cz X, and note that
(X*2)s* = X5*. By a result of Edmonds [1], the manifold X%2 is orientable.
Therefore, applying the argument again shows that sig(X) = sig(X22).
However, by (3.7) we have that dim(X22) = 2 (mod 4), and so
sig(X*2) = 0. 
This result is analogous to the fact that on manifolds which admit
orientation reversing diffeomorphisms, all the Pontrjagin numbers are
zero. For S^-actions on spin manifolds which are free outside the fixed-
point set, there are additional characteristic classes which vanish (see
Landweber and Stong [1]). This leads into the fascinating area of elliptic
genera (cf. Chudnovsky and Chudnovsky [1]).
Proof of Theorem 3.4. Recall that any spin manifold X which carries a
metric of positive scalar curvature has л/(Х) = 0 (see 11.8.12). Thus Theo-
rem 3.4 is a corollary of the following more general result. 
Theorem 3.9 (Lawson and Yau [1]). Any compact spin manifold which ad-
mits a non-trivial S3-action, carries a riemannian metric of positive scalar
curvature.
When the action is free, the proof of this theorem is rather easy. In this
case, the manifold is the total space of a smooth principal S3-bundle X Л
M = X/S3. Introduce an invariant splitting TX = i ' @ УТ where
denotes the field of 3-planes tangent to the orbits. The bundle iT has a
natural trivialization / ’ X x R3 given as usual by choosing an ortho-
normal basis ei9e2.e3 of TJS3) and defining sections Ej of / ’ by Ej(x) =
d/dt [exp(tej) • x\ = o- We fix a metric on M and for each e > 0 we construct
a metric gr on X as follows. We declare and Ж to be everywhere
orthogonal; we lift the given metric on M to # via n; and we set <Eh E;> =
с2ди. This uniquely determines the metric gr. In this metric the orbits of
S'3 are all totally geodesic and isometric to the euclidean sphere of radius
v. Applying the fundamental equations of O’Neill [1] for a riemannian
submersion, one easily sees that the scalar curvature ке of the metric gc
is of the form ке(х) = 6/e2 + a(x) + c2b(x) for continuous functions a and
h on X. Taking c sufficiently small completes the proof for this case.
The general case, where the action of S'3 is not free, is more difficult
and is handled as follows. Consider the diagonal action of S'3 on X x S3.
This action is free with quotient X. (Of course, this orbit space map
§4. COMPACT MANIFOLDS WITH к > 0	297
X x S3 X is different from the “product” projection pr: X x S3 -> X,)
We now fix an S3-invariant metric of X, and using this metric we con-
struct, for each £ > 0, a metric ge on X x S3 in the manner above. The
metric ge is, in fact, S3 x S3-invariant and thus makes the product pro-
jection pr: X x S3 -> X into a riemannian submersion. Detailed calcula-
tions show that for £ sufficiently small, this submersed metric has positive
scalar curvature. The estimates in this calculation are most delicate at the
fixed point set.
Theorem 3.9 actually provides a wealth of non-existence theorems for
S3-actions. We clearly have the following:
Corollary 3.10. Any compact manifold which cannot carry a riemannian
metric of positive scalar curvature has no S3-actions.
In the next section we shall produce large families of manifolds which
do not carry positive scalar curvature. One consequence of this will be a
proof of the fact (due originally to Browder and Hsiang [1]) that if a spin
manifold X admits an S3-action, then all the “higher A-genera” of X are zero.
§4. Compact Manifolds of Positive Scalar Curvature
One of the most important applications of the Atiyah-Singer operator in
riemannian geometry is in the study of manifolds of positive scalar cur-
vature. Recall that the scalar curvature of a riemannian manifold X is a
function k:X R defined at each point x by averaging all the sectional
curvatures at x. Specifically we define
«*) = t <*e<,e^U>	(4.1)
»J=1
where {el9... 9en} is an orthonormal basis of TxX and R is the Riemann
curvature tensor. Note that к can also be written as
k(x) = trace(Ricx)	(4.2)
where Ricx denotes the Ricci transformation at x. Thus, the condition of
positive scalar curvature, i.e., к > 0, is the weakest of the various “positive
curvature” assumptions.
There has already been some discussion of manifolds with к > 0 given
in §8 of Chapter II. We recall the basic results proved there. There is a
graded ring homomorphism
^:QSPin----> KO"*(pt)	(4.3)
which in dimensions 4k is a multiple of the Л-genus. This homomorphism
is defined in (3.22) of Chapter II. It coincides with the KO-index of the
298
IV. APPLICATIONS
Atiyah-Singer operator, and can be computed in terms of the space of
harmonic spinors (see Theorem II.7.10). Applying the Bochner-type for-
mula (II.8.18) gives the following result of Atiyah, Hitchin, Lichnerowicz
and Singer.
Theorem 4.1. Let X be a compact spin manifold with к > 0, then «$/(%) = 0.
This theorem implies, in particular, that a spin 4/c-manifold with
non-vanishing Л-genus cannot carry a metric of positive scalar curvature.
There are many examples of such manifolds. For example, let V2k(d)
denote a non-singular complex hypersurface of degree d in IP2k+ ^C). Then
V2k(d) is spin if and only if d is even. (See II.2.7.) However, we have that
(«)
so that for all even d > 2k, and all k 1, the 4/c-dimensional spin manifold
V2k(d) cannot carry к > 0, and therefore cannot carry metrics of positive
Ricci or positive sectional curvature.
It is a consequence of the Calabi Conjecture, proved by S. T. Yau [2],
that each of the hypersurfaces V2k(d) for d 2k + 1 carries a metric of
positive Ricci curvature. This shows that the spin condition in Theorem
4.1 is necessary. It also shows, together with Theorem 4.1, that the poly-
nomial^) = 4(K2k(d)) must have zeros ford = 2,4,... ,2k. It is interesting
that with this information and some general considerations, it is rather
easy to compute formula (4.4) for p(d).
An important phenomenon occurs in the borderline case V2k(2k + 2)
where Yau proves that there exist metrics of zero Ricci curvature. In this
case we see that A = 2. It follows from Corollary II.8.10, that in any Ricci
flat metric, V2k(2k + 2) carries at least two parallel spinor fields. Such
fields automatically reduce the holonomy group of the metric. We shall
pursue these considerations further in §10. We now return to the discus-
sion of Theorem 4.1.
Recall that the invariant «я/ is non-zero also in dimensions 1 and 2
(mod 8). Here it takes values in Z2. Among the spin manifolds on which
it is non-zero, are half of the exotic spheres in dimensions 1 and 2 (mod 8).
(A general discussion of positive curvature metrics on exotic spheres is
given in II.8.) This fact has the following consequence:
Corollary 4.2. In dimensions 1 and 2 (mod 8), every compact spin manifold
is homeomorphic to a manifold which cannot carry positive scalar curvature.
Proof. Let X be a spin k-manifold with k = 1 or 2 (mod 8), and let E be
an exotic к-sphere with л/(Е) / 0. Since ^{X # E) = <s£{X} 4- л/(Е),
we may apply Theorem 4.1 to either X or X ft E. 
§4. COMPACT MANIFOLDS WITH к > 0
299
To keep these non-existence theorems in perspective it is useful to
examine manifolds which are known to carry positive scalar curvature.
The classical examples of spaces with curvature ^0 are the homogeneous
spaces with normal metrics given as follows. Let G be a compact Lie
group with Lie algebra g, and let (•, ) be any AdG-invariant inner prod-
uct of g. Then for any closed subgroup H c G, there is a G-invariant
metric naturally introduced on the coset space X = G/H. The sectional
curvatures of this metric are always ^0, and apart from the case of the
flat torus, the scalar curvature is always >0. (For details, see Kobayashi-
Nomizu [2] or Cheeger-Ebin [1].)
These homogeneous spaces represent special cases of the general pheno-
menon of Theorem 3.8. Any compact manifold which admits a compact
connected non-abelian Lie transformation group, carries a metric with
к > 0.
There is a general surgery procedure for constructing metrics with к >
0. Recall (cf. Milnor [8]) that a surgery of codimension q on an и-manifold
X is a modification of X of the following type. Let '£n~q <= X be a
smoothly embedded (n — q)-sphere with a trivialized tubular neighborhood.
By this we mean a neighborhood U of together with a diffeomorphism
/: U -► Sn~q x Dq such that f(Ln~q) = Sn~q x {0}. The surgery operation
now consists of removing the neighborhood U = Sn~q x Dq and replacing
it with the product Dn~q+l x Sq~r by gluing in the obvious, canonical
way along the boundary Sn~q x S*-1.
Proposition 4.3 (Gromov-Lawson [2] and Schoen-Yau [2]). Let X be a
manifold which carries a metric of positive scalar curvature. Then any
manifold obtained from X by surgery in codimension 3, also carries a
metric of positive scalar curvature.
The method of proof in Gromov-Lawson [2] is an elementary con-
struction, the proof in Schoen-Yau [2] uses techniques of partial differ-
ential equations.
Combining this result with techniques of the h-cobordism Theorem (cf.
Milnor [8]) and facts concerning the cobordism ring Q^°, it is possible
to prove the following result:
Theorem 4.4 (Gromov-Lawson [2]). Let X be a compact simply-connected
manifold of dimension 5.
(A)	If X is not spin, then X carries a metric with к > 0.
(В)	If X is spin, and is spin cobordant to a manifold with к > 0, then X
carries a metric with к > 0.
Remark. Contrasting part (A) of this theorem with Corollary 4.2 makes it
evident that the question of positive scalar curvature is intimately related
to spin geometry. This relationship will soon be examined in detail.
300
IV. APPLICATIONS
Proof of Theorem 4.4. Suppose X is spin and spin cobordant to a mani-
fold XQ which carries к > 0. By a sequence of surgeries (on embedded
circles) we can make XQ simply-connected while preserving the spin
corbordism class. By Proposition 4.3, the new XQ will also carry к > 0.
Our cobordism assumption means that there exists a compact spin
manifold W with dW « X II Xo (as spin manifolds). As above, we can kill
by surgery. Moreover, since dim W = n 6 and W is spin, every
2-sphere embedded in W has a trivial normal bundle (cf. II.2.11), and we
can kill n2(W) by surgery. It follows that the groups Hn_k(W,X0) =
Hk(W,X) are zero for k = 1,2, and torsion-free for k = 3. By the basic
theory of S'. Smale (cf. Milnor [8]), there exists a smooth function f:W-+
[0,1] with the following properties:
(i)	/|xo = 0 and f\x = 1,
(ii)	all critical points of f are non-degenerate and lie in the interior
of IK,
(iii)	all critical points of f have index n — 3.
It now follows from the fundamental theorem of Morse Theory (cf.
Milnor [6]) that X can be obtained from Xo by surgeries in codimension
3. We then apply Proposition 4.3.
When X is not spin, the argument is similar. From detailed knowledge
of the oriented cobordism ring we know that there exists a compact
oriented manifold W with d W = X II Xo, where Xo carries к > 0. We
may assume that fti(X0) = 0 as before. We now proceed to kill rc^IK) by
surgery. We then have H2(VH) я2(И/), and the second Stiefel-Whitney
class gives a homomorphism
w2:7t2(W9----->Z2.	(4.5)
Since dim W 6, the group я2(1У) is generated by embedded 2-spheres,
and w2 exactly measures the non-triviality of the normal bundle to these
embeddings. Thus, we can kill the subgroup ker(w2) с я2( W). The homo-
morphism (4.5) then becomes injective. Observe now that w2 restricts to
be the second Stiefel-Whitney class of X, which is non-zero since X is not
spin. It follows that (4.5) is an isomorphism and that the map H2(X0) £
я2(Х0) -► n2(W) ~ H2(IV) is surjective. In particular, Hk(W,X0) = 0 for
k = 1,2. Consequently Hk(W,X) = 0 for к = 1,2, and H3(VF,X) is torsion
free. The argument now proceeds as before. 
Let X be a compact manifold of dimension = 5. Theorem 4.4 states that
there is an obstruction to the existence of a positive scalar curvature metric
on X only if X is spin, and, furthermore, that obstruction depends only on
the spin cobordism class of X. This leads us to consider the set of spin
cobordism classes a g Q^pin with the property that some manifold in a
carries positive scalar curvature. We claim that = ф is an ideal in
§4. COMPACT MANIFOLDS WITH к > 0
301
QSpin Closure under addition (which corresponds to disjoint union) is
clear. Suppose now that a g and choose any class Д g Q^p,n. Represent
a by a spin manifold X with a metric gx with kx > 0. Represent fl by a
spin manifold Y with some metric gY. Then for all g > 0, the product
metric g = cgx -I- gY on X x Y has scalar curvature к = гкх + ky which
is >0 for all sufficiently small g. This shows that fl * a g ^и+ш, and is
an ideal as claimed.
Consequently, setting to* = Q*pin/^₽*, we get a graded ring homo-
morphism
fi.QSpin --(4.6)
Note that in each dimension n, fo„ is a finitely generated group. Theorem
4.4 states the following.
Corollary 4.5. The classes a* constitute a complete set of invariants for de-
termining whether or not a simply-connected manifold (of dimension 5) can
carry a metric of positive scalar curvature.
The fundamental Theorem 4.1 implies that there is a factoring of graded
ring homomorphisms
KO-*(pt)
and it seems likely that n is an isomorphism, i.e., that a £ (We know
и ® Q to be an isomorphism; see Gromov-Lawson [2].) If this were so,
then the .o/-class would be exactly the obstruction to the existence of posi-
tive scalar curvature metrics in the simply-connected case.
Since Q;Jpin = о for n = 5, 6 and 7, Theorem 4.4 has the following con-
sequence.
Corollary 4.6. Every compact simply-connected manifold of dimension 5, 6,
or 7 carries a metric of positive scalar curvature.
We shall next take up the question of positive scalar curvature on non-
simply-connected manifolds. Here the situation can be quite subtle, as the
following example, due to L. Berard-Bergery [2], illustrates. Let E be an
exotic 9-sphere such that j/(E) ± 0, and consider the spin manifold X =
(P>7(IR) x S'2) ft E, with nrX Z2. This manifold does not carry positive
scalar curvature (since j/(X) = ^/(E) / 0), however its two-fold universal
covering manifold does. To see this, one uses the fact that Q|p,n = Z2 ф Z2
(cf. Stong [1]) which shows that the universal covering X = (S'7 x S'2) #
E#E S7 x S'2.
302
IV. APPLICATIONS
§5. Positive Scalar Curvature and the Fundamental Group
We now take up the case of manifolds with non-trivial fundamental group.
To set the stage we note that the product of any manifold with S2 carries
a metric with к > 0. Consequently, in dimensions ^6, the presence of
positive scalar curvature places no restriction on the isomorphism class of
the fundamental group. (This is also true in dimensions 4 and 5; cf. R.
Carr [1].) Nevertheless, if one takes into account the interaction of the
fundamental group with the geometry of the manifold, then tremendous
restrictions arise.
The interaction we are seeking lies in the geometry of the covering
spaces. In particular, we shall be interested in manifolds with covering
spaces which are “large in all directions, or put another way, whose uni-
versal covering spaces contain big “cubes.” To give an idea of what this
should mean, we examine a notion due to Gromov. Let Г denote the
standard metric cube in R". We shall say that a riemannian n-manifold X
contains a cube of size L if there exists a continuous map f:In-*X such
that dist(y(S)/(S')) = L for each pair of opposite faces E,E' of the cube Г.
We can then say that a riemannian manifold X has big covering spaces, if
for any L > 0, there is a covering space which (in the lifted metric) contains
a cube of size L. For a compact manifold this concept is independent of
the metric chosen (since any two metrics are uniformly commensurate).
A good example of a manifold with big covering spaces is a torus Tn.
This idea can be usefully generalized. However, for the purposes here
it is better to “dualize” the concept by mapping out of the manifold rather
than into it.
Definition 5.1. A Cx-map f:X-*Y between riemannian manifolds is
s-contracting (for a given £ > 0) if ||/*t>||	£||v|| for all tangent vectors v
to X. This means that for any piecewise smooth curve у in X, one has
length(/(y)) S dength(y).
Such £-contracting maps are not hard to find (consider the constant
maps). Suppose, however, that X and Y are compact oriented manifolds
of the same dimension. Then the existence of an ^contracting map f: X -►
Y of non-zero degree means that in a strong sense X is “bigger than У” on
an order of at least l/£. Now every oriented n-manifold admits maps of
degree 1 onto the n-sphere. Thus, we let S"(l) denote the standard riemann-
ian n-sphere of constant curvature 1, and we replace the above notion of
big covering spaces with the following (cf. Gromov-Lawson [1],[3]):
Definition 5.2. A compact riemannian n-manifold is said to be en-
largeable if for every £ > 0 there exists an orientable riemannian covering
space which admits an £-contracting map onto S"(l) which is constant at
infinity and of non-zero degree.
§5. к > О AND THE FUNDAMENTAL GROUP	303
If for each £ > 0, there is a finite covering space with these properties, we
call the manifold compactly enlargeable.
Note. A map is constant at infinity if it is constant outside a compact set.
The degree of such a map f: X Sn is defined as
f
deg(/) = ^------- (5.1)
co
JS"
where co is an n-form on Sn with non-zero integral. The degree can also be
defined as usual in terms of regular values of f.
The square flat torus Tn = is certainly enlargeable since the uni-
versal covering space has the required mappings for all £ > 0. This torus
is, in fact, compactly enlargeable. We see this as follows. For each k e Z*,
the lattice (k • Z)" <= Z" gives a fc"-fold covering torus Tn = Un/(k • Z)",
which admits the (7i//c)-contracting map to S"(l) of degree 1 pictured below.
Theorem 5.3. The following statements hold in the category of compact
manifolds:
(A)	Enlargeability is independent of the riemannian metric.
(B)	Enlargeability depends only on the homotopy-type of the manifold.
(C)	The product of enlargeable manifolds is enlargeable.
(D)	The connected sum of any manifold with an enlargeable manifold is
again enlargeable.
(E)	Any manifold which admits a map of non-zero degree onto an en-
largeable manifold is itself enlargeable.
Proof. It is evident that (E) => (B) => (A) and that (E) => (D). To prove (E)
we consider two compact oriented riemannian n-manifolds X and У, and
a map F : X -► У of non-zero degree. By compactness there exists a c > 0
so that ||dF|| c on X (i.e., F is c-contracting). Given £ > 0, there is a
304
IV. APPLICATIONS
riemannian covering space p: Y -► Y which admits a (e/c)-contracting map
f: Y -► S"(l) which is constant outside a compact set К a Y and of non-
zero degree. Taking the fibre product of p and F gives a covering space
p': X -► X and a proper mapping F: X -+ Y so that the diagram
X	Y
p'	p
X	Y
commutes. Since F is a lifting of F, we have ||VF|| c on X. Hence, the
composition f ° F:X -► S"(l) is e-contracting. Since F is proper, we see
that f о F is constant outside the compact set F X(K). It is easy to see that:
deg(/ ° F) = deg(/)deg(F) 0. Hence, X is enlargeable as claimed.
To prove (C), we fix a degree-1 map </>:S"(l) x Sw(l) -► S"+w(l) and let
c = sup||d</>||. This map is chosen to be constant on the set (S"(l) x {*}) и
({*} x Sm(l)), where each denotes a distinguished point in the sphere.
Suppose now that we are given (e/c)-contracting maps, f: Xn -► S"(l) and
g :Xm -► Sw(l), which are constant (= *) at infinity and of non-zero degree.
Then the map ф ° (/ x g): Xn x Ym--------► S" + m(l) is e-contracting, con-
stant at infinity and of non-zero degree. From here the argument is
straightforward. 
Theorem 5.3 says that the category of enlargeable manifolds is closed
under products, connected sums (with anything), changes of differentiable
structure, etc. The category is also quite rich. It contains, as basic building
blocks, many families of important K(7i,l)-manifolds.
Theorem 5.4 (Gromov-Lawson [1]). The following manifolds are enlarge-
able:
(A)	Any compact “solvmanifold”, i.e., a compact manifold diffeomorphic to
G/Г where G is a solvable Lie group and Г is a discrete subgroup.
(B)	Any compact manifold which admits a metric of non-positive sectional
curvature.
(C)	Any ''sufficiently large” 3-manifold (X is sufficiently large if there is
a compact surface of positive genus IcX with	-► лгХ injective.)
Proof. We begin with (B). It is easy to see that a manifold X of non-
positive curvature is enlargeable. By the Cartan-Hadamard Theorem (cf.
Milnor [6]), the exponential map at any point of the universal covering
X is a diffeomorphism whose inverse exp71: X -► TxX is distance de-
creasing onto TxX z R" with its euclidean metric. Composing with a fam-
ily of e-contracting maps fE: R" -► S"(l) as above completes the proof of (B).
§5. к > О AND THE FUNDAMENTAL GROUP
305
To prove (A) we invoke the basic fact that every compact solvmanifold
X admits a fibration it: X -► Tk over a torus of positive dimension with
a (compact) solvmanifold Xo as fibre (see Raghunathan [1] for example).
We introduce the standard flat metric on Tk = Rk/Zk, and we fix a metric
g on X. By replacing g with tg for t > 0 sufficiently large, we can assume
that я is distance-decreasing.
Consider now the covering X -► X induced by lifting it over the cov-
ering Rk -► Tk. The induced fibration n: X -► Rk is again distance de-
creasing. Furthermore, there exists a diffeomorphism
q>\X-----► Rk x Xo
such that я = prx ° <p where pr!: Rk x Xo Rk is the projection map. To
see this when k = 1, lift a non-vanishing vector field from Tl to X and
integrate to get the coordinate transverse to the fibres. When k > 1, apply
induction.
Suppose that £ > 0 is given. Consider the ball В = {x g Rk: ||x|| < я/е}
and fix a map f: Rk -► Sk(l) which is ^contracting, constant on Rk- В
and of degree 1 (as pictured in the figure above). Choose a metric g0 on
Xo so that the composition
X R x Xo Xo
is distance decreasing on the compact subset я-1(В) с: X. (A sufficiently
small multiple of any metric will do.) We may assume that Xo is en-
largeable by induction on dimension. Hence there exists a covering
я0: Xo -► Xo and an ^contracting map g: Xo -► S"-k(l) which is constant
at infinity and of non-zero degree. Let q: X -► X be the covering induced
by lifting (p over Id x я0, so there is a commutative diagram (below).
306
IV. APPLICATIONS
We now take the “smash product”
* F л G
X-^-+ S"(l)
exactly as we did in the proof of Theorem 5.3(C). This map is ^-con-
tracting, constant at infinity and of non-zero degree. This completes the
proof of part (A). For part (C), the reader is referred to Gromov-Lawson
[1]- 
Note that the tori Tn = S1 x • • • x S1, n 1, play a special role in
this theory. For example, they belong to each of the three classes of
manifolds in Theorem 5.4. Furthermore, there is the following “stability”
phenomenon:
If X is an enlargeable manifold, so is X x Tn for any n 1.	(5.2)
If a manifold X carries a metric with к > 0, so does X x Tn	(5.3)
for any n 1.
There is an (adversary) relationship between enlargeability and positive
scalar curvature, and the mediating agent is the Atiyah-Singer operator.
Here, however, the operator has coefficients in a vector bundle. The first
main result is the following:
Theorem 5.5 (Gromov-Lawson [1], [3]). An enlargeable spin manifold X
cannot carry a metric of positive scalar curvature. In fact any metric with
к^ОопХ must be flat.
Note. Here the spin assumption is not vital. It suffices to know that an
appropriate covering space is spin.
Note. Results of this type were first proved in dimensions ^7 by R.
Schoen and S. T. Yau [1], [2] using minimal surface techniques.
Theorems 5.4 and 5.5 together produce the following nice “exclusion
theorem.”
Corollary 5.6. A compact manifold which carries a metric with sectional
curvature ^0 (or <0), cannot carry a metric with scalar curvature >0(^0
respectively).
Proof of 5.5. For clarity’s sake we shall present here a proof for the case of
compactly enlargeable manifolds. The more general result requires some
tools from analysis (the Relative Index Theorem) and will be proved with
more general results in the next section.
Let X be a compactly enlargeable n-manifold, and suppose X carries a
metric with к kq for a constant к0 > 0. From (5.2) and (5.3) we may
assume that X has even dimension 2n. (If not, replace X by X x S1.)
§5. к > О AND THE FUNDAMENTAL GROUP	307
Choose a complex vector bundle Eo over the sphere S2"(l) with the prop-
erty that the top Chern class c„(E0) / 0. This is certainly possible, since
on S2n the Chern character
gives an isomorphism ch: K(S2n) -► Z) (cf. Atiyah-Hirzebruch [2]).
We now fix a unitary connection VEo on Eo and we let REo denote the
curvature 2-form.
Let £ > 0 be given and choose a finite orientable covering X -► X which
admits an ^contracting map f: X -► S2"(l) of non-zero degree. Using /,
we pull back the bundle Eo, with its connection, to X. This gives us a
bundle E = y*E0 with connection Vе = f*VE°. We then consider the com-
plex spinor bundle $c of X with its canonical riemannian connection, and
consider the Atiyah-Singer operator 0 on the tensor product $c ® E (as in
II.5.10). We know from Theorem II.8.17 that there is a “Bochner formula”
0l = v*V +	(5.4)
where 9t£(a ® c) =	® (^ ,eke) depends universally and lin-
early on the components of the curvature tensor RE of E. The operator
is a symmetric bundle endomorphism of $c ® E, and if its pointwise norm
everywhere satisfies the inequality
IMI<^o,	(5.5)
then we will have Ц>1 > 0, and the index of the operator
: Г($с ® E)-----> Г($г ® E),	(5.6)
will be zero. To achieve this estimate we first note that there is a constant
kn depending only on dimension, such that
||«£|| S M4	(5.7)
We then observe that the curvature of E is the pull-back of the curvature
of Eo, and therefore
(5.8)
To prove (5.8), we fix a point x g X and let {ф19...	be an ortho-
normal basis of \2TxX which diagonalizes the symmetric bilinear form
Р(ф9ф') =	This means that	= tfdjk where, since
f is £-contracting, we have |zj £2 for all j. It follows that there exists an
orthonormal basis {ф19... ,^и(и_1)/2} °f h2T r(x}S2n such that /*ф^ =
for all j. We then compute that ||RE||2 = £ ||Kj ||2 = £ ||KE.°J|2 =
308
IV. APPLICATIONS
Combining (5.7) and (5.8) shows that the positivity condition (5.5) is sat-
isfied for all £ < \lKQ/4kn. The index of the operator (5.6) must then vanish.
However, this index is given by
index(0£) = {ch E • A(X)} [X]	(5.9)
= {(d + (7LHj!C"(£)) Aw} Й
= d^) + (7=4j!c"(£)[^]
= ^4iyde^"(Ms2"]
/0.
This contradiction completes the proof of the first statement of the theorem.
To prove the second statement, we call upon the following result which
appears in Kazdan-Warner [1]:
Theorem 5.7 (J. P. Bourguignon). Let X be a compact manifold which car-
ries no metric with к > 0. Then any metric with к 0 on X is Ricci flat.
To complete the argument we now apply the following corollary of the
Splitting Theorem of Cheeger and Gromoll [2].
Proposition 5.8. A Ricci flat enlargeable manifold is flat.
Proof. Let X be a (compact) enlargeable и-manifold with a Ricci flat
metric. The Splitting Theorem implies that the universal cover of X splits
as a riemannian product x Y of a euclidean space with a compact
simply-connected (Ricci-flat) manifold Y. We claim that dim(T) = 0. If
not, set d = diameter( Y) > 0, and choose a covering X of X which admits
a (7i/4<5)-contracting map f .X -► S"(l) which is constant at infinity and
of non-zero degree. This covering is a quotient X = (Rm x У)/Г by a
subgroup Г of the deck group. The deck group acts freely and properly
discontinuously by isometries which preserve the factors. By passing to a
subgroup of finite index we may assume that Г acts freely and properly
discontinuously on Rm. Then X is a fibre bundle p: X -► Rm/r over the
quotient manifold with fibre Y.
Since f is (7i/4<5)-contracting, we see that each fibre is mapped into a ball
of radius я/4 in S"(l). Hence, there is a continuous map /0: Rw/T -► S"(l)
§5. к > О AND THE FUNDAMENTAL GROUP
309
so that
dist(/0(p(x)),/(x)) < я/2
for all x g X. (Choose a section of the disk bundle whose fibre at у g Rm/T
is the convex hull of /(Ур) = /(p-1(y))) This map can be chosen to agree
with f at infinity. By pushing along the unique geodesic arc joining /(x)
to /o(P(x))> we construct a homotopy from f to /0 ° p which is constant at
infinity. Since dim(K) > 0, it follows that deg(/) = deg(/0 0 p) = 0 contrary
to assumption. Therefore, we conclude that dim(K) = 0, and both the
proposition and the theorem are proved. 
The argument given for Theorem 5.5 actually proves much more than
is claimed. To use the full force of the argument we only need to readjust
our definitions. In doing this we shall unify these results with the previous
ones concerning the A-genus.
We begin by generalizing the notion of the degree of a map. Let f: X -► Y
be a smooth map between compact oriented manifolds, and suppose
dim X — dim Y = 4k 0. The inverse image /-1(y) of a regular value
у g У is a compact 4k-manifold whose oriented cobordism class is indepen-
dent of y. To see this, join two regular values у and y' by a smoothly
embedded are y, and wiggle /, away from f~l(y) u f~l(y'), so that it
becomes transverse to y. The inverse image of у provides a cobordism
between/’/у) and f~i(y,\ It now follows that the Л-genus Л(/-1(т))
is independent of y. We call this number the A -degree of f.
The Л-degree can be computed by a formula of type (5.1). Let co be a
volume form on У with non-zero integral, and let Ak(X) be a de Rham
representative (a closed 4k-form) of the kth-component of the total Л-class
of X. Then
Г/*солЛк(Х)
^-deg(/) = J----------------	(5.10)
Jr"
Notice that the Л-degree makes sense when X is not compact, provided
that f is constant at infinity.
We now generalize the notion of enlargeability by replacing the word
“degree” in 5.2 by the word “Л-degree”.
Definition 5.9. A compact riemannian n-manifold is said to be Л-
enlargeable if given any £ > 0, there exists a riemannian covering space
which admits an e-contracting map onto Sw(l) (for some m = n (mod 4))
which is constant at infinity and of non-zero Л-degree. The basic facts
are as before.
Theorem 5.10. The statements (Л), (C) and (D) of Theorem 5.3 remain true
with the word “enlargeability” replaced by the word “A-enlargeability.”
310
IV. APPLICATIONS
Proof. The proof of part (C) of Theorem 5.3 goes through after making
the obvious linguistic changes. Parts (Л) and (D) are easy. 
Of course, an enlargeable manifold is Л-enlargeable. Moreover, an ori-
entable manifold of non-zero Л-genus is also Л-enlargeable since the con-
stant map to 5° has non-zero Л-degree. More generally we have the
following:
Corollary 5.11. A compact manifold which admits a mapping of non-zero
А-degree onto an enlargeable manifold is A-enlargeable. In particular, the
product of an enlargeable manifold and a manifold of non-zero A-genus
is A-enlargeable.
We now have a unification of many of the previous results.
Theorem 5.12 (Gromov-Lawson [1], [3])). An A-enlargeable spin manifold
cannot carry a metric with к > 0. In fact, any metric with к 0 on such a
manifold is Ricci flat.
Proof. The proof of Theorem 5.5 goes through with no essential changes.
The main point is that the computation (5.9) now becomes
ind(^) = f(d + —!— cm(£)) • A(X)l [X]	(5.11)
(A (m - 1)'	/ J
= dA(X) + —{cm(£) • Л»(Х)}[Х]
(m - 1)!
= (^Hn)!(x’deg/)c’"(£o)[s2’"]
/0. 
This theorem implies, in particular, that a spin manifold of the form
X x Y where X is enlargeable and Л(У) / 0, cannot carry positive scalar
curvature. In a sense, it interpolates between the two previous results
which concerned enlargeability and A separately.
A good interpretation of Theorem 5.12 can be made in terms of the
higher Л-genera. These are defined for any compact differentiable mani-
fold X as follows. Fix a K(n,l)-space К and a map f :X->K (corre-
sponding to a homomorphism nxK). For each cohomology class
и g H*(K; Z) Н*(л; Z), the higher Л-genus associated to u is the number
Л/1В(Х) = {/*мА(Х)}[Х]
§5. к > О AND THE FUNDAMENTAL GROUP	311
where, as usual, A(X) is the total Л-class of X. We could, of course, pass to
the “universal” space Kx =	and the map fx:X -► Kx through
which every f :X -► К(л,1) can be factored. These “universal” higher
A -genera are simply indexed by the cohomology classes of the group л x(X).
Example 5.13. When the space К = К(л,1) is a compact oriented m-
manifold and when и e Hm(K\ Z) is the fundamental cohomology class,
then it is evident that:
ЛЛи(Х) = (Л-degX/).	(5.12)
Observe now that many fundamental examples of enlargeable manifolds
(such as solvmanifolds, manifolds of non-positive curvature, and suffi-
ciently large prime 3-manifolds) are manifolds of K(7t,l)-type. Theorem
5.12 states that for a spin manifold X of positive scalar curvature, the
higher A -genera, coming as in (5.12) from maps to an enlargeable mani-
fold, must be zero.
It is at the moment unknown whether every compact K(n,l)-manifold
is enlargeable; however, any counterexample to this statement is likely to
be quite complicated. Assuming it is so, one might be led to conjecture
the following:
Conjecture 5.14. For any compact spin manifold with positive scalar curva-
ture, all the higher A-genera must vanish.
There is much evidence for this conjecture if one restricts to “geometric”
K(n,l)-spaces. At very least, the conjecture serves as a rough map of the
wilderness.
There is, in fact, a general theory which allows one to transform this
conjecture to a more general conjecture in the К-theory of C*-algebras.
Again it involves the Atiyah-Singer operator.
To each discrete group Г there is a canonically associated C*-algebra,
denoted С*(Г) and called the C*-aIgebra of Г. Over any manifold with
fundamental group Г we can construct the flat C*(T)-bundle -> X
corresponding to the representation of Г on С*(Г) by right multiplication.
This is just the associated bundle
£ = X xrC*(T)	(5.13)
where X is the universal covering space of X (a principal Г-bundle). Note
that is a bundle of left C*(T)-modules. Each fibre is free of rank one.
At this point, our discussion becomes rather general. A guide to the
formidible technical details required to make it rigorous is found in J.
Rosenberg [1]. Suppose the manifold X is spin and consider the twisted
Atiyah-Singer operator
0+: Г0 + ® g)------► Ftfa ® S)	(5.14)
312
IV. APPLICATIONS
where 8 carries its canonical flat connection. The kernel and cokernel of
0 + are (after possibly a compact perturbation of $+) finitely generated pro-
jective C*(T)-modules. The formal difference gives an element in the alge-
braic К-group, К0(С*(Г)). This element depends only on the homotopy
class of the operator Ц) + and is called the analytic index of#+. Mishchenko
and Fomenko [1] have given a formula for this index of the Atiyah-Singer
type:
ind(0+) = {ch <? • A(X)}[X] g Ко(С*(Г)) ® Q (5.15)
The claim is that for “reasonable” fundamental groups Г, this index carries
all the higher Л-genera of the manifold X. To prove this it is necessary
to study a certain universal map
er: К*(К(Г,1))----->К+(С*(Г)),	(5.16)
defined by using the analogous bundle <? over К(Г,1). (Here K* denotes
K-homology theory extended to infinite CW-complexes.) Our assertion is
true whenever this map is injective after tensoring with Q.
Observe now that since the bundle S -► X is flat, the curvature term
9?^ in the Bochner formula (5.4) is zero. This implies (by Rosenberg [1])
that if X carries positive scalar curvature, then ind(0+) = 0. Conse-
quently, Conjecture 5.14 is true whenever the map er ® Q is injective.
The injectivity of er has been established by Kasparov [1] for any Г
which arises as a discrete subgroup of a connected Lie group.
Readers may have noticed the direct parallel between the higher
Л-genera and the Novikov higher signatures, or “higher L-genera.” (Such
classes can be formulated for any multiplicative sequence.) S. P. Novikov
has conjectured that these higher signatures are, like the signature itself,
homotopy invariants of a manifold. A proof of this conjecture for certain
fundamental groups has been given by G. Lusztig [1] using a family of
Dirac operators of the form D* :(C£+(X) ® Et) -► Г(С£ “(X) ® Et) where
Et is a family of flat bundles over X induced from К(Г,1). Much more
generally, one can consider the operator
D+:T(C£+(X) ® <?) -> Г(С£"(Х) ® <?)
in analogy with (5.14). The assertion here is that the index
ind(D+) = {ch <? • L(X)}[X] g К*(С*(Г))
carries all the higher signatures of X. This again will follow from the in-
jectivity of the map er ® Q. For this reason, the conjecture concerning
er is called the Strong Novikov Conjecture.
We shall end this section with some remarks on a possible classification
of manifolds with = Г which admit positive scalar curvature. The
homomorphism j/ :Q*pin -> KO*(pt) induces a transformation of general-
§6. COMPLETE MANIFOLDS WITH к > 0	313
ized homology theories. Thus, setting й*(Г) = й*(К(Г,1)), we get a trans-
formation
з/:П^п(Г)------► КО*(Г).	(5.17)
Given any compact spin и-manifold X with п^Х Г, there is a canonical
mapping X -> К(Г,1), inducing an isomorphism on ni. This map deter-
mines an element [X] g О^р,п(О-
Again as a guide to the forest, one could conjecture that as in the case
where Г = {e}, the classes з/([Х]) constitute a complete set of invariants
for the existence of positive scalar curvature on X. There is some evidence
for this. The analogue of Theorem 4.3 has been proved by Rosenberg [2]
and Miyazaki [1]. They show that the question indeed only depends on
the fundamental class [X] g ^‘"(т^Х) in the spin case. In the non-spin
case, it depends only on the class [X] g ^(tcjX), and often, for example
when nxX Z or one can show that such X always carry к > 0.
Unhappily, the conjecture that sS = 0 in the presence of positive scalar
curvature fails for torsion groups Г. Rosenberg [2] shows that every spin
5-manifold with Z3 carries к > 0, but that Im / 0 in this case.
Nevertheless, for torsion-free groups Г there is reasonable evidence for
the following:
Conjecture. For torsion free groups Г, the classes з/([Х]) (from (3.17))
constitute a complete set of obstructions to the existence of positive scalar
curvature on compact spin manifolds X with ntX = Г.
§6. Complete Manifolds of Positive Scalar Curvature
We now consider the question of the existence and structure of complete
metrics of positive scalar curvature on non-compact manifolds. We shall
touch only a few results. The reader is referred to Gromov-Lawson [3]
for a much more extensive discussion.
The first important concept here is the following:
Definition 6.1 A bundle with compact support on a manifold X is a
vector bundle E -► X which is trivialized at infinity together with a con-
nection which is compatible with that trivialization.
This means that outside a compact subset of X, there is a given iso-
morphism of E with the product bundle. This isomorphism identifies the
given connection with the canonical flat one on the product. The con-
nection will always be assumed to be orthogonal or unitary (depending
on whether E is real or complex).
Suppose now that X is a complete riemannian spin manifold of even
dimension n — 2k. Let $c denote the spinor bundle of X with its canon-
ical riemannian connection, and let E -> X be any bundle with compact
314
IV. APPLICATIONS
support. Then the Atiyah-Singer operator 0£ on bisections of $c ® E
is essentially self-adjoint and satisfies ker 0E = ker Ц)Е (see Theorem 11.5.7).
This operator satisfies the pointwise formula
0| = V*V + £k +	(6.1)
derived in II.8.17. Since E is flat at infinity, this reduces to the equation
0j = V*V 4- outside a compact subset of X. Using this fact and some
standard arguments from analysis, we prove the following. We say that
a function к is uniformly positive on a set A if к k0, for some constant
kq > 0, on A.
Proposition 6.2. Suppose X has uniformly positive scalar curvature outside
of a compact set. Then the Dirac operator 0E on L2($c ® E) has a
finite-dimensional kernel and a bounded Green's function.
Corollary 6.3. The operator
0e : L2($c ® E)--------► L2($c ® E)	(6.2)
and its adjoint 0E have finite-dimensional kernels. Hence the index
ind(0£) = dim(ker Ц)Е) — dim(ker 0£)	(6.3)
is a well-defined integer.
Proof of 6.2. Let К cz X be a compact set outside of which E is trivialized
and k0 for some k0 > 0. Let c > 0 be a constant such that
c Id — (|k + 9i£) over K. Then for any element (p g ker(0£), formula
(6.1) implies that V*V<p = —— 9?K(<p). Integrating by parts, and es-
timating gives the inequality
+	(6.4)
Let ||-||a denote the L2-norm over a set A cz X. If we assume
IK (= ||K + ||ф||х-к) = 1, then (6.4) implies that
Kq T C	Kq -f C
Consider now the uniform Cx-norm ||<p||citK over K. By Theorem III.5.4
we know that there exists a constant C > 0 such that
lk||c‘.K С||ф||х	(6.6)
for all (p g ker(D£). Let us fix a number e > 0 and consider and e-dense
subset of points {xm}^,= 1 in K. If dim(ker DE) d, then there exists an
element (p e ker DE with ||<p||x = 1, such that <p(xy) = 0 for j = 1,... ,d. It
§6. COMPLETE MANIFOLDS WITH к > 0	315
then follows from (6.6) that the pointwise norm of (p is uniformly less than
Cc on K, i.e. ||<р||со,к < Ce. For e sufficiently small, this violates (6.5), and
we conclude that dim(ker DE) is finite.
The invertibility of the Green’s function will not be used directly here.
We refer to the reader to Gromov-Lawson [3] for the proof of this fact.	
Proof of 6.3. With respect to the bundle splitting $c ® E = ($£ ® E) ф
(#c ® E) the operator 0E can be written (pointwise) as
( 0
where $Е : Г(#с ® E) -► Г(#£ ® E). Passing to bisections, we get an
orthogonal decomposition L2($c ® E) = L2($£ ® E) ф L2($q ® E),
and Ц)Е continues to interchange factors as above. Consequently
ker f)E = ker Ц)Е ф ker 0E
and the finite dimensionality of ker f)E is clear.
Since f)E is self-adjoint, the operators f)E are adjoints of one another,
and so ker 0E = coker f)E . 
Of course we will need a version of the vanishing theorems from the
compact case.
0Е
О
Proposition 6.4. Let X and E be as above, and suppose there is a constant
c > 0 so that
±к + ЫЕ^с Id.	(6.7)
on X. Then ker f)E = 0, and in particular, ind(0^) = 0.
Proof. As in the proof of II.5.7, the completeness of X allows us to inte-
grate by parts. Thus formula (6.1) implies that
||1>еф||х = l|v<jo||i +	+ э?£)(ф),ф> c||<je>||i
for all (p g L2($ ® E), and it is evident that ker Ц>Е = 0. 
In order to apply the arguments of the previous section, we need some-
thing to play the role of the Atiyah-Singer Index Theorem. This will be
done by a Relative Index Theorem. We will present here only the special
case that we need. More general versions of this result have been proved
by J. Cheeger and by H. Donnelly [3].
Theorem 6.5 (Gromov-Lawson [3]). Let X be a complete even-dimensional
spin manifold whose scalar curvature is uniformly positive at infinity. Let
Eo and Er be two bundles with compact support on X, and assume that
316
IV. APPLICATIONS
dim Eo = dim Ev Then the difference of the indices of the Dirac operators
0eo and is a topological invariant, given by the formula
ind - ind 0+ = {(ch Er - ch £0) • A(X)} [X].	(6.8)
where, via the trivializations at infinity, the class ch Er — ch Eo has com-
pact support on X.
Via Chern-Weil Theory (see Kobayashi-Nomizu [1].). the classes ch Eo,
ch En and A(X) can be represented canonically in terms of the curva-
ture forms of the given connections. Formula (6.8) can then be rewritten as
ind fa, - ind 0£+o = J*x (ch Er - ch Eo) л A(X).	(6.8)
(Recall that Eo and Er are flat outside a compact set.)
Another method to compute the relative index is as follows. Chop off
the manifold X outside the support of Eo and Et to obtain a compact
manifold X with boundary. Let Y be the double of X and let E be the
bundle on Y given by Eo on one piece and Et on the other piece with £0
and Ex glued together at the “seam” dX by the trivialization. Then the
right hand side of (6.8) becomes {ch E • А(У)}[У].
An important special case of Theorem 6.5 is where Ег = E is some
bundle with compact support on X and where Eo = X x Cfc is the triv-
ialized bundle of dimension к = dim E. Note that 0Eo = 0	’ ® 0
(к-times) where f) is the usual Atiyah-Singer operator. We then define the
reduced Chern character of E to be
cK E = ch £ - ch Ck = cl?E + ch2E 4- ...	(6.9)
The formula (6.8) then becomes
ind(^) - к ind(0+) = {cRe • A(X)}[X].	(6.10)
For the proof of Theorem 6.5 the reader is referred to Gromov-Lawson
[3].
Completion of the proof of Theorems 5.5 and 5.12. Armed with Theorem
6.5 we are able to extend the arguments of the last chapter from the
“compactly enlargeable” to the “enlargeable” case. Let X be a compact
riemannian manifold with к kq for a constant k0 > 0. Let e > 0 be
given and suppose X is a spin covering space which admits an e-con-
tracting map f:X -> S2m(l) of non-zero Л-degree. Fix a bundle Eo with
connection over S2m(l) with ст(Е0) Ф 0 and set E = /*E0 with its induced
connection. Note that E is a bundle with compact support on X, and so
we can construct operators 0 and as above.
§6. COMPLETE MANIFOLDS WITH к > 0	317
Since к k0 > 0, we have ind(0 + ) = 0. However, for £ sufficiently small,
the inequality (6.7) will be satisfied, and we conclude that ind(^) = 0.
Consequently, by (6. 10) we have
0 = {ей E • A(X)}[X] = {cm(E) • A(X)} [X]
= {/*cm(E0)-A(X)}[X]
= (4-deg)(/)cm(E0)[52w]
contrary to assumption. 
Theorem 6.5 allows us to prove a number of results about complete
metrics on non-compact manifolds. We begin by focusing on some impor-
tant facts. The first fact is that in all the arguments of §5 we only used
the fact that the mappings f were uniformly contracting on the curva-
ture 2-form of a connection. This leads to the following notion:
Definition 6.6. A smooth map f: X -> Y between riemannian mani-
folds is said to be e-contracting on 2-forms or (c,A2)- contracting if
IIA4s«l
for all elements £ g A2TX, or equivalently if
sup ||/*^|| £ sup ll^ll
X	Y
for any differential 2-form ф on У.
The strength of this is that to be (e,A2)-contracting the map needs only
to be contracting in (и — 1) of the directions at each point, where n =
dim(X). For example, the linear map L: R" -> R" with matrix
is (e,A2)-contracting but not c-contracting.
All of the discussion of §5 remains valid if the hypothesis “c-contracting”
is replaced by “(e,A2)-contracting”. The real power of this observation
will be seen in the non-compact case.
Definition 6.7. An orientable riemannian manifold X is called (c,A2)-
hyperspherical if there exists an (c,A2)-contracting map f:X -► Sm(l)
which is constant at infinity and of non-zero 4-degree.
318
IV. APPLICATIONS
The basic argument given above proves the following result.
Theorem 6.8 (Gromov-Lawson [3]). There is a constant £n> 0 so that any
spin n-manifold carrying a complete metric with к 1, cannot be (cn,A2)-
hy per spherical.
The number can be explicitly estimated for each n.
We now introduce a topological property of manifolds.
Definition 6.9. A manifold X is called weakly enlargeable if for each
riemannian metric on X and each £ > 0, there exists an orientable covering
space of X which is (A2,£)-hyperspherical in the lifted metric.
Any (compact) enlargeable manifold is weakly enlargeable. However,
there are many non-compact examples .
Proposition 6.10. If X is enlargeable, then X x R is weakly enlargeable.
Proof. Fix a metric g on X x R and consider the composition X x R ->
R -► S\l) where the first map is projection and where the second map is
constant (= ♦) outside the interval (—1,1), and of degree 1. Call this com-
position д. Since X is compact, p is ^-contracting for some 5^1.
Consider also the projection X x [—1,1] -> X, and choose a metric
g0 on X so that this map is 1-contracting from the given metric g on
Xx[—1,1].
Fix now an c > 0 and choose a riemannian covering (X,g0) of (X,g0)
which admits an (c/<5)-contracting map f: X -► Sm(l) which is constant
(= ♦') at infinity and of non-zero Л-degree. Extend the map f trivially to
X x R and note that for the lift g of the original metric g, this map is
(c/(5)-contracting over X x [—1,1].
Fix now a 1-contracting map a:Sm(l) x Sx(l) -► Sm+1(l) which is con-
stant on the set (Sm(l) x {*})o ({♦'} x Sx(l)). Then the composition
X X R	X S1(1)-^S”, + 1(1)	(6.11)
is (a,A2)-contracting with respect to the metric g, constant at infinity and
of the same Л-degree as f. 
We now have, for example, that any manifold of the form Xn x R,
where Xn is compact and carries non-positive curvature, is weakly
enlargeable. A basic case is that of Tn x R.
The property of weak enlargeability is contagious.
Proposition 6.11. Let U be an open submanifold of X such that the map
7trU -► 7trX is injective. IfU is weakly enlargeable, so is X.
§6. COMPLETE MANIFOLDS WITH к > 0	319
Proof. This is an immediate consequence of the definition of weak en-
largeability. 
Corollary 6.12. Any manifold X which contains a (compact) enlargeable
hypersurface Xo with лгХ0 -► nYX injective is weakly enlargeable.
Proof. Let U be a tubular neighborhood of Xo and apply 6.10, 6.11. 
As an example, let X = Tn — A where A is any closed subset such that
A n T"-1 = 0 for some linear hypertorus T"-1 c= Tn.
This example gives substance to the following:
Theorem 6.13 (Gromov-Lawson [3]). A weakly enlargeable manifold can-
not carry a complete metric of positive scalar curvature.
Remark. J. Kazdan [1] has proved a version of Theorem 5.7 for non-
compact complete manifolds. This strengthens Theorem 6.13 by adding
the statement: Any complete metric with к 0 on a weakly enlargeable
manifold must be Ricci flat.
Proof. Note that if the scalar curvature were uniformly positive, the result
would follow easily from Theorem 6.8. To achieve the uniform positivity
of к we shall multiply by a large 2-sphere. The estimates then become
more delicate.
Let X be weakly enlargeable and suppose X carries a complete metric
g with к > 0. For any given e > 0, we can find a covering X of X which
admits an (c,A2)-contracting map f:X -► Sm(l), constant at infinity and
of non-zero Л-degree. In fact, we may assume this map to be (c,A2)-
contracting with respect to the (not necessarily complete) metric д' = кд.
Hence, the map f is pointwise ck(x) contracting on 2-forms with respect
to the (lift of the) metric g.
We now take the riemannian product X x S2(r) and consider the com-
position
/*(i)
X X S2(r)----Sm(l) X S2(l) —Sm+2(1)	(6.12)
F	S
where a is a fixed “smashing” map as above (cf. (6.11)). Call this composi-
tion F. We want to derive a pointwise estimate for the contraction of F on
2-forms. Since f has compact support on X, it is c-contracting on tangent
vectors for some c > 0. The dilation map on S2(r) is of course (l/r)-con-
tracting on tangent vectors and (l/r2)-contracting on 2-forms. We may
assume a to be essentially 1-contracting. It follows that at any point
320
IV. APPLICATIONS
(x,y), the map F is e(x>contracting on 2-forms, where e(x) = тах{ск(х),
c/r, 1/r2}. The map F is supported in a compact set K. Set к0 = inf{K(x):
(x,y)eK some y} >0, and choose r > 1 so that max{c/r, 1/r2} гк0.
Then the map F will be cx(x)-contracting on 2-forms at every point (x,y).
We now proceed as before to pull back a fixed bundle with connection
from Sm + 2(1), via the map F. The curvature term (6.7) will be of the form
}к(х) + Д + 91£	(6.13)
where E = F*(E0) is the induced bundle. There is a constant у depending
only on dimensions so that
IM 7MI
pointwise on X x S2(r). However, since F is pointwise £K(x)-contracting
on 2-forms, we see that
IM у£к(х)||Я£°||00
pointwise, where [[R^H^denotes the C°-norm over Sm + 2(1). We may as-
sume that c satisfies с < 4у||К£°||00 since these constants are fixed at the
beginning. The curvature term (6.13) then satisfies the inequality
1	2	2
4 Ф) + p + *E
on all of X x S2(r). Hence, Proposition 6.4 applies to the twisted Atiyah-
Singer operator and we conclude that indexes) = 0. Of course,
index(0 + ) = 0 for the standard Atiyah-Singer operator. We now apply
the Relative Index Theorem as before to complete the proof. To do this,
it is necessary that the bundle Eo over Sm + 2(1) have a non-trivial Chem
character, and therefore that m be even. If m is odd, however, we may
simply replace S2(r) in the construction above, with S3(r), and everything
goes through.
We now complete the argument. Let к = 2 or 3, so that m + k = 2N,
for some integer N. Then since ind(0£) = ind(0+) = 0, Theorem 6.5
together with formula (6.10) implies that
0 = {ch E • A(X x Sk)}[X x S*]
= {Cjv(E) A(X x S‘))[JVxS‘]
= {FM£o)'LVxSW><S4
= (l-deg)(F)C„(E0)[S2i4
= (l-deg)(/)Ctf(Eo)[S2,v]
contrary to assumption. 
§6. COMPLETE MANIFOLDS WITH к > 0	321
Examples. As a consequence of Theorem 6.13 we know that there is a
large collection of manifolds which cannot carry complete metrics with
к > 0. This includes X x IR for any enlargeable manifold X. It also in-
cludes manifolds of the form Tn — C where C is any closed subset which
misses some geodesic subtorus T"-1 of codimension one. (For example,
let C be finite.) One can also replace (Tn,Tn-1) here by any compact pair
(Xn, X"-1) where Xn carries a metric of non-positive sectional curvature
in which Xn~1 is totally geodesic. The property of not admitting complete
metrics with к > 0 remains after taking products of these manifolds and
then taking connected sums with countable families of arbitrary spin man-
ifolds. It persists also after “boundary” connected sums with any open
spin manifold.
The “exclusion theorem” 5.6 of the last chapter has a nice generaliza-
tion to the open case.
Corollary 6.14. A manifold which carries a complete hyperbolic metric of
finite volume cannot carry a complete metric with positive scalar curvature.
Proof A hyperbolic manifold X of finite volume has ends of the form
N x IR where N has a finite covering by a nilmanifold and t^N -► n^X
is injective. Since N x IR is weakly enlargeable, so is X by Proposition
6.11, and Theorem 6.14 applies. 
By introducing one further trick we are able to “localize” the results
above. To do this we recall the notion of a warped product. Let Xr and
X2 be riemannian manifolds with metrics g{ and g2 respectively, and let
f : Xi -►	be a smooth function. The warped product of Xj and X2 with
warping function /is the cartesian product manifold X = Xt x X2 with
riemannian metric
g = gi+f2g2	(6.14)
It is an elementary exercise to compute that for such a warped-product
metric, the scalar curvature is
K = iq +p-|»<2-2n/V2/'-n(n- 1)||V/||2}	(6.15)
where is the scalar curature of X; and n = dim(X2). If we let X2 =
S"(l), the euclidean и-sphere of radius 1, then onX = Xt x S"(l) we have
* = *1 +	(1 - HATH2) - 2n	(6.16)
By choosing/small and constant over a compact set, we can make к
positive there. On a compact manifold Xo we can then slowly change /
to suit our purposes. This is the basic idea in proving the next result.
322
IV. APPLICATIONS
Theorem 6.15 (Gromov-Lawson [3]). Let X be a non-compact, connected
spin n-manifold containing a compact, connected, orientable hypersurface
Xo c= X. Suppose that Xo separates X into two components. Suppose fur-
ther that there is a map F:X -► Y, onto an enlargeable (n — l)-manifold
which, when restricted to Xo, has non-zero degree. Then X carries no com-
plete metric with |Ric| constant and with к uniformly positive outside a
compact subset.
Proof. Suppose that X carries a complete metric with к 2 outside a
compact set. Let X+ and X_ denote the two connected components of
X — Xo. Since X is non-compact, at least one of these components, say
X+, is unbounded in the given metric. Define a function p :X -► R by
setting
dist(x, Xo) for x e X+
— dist(x, Xo) for x € X _.
By the assumption on Ric there is a smoothing p of p with ||Vp||	2 and
with |V2p| C for some constant C 2. Fix a number p0 so that к 2
outside the compact set K = {x6 : |p(x)| S p0}. Let cn be the number
given by Theorem 6.8, and choose R > 0 so that 1/R < £n. Let Kinf =
inf{K(x):x € K}, and choose r, 0 < r < 1, so that 1/r2 > 1 + |Kinf|. Choose
now a C00 function ф: R -► R so that
P(*) =
0(t) = r
0(t) = R
0 |0'| £
for |t| p0
for |t| > Po + 2R/c
(6.17)
\Ф"\ £
where e is the constant
£ = r/3C.
We take the warped product of X with S2(l) using the function f on X
given by
/(x) = ф(р(х)).
By (6.16) the scalar curvature k of this metric satisfies: k |Kinf | + 2/r2 > 1
over К x S2, and k > 2 -I- 6е2С2г~2 > 1 over (X — K) x S2. Hence, k > 1
on all of X x S2.
We shall now find a covering of X x S2 which is (£n,A2)-hyperspherical,
in contradiction to Theorem 6.8. To begin we fix a number b > p0 + 2R/e
and we consider the compact set Q = {x е X : b p(x) b + 4л}. Fix a
metric on Y. Let S = sup{||F#||x: хей} and choose a riemannian cover-
ing л:У-> У which admits an (e„/5)-contracting map g:Y ->S"-1(1)
which is constant at infinity and of non-zero degree. Taking the fibre
§6. COMPLETE MANIFOLDS WITH к > 0
323
product of F and л gives a commutative diagram
X	Y
it	it
X	Y
where it is a covering map and F is proper. Let g = g о F and let
h = p°p°ii:X -► S^l) where p : R -► S^l) is the degree-1 map given
by collapsing everything outside the interval (b,b + 4л) to a point. (This
map is (l)-contracting). Let G denote the composition
X
S"-1(l) X Si(l)
G
S"(l)
where a is a “smashing” map which collapses the axes Sn-1(l) x {♦} о
{*} x S^l) to a point. The map G is a constant outside a compact set
contained in П = л“ ^П). Since h is 1-contracting and g is ^-contracting,
it follows that G is, at every point, >/0 + ^-contracting and ^-contracting
in the hyperplane ker(A#).
We now lift the warped metric to X x S2 and let H denote the
composition
S2 $"(]) x S2(1)
H
Sn + 2(1),
where a' is again a “smashing” map. Since the warping function f =
f ° ft — constant = R on the support of G, and since 1/R < en, we see that
H is again, like G, 1-contracting and ‘^„-contracting in a hyperplane” at
each point. In particular, the map H is ^„-contracting on 2-forms.
To complete the proof we must show that H is of non-zero degree.
The main point here is the following. Let t be any regular value of p and
consider the compact manifold Xt = p~l(t). Clearly Xt is homologous to
Xo = p-1(0), and so by the hypothesis on F, its restriction gives a map
F: Xt -► Y of non-zero degree. The rest is straightforward. Set Xt =
n~\Xt) and note that the lift F is proper on Xt and has the same
degree as F. Now the degree of g x h equals the degree of g restricted to
a regular level set of A, and deg(g) = deg(g)deg(F) 0. Hence, deg(G) =
deg(H) / 0. 
There are many interesting applications of Theorem 6.15. For exam-
ple, let X be an open и-manifold and XQ <= X a compact hypersurface.
324
IV. APPLICATIONS
Suppose Xo disconnects X and let S be an unbounded component of
X — Xo with dS = Xo (see diagram below).
Definition 6.16. We say that S is a bad end if there exists a map
F.£^>Y onto an enlargeable manifold У so that F|Xo is of non-zero
degree.
Theorem 6.17 (Gromov-Lawson [3]). Suppose X is a spin manifold with a
bad end £. Then there is no complete metric on X which has |Ric| bounded
and к uniformly positive on S.
Note. Actually, only the end 3 needs to be spin.
Proof. Let X be given a complete metric. Consider the double D(<f) =
S иХо S with a metric which agrees, outside a neighborhood of the “seam”
дв = Xo, with the given one. This manifold satisfies all the hypotheses of
Theorem 6.15, and therefore cannot have |Ric| bounded and к uniformly
positive outside the compact neighborhood of X. 
Theorem 6.18. A compact 3-manifold of К(л, l)-type cannot carry a metric
with positive scalar curvature. In fact, no compact 3-manifold which can
be written as a connected sum with а К(л, l)-manifold, can carry positive
scalar curvature.
Proof. Let X be a compact 1) 3-manifold. We may assume that X
is orientable and therefore spin. Choose an embedded curve у c= X which
is not homotopic to zero, and consider the covering space X X cor-
responding to the cyclic subgroup of nrX generated by [у]. There is a
lifting of у to an embedded curve у c= X which generates щХ. Note that
X is а K(ti{X, l)-manifold. In particular, т^Х 7L and the inclusion у cz X
is a homotopy equivalence. (It is not possible that лгХ = 7Lm since
H2WwJ); Zm) = for all fc.)
Suppose now that X carries a metric g with к > 0. Then g lifts to a
complete metric g on X of uniformly positive scalar curvature with |Ric|
bounded.
§6. COMPLETE MANIFOLDS WITH к > 0	325
Choose a tubular neighborhood U of y, and set g = X — U. We claim
that g is a bad end. To prove this, we first show that the inclusion dg S
induces an isomorphism on This is an easy consequence of the
Mayer-Vietoris sequence for X = U о S which gives
о = h2(x) —►	—> Hrg ® hJj —► hyx —► o.
(Recall that the inclusions у cz U c= X are homotopy equivalences.) By
general theory the surjective homomorphism n^g -> H^g) = Z ® Z is
represented by a continuous map F:<? -► S1 x S1 = K(Z ® Z,l). Re-
stricted to dg S1 x S1 this map induces an isomorphism on яь and
therefore has non-zero degree. Thus, g is a bad end, and the first state-
ment of Theorem 6.18 is proved.
For the second statement consider a compact oriented 3-manifold Z =
X # Y where X is as above. Let s:X # Y X be a map given by collapsing
the У-summand to a point ♦ € X. Choose у c= X — {♦} and pass to the
covering я : X X with the lifted curve у cz X as above. Taking the fibre
product of s and я gives a commutative diagram
—s—+ x
X#Y X
where я is a covering map and s is proper. The map s simply consists of
collapsing a family of “У-summands” to the discrete family of points я” 1(*).
The neighborhood IJ of у can be chosen to lie in X — я" \*) where s is a
diffeomorphism. Thus we can lift U back to X ft Y and consider the end
g’ = X ft Y — s-1((J). By restriction we have a proper degree-one map
s:g' -► g onto the end g constructed above so that s:dg' Л dg is a
diffeomorphism. Composing with the map F:g -► S1 xS1 above shows
that the end g' is a bad one, and Theorem 6.17 applies as before. 
Recall that every compact orientable 3-manifold has a “prime decom-
position”
X =	xS2) #...#($*
where the manifolds E, have finite fundamental groups (in SO4) and where
each Kj isa K(a,l)-manifold (see Milnor [4] and Hempel [1]). The theorem
above says that if X carries positive scalar curvature, then there are no
K(nJ)-factors in the prime decomposition of X. There are alternative proofs
of this fact using minimal surfaces (see Gromov-Lawson [3], Schoen-Yau
[1]). If standard conjectures in the theory of 3-manifolds were true, this
result combined with Proposition 4.3 would settle the question of which
3-manifolds carry к > 0.
326	IV. APPLICATIONS
Arguing as in the proof of Theorem 5.5, we see that a compact 3-mani-
fold X with к 0, which does not carry к > 0, must be flat. There are
six such manifolds up to diffeomorphism. (See Calabi [1].)
Results for non-compact 3-manifolds M can be obtained by these
methods. We say that M contains an incompressible surface if there is an
embedding E M of a compact surface of genus > 0, such that the
induced homomorphism яДЕ) -> nfM) is injective. We say that M carries
a small circle if there is an embedding S1 <-+ M whose class is of infinite
order in	and such that the class of the small normal circle is of
infinite order in — S1; T}. Thus, S1 x R2 carries a small circle, but
S1 x S2 does not. From 6.12 and 6.13 we see that any 3-manifold which
carries an incompressible surface cannot carry a complete metric with к > 0
(a result due originally to Schoen and Yau [6]). Furthermore, any
3-manifold which carries a small circle cannot carry a complete metric with
к 1. In particular, the manifold S1 x R2 carries no such metric (see
Gromov-Lawson [3]).
§7. The Topology of the Space of Positive Scalar Curvature Metrics
Fix a compact m-dimensional manifold X and let Л^Х) denote the space
of all riemannian metrics on X. Note that Jt(X) is a convex cone in the
linear space Г(Т*Х ® T*X) and is therefore contractible. It is acted on
naturally by the group Diff(X) of difleomorphisms of X.
We consider here the subspace &>(X} cz Л(Х} of those metrics which
have positive scalar curvature. This subspace is stable under Diff(X), and
as we have seen, could possibly be empty. The first results on the topology
of &(X) were due to Hitchin [ 1 ] who defined for spin manifolds X a homo-
morphism
h^^X))-----------►KO-"-'"(pt)	(7.1)
for each n 1 as follows. Suppose a e я„_ fjP(X)) is represented by a map
f:Sn~i -► &>(X). Since ^f(X) is contractible, we can extend this to a map
f :Dn-+ Л?(Х) where Sn~l = dDn. (This extension process realizes the
isomorphism яи_1(^>(Х))	яп(^(Х),^(Х)).) Fix a spin structure on X
and associate to each metric f(y) the canonical C£m-linear Atiyah-Singer
operator Ф/ for that metric. This gives a family of elliptic operators over
Dn which at each point of dDn are invertible (since the scalar curvature is
positive). The invertible operators form a contractible space, so we can
deform /to be constant on S"-1. Taking indm of the resulting family gives
an element h(f) e K0~m(D\Sn~l) KO"W(S") = XO"w"n(pt) which de-
pends only on the homotopy class of/.
Hitchin is able to show that his homomorphism (7.1) is often non-
trivial by the following means. Fix a metric у e &>(X) without isometries
and consider the embedding iy: Diff(X) 0>(X) given by ifg) = g*(y).
§7. SPACE OF METRICS WITH к > 0
327
Composing with h leads to a homomorphism яи_ ^DiflfX)) -► KO"n"m(pt)
whose value on an element и € nn-i(Diff(X)) is j/(ZM) where Zu -► Sn is
the fibre bundle with fibre X constructed by clutching together two copies
ofD" x X along Snl x X via u. Using deep results from topology, Hitchin
is able to construct sufficiently many non-trivial examples to prove the
following:
Theorem 7.1 (Hitchin [1]). Let X be a compact spin manifold of dimension
n with 0>(X)	0, then
ло(0>(Х))	0	if n = 0 or 1 (mod 8)
л^(Х))	0	if n = 0 or - 1 (mod 8)
Note that these classes do not survive to the quotient &(X)/Diff(X).
A slightly different approach to the study of the topology of &(X) was
given in Gromov-Lawson [3]. The cleanest statements require the Rela-
tive Index Theorem 6.5. However, the basic idea is based on the following
elementary construction. Fix у e 0>(X) and suppose У is a compact
manifold with d Y = X. We say that у extends to Y if there exists a metric
уе^(У) which coincides with the product metric у + dt2 on a collar
neighborhood!/ % X x (0,1] of the boundary. This extendability depends
only on the homotopy class of у in 0>(X). To see this, consider a smooth
family of metrics yt e &(X), t e R, which is constant for t 0 and for t 1.
It is easily checked that the metric ytfc + dt2 on X x R has к > 0 for all suf-
ficiently large constants c. Adding the collar X x [0, c] (with this metric)
to Y shows that y0 extends to Y if and only if yj does.
Proposition 7.2. Suppose that X is a spin manifold. Let {уа}аеЛ c= &*(X)
be a family of metrics, and suppose that there exists an associated family
{Уаел °f compact spin manifolds with dYa = X such that ya extends over
Yafor each a e A. If
Л(Уаох(-У/?))^0 for all a *0,
then ya is not homotopic to yp in 0*(X) for all a Д.
Proof. If ya is homotopic to yp in 0\X), then ya extends over both Ya
and Yp. The extended metrics fit together to give a metric of positive scalar
curvature on Ya о (— УД and therefore A(Ya u (— Yp)) = 0. 
The “extension pairs” (ya,Xz) discussed above are not difficult to con-
struct. There are two good sources.
Lemma 7.3. Let n:V -► X be a riemannian vector bundle of (fibre) dimen-
sion k over a compact manifold X, and let D(V) = {ue P:||v|| =1}. If
k > 2, then there exists a metric у € 0>(dD(V)) which extends over D(K).
328
IV. APPLICATIONS
Proof. Choose an Ok-invariant metric y0 on Rk which has к 1 and is
isometric to the standard product metric on Sk-1 x [l,oo) outside the
unit disk. Choose an orthogonal connection on V and let Ж denote the
corresponding field of horizontal planes on V. Choose any metric on
X and lift this metric to via я. Introduce the metric y0 on the fibres
of V. Using the formulas of O’Neill [1], the sum + t2y0 is seen to have
positive scalar curvature for all t > 0 sufficiently small. 
More elaborate arguments prove the following in any manifold Y:
Theorem 7.4 (Carr [1]). The boundary of a regular neighborhood U of any
smoothly embedded finite subcomplex of codimension > 2 in У, admits a
metric of positive scalar curvature which extends over U.
Example 7.5. Let V -► S4 be a real vector bundle of dimension four with
Euler number / and Pontrjagin number pP It is elementary that if / = ± 1,
the manifold = dD(V) is a homotopy sphere. Milnor showed that
= S7, the standard 7-sphere, if and only if pj = 4 (mod 896). Let Vk
be the bundle with z = 1 and Pi = 4 + 896k. Calculations in Milnor [7]
show that
A(Vk о D8) = k for each k.
From Lemma 7.3, for each к we can construct a metric yk on S7 which
extends over Vk. By Proposition 7.2 these metrics are mutually non-
homotopic in &>{S7) and so we have that
л0(^(57)) is infinite.
Since rc0(Diff S7) is finite, we also have that
rc0(^>(S7)/Diff(S7)) is infinite.
The analogous comments apply to each of the non-standard Milnor
spheres
Example 7.6. Fix an integer n > 1 and let Y denote the compact 4n-
manifold with boundary constructed by plumbing together eight copies
of the tangent disk bundle of S2n according to the Dynkin diagram for E8.
Let Yk denote the boundary connected sum of к-copies of Y. It is ele-
mentary to see that E = dY is a homotopy sphere. By the finiteness of
the group of homotopy spheres, there is an integer m = m(n) such that
dYkm = E #  •  # E (km-times) = S4"-1 for all k. Using Theorem 7.4, we
can construct for each к a metric yk e ^(S4"-1) which extends over Ykm.
On the other hand, as shown in Carr [1],
Л(Укти(-УГт))/0	if k/k'.	(7.2)
§7. SPACE OF METRICS WITH к > 0	329
In fact, since Ykm u (- yk,m) is a (2n — 1)-connected 4n-manifold, its Л-genus
and its L-genus are both certain universal (non-zero) multiples of p„, the
nth Pontrjagin number. It suffices therefore to show that the signature
0. Since sig(K)=l, we see that sig(Укт и (-Ук,т)) = (k — k')m. This
establishes (7.2) and from Proposition 7.2 we conclude that
n0(^(S4"*‘)) and	are infinite
for all n > 1. From this we can deduce the following:
Theorem 7.7. Let X be any compact spin manifold of dimension 4n — 1	7
with &(X)	0. Then ло(0>(Х)) is infinite.
Proof. Let Zk = (X x [0,1]) t| Ykm where h denotes boundary connected
sum. Note that dZk = XII (X # S*n~J) % X II X. Fix a metric у e 0>(X)
and take connected sum with the metric yk constructed above on the
component “X	(cf. 4.3). This metric extends over Zk. (Essen-
tially this extension is constructed by taking the connected sum of the
product metric on X x [0,1] with that on Ykm and modifying it; see
Carr [1] for example.) Since Zk u (-ZJ = (X x S1) # (Ykm и (- Ук,те)),
we have A(Zk и (-Zk,)) = A(Ykm и (-Yk,m)) * 0 and 4.3 applies. 
The arguments given above can be nicely encapsuled by using the
Relative Index Theorem. Let X be a compact spin manifold of dimen-
sion 4n — 1. For each pair of metrics д0,дг € ^(X) we define a relative
index i(0o,0i) Z as follows. Introduce on X x R a metric g which equals
the product metric g0 + dt2 on X x ( — oo,0] and gr + dt2 on X x [l,oo).
Set
i(0o.0i) s ind(0+)
where $+:Г($ + ) -► Г($~) is the Atiyah-Singer operator on X x R. By
Theorem 6.5 this index is independent of the choice of g in X x [0,1]. If
g0 is homotopic to g{ in ^(X), then we can choose g to have к > 0 (as
above) and so i(0o,0i) = 0- Thus, Цд0,дх) measures components in ^(X).
It is shown in Gromov-Lawson [3] that
*(0o,0i) + *(0i,02) + *(02,0o) = 0
for all 0o,0p02 e ^(X).
Suppose further that X = d Y. Then given g e ^(X) we introduce a
complete metric on У — dY which is the product g + dt2 on the boundary
collard У x [0,oo) = X x [0,oo). Using the Atiyah-Singer operator for this
metric we define an invariant:
i(0,y) = ind(0+)
330
IV. APPLICATIONS
which by 6.5 is independent of the extension and which has the property
that
i(g,Y) = 0 if g extends over Y.
By the Relative Index Theorem we have that
^,Г)-^,Г) = Л(Ги(-Г)).
Using Browder-Novikov Theory and proceeding as above, one can
detect non-trivial elements in	for higher values of j.
§8. Clifford Multiplication and Kahler Manifolds
Until now we have paid little attention to complex manifolds. This is not
because the topic is irrelevant to spin geometry. In fact combining com-
plex and spin structures yields the richer study of Spinc manifolds, which
are discussed in Appendix D. In this chapter we restrict attention to the
fundamental case of Kahler manifolds.
If Clifford multiplication enters naturally into riemannian geometry and
leads to basic identities, then in Kahlerian geometry it should enter in an
even more interesting way. This is indeed true. For Kahler manifolds there
is a rich algebraic formalism which relates Clifford multiplication and the
complex structure. We shall sketch here the principal results. We refer the
reader to Michelsohn [1] for complete details.
Let X be a 2n-dimensional manifold equipped with an almost complex
structure, that is, equipped with a bundle automorphism J:TX -► TX
such that J2 = — Id.
Definition 9.1. A riemannian metric < •, • > on X is said to be Kahlerian if
J is pointwise orthogonal, i.e., <JF, JW) = <F, IV> for all V,Wg TxX at
all points x, and if
VJ = 0
where V denotes the canonical riemannian connection.
Note. It is a basic fact that if X admits a Kahler metric, then the al-
most complex structure is integrable, i.e., it comes from a system of
holomorphically related coordinate charts on X.
Suppose now that X carries such a metric and let D:F(C£(X)) -►
F(C£(X)) denote the associated Dirac operator on the complexified Clif-
ford bundle Ct(X) = Ct(TX) C of X (considered as a real manifold).
The first interesting fact is that there is a natural decomposition
D = 3 + 3
where 3 and 3 are first-order operators which are formal adjoints of one
another (®* = 3} and satisfy
®2=0,	= 0	(81)
§8. KAHLER manifolds
331
Similarly, the zero-order operator L defined in Chapter II can be expanded
into a pair of operators and 3? which are formal adjoints of one
another and which canonically generate an sI2(C)-subalgebra. In partic-
ular, if we set = [^, J?"], then the three endormorphisms of Ct(X)
satisfy the identities
[J^,	= 2^,	[Ж, ^] = - 2^,	[^,	= Ж. (8.2)
These operators are defined by setting
= Z	S)<p = £ EjS7ej(p
j	J
&<p = - z &<? = - Z
j	j
where
£j = 2 (ej “ iJej) an<i = 2 (ej + iJej)
for any local orthonormal frame field of the form ei9Jei9... These
complex vector fields satisfy the “supersymmetry” relations
Efij + Efii =	4“ Efii = 0
+ Efii = - dij
The endomorphism J:TX -► TX extends to Ct(X) as a C-linear map
which is a derivation with respect to Clifford multiplication. If we set
# = —iJ, then the commuting endomorphisms $ and Ж are each
diagonalizable with integral eigenvalues, and we can define the sub-
bundles
C£™(X) = {ф e C£(X): /(ф) = рф and е>Г(ф) = qq>}
The representation theory for $I2(C) is used to show that these bundles
are non-zero exactly for pairs (p,q) e IL x IL with |p 4- q\ n and p 4- q =
n (mod 2). This gives us the bigraded “Clifford diamond” below.
q
n
3/
332
IV. APPLICATIONS
with “raising” and “lowering” operators:
<£: C£p’fl(X)----►	+ 2(X),	<£: CU^X) -------► C£™ " 2(X),
and with the differential operators 2 and 2 acting diagonally
©:Г(С£Р’*)------► T(C£p+1’fl+1),	©:Г(С£™)--------► ЦСР’1’4-1).
The bigrading is nicely related to multiplication. One has that
C£p’* • C£p'’* c: C£p+p'’*
for all p,p', and furthermore that
• C£p,’«' = {0} if q - q' * p + p', and
|Ctp,fl • ctp,,fl'”(p+p,) cz Ctp+p,’q~p'
for all p,p',q,</. These facts follow easily from the identities
#(p = a)(p — (par, Wtp — a)(p + (pa)
where by definition 2ia) - £ е; -Je; for any orthonormal tangent frame
{ei,Jei,... ,en,Jen} as above.
We now examine the differential operators. It is easily seen that when
restricted to any diagonal line in the Clifford diamond, © gives an elliptic
complex
..	r(Ct₽'’)-^- Г(С€₽+1’,+ 1) —.
Definition 8.2. The quotient Ж*’*(Х) = ker ©/Im © is called the
Clifford cohomology of X.
The general Hodge Decomposition Theorem gives us the following. We
define a ©-laplacian A by
A = 22 + 22.
and consider the associated harmonic spaces
H™(X) = (ker A) n Г(С£Р’*(Х)).
Theorem 8.3. //X is compact, then J^*’*(X) is finite dimensional and there
are natural isomorphisms
#fp*(X} z HP’4(X)
for all p,q.
Remark 8.4. If W is a holomorphic vector bundle over X, then the
algebraic formalism above carries over easily to C£(X) ®c W and leads to
Clifford cohomology groups Jtfp’4(X;W) with coefficients in W. If X is
compact, Hodge Theory applies to give finite dimensional harmonic spaces
Hp,fl(X;IV) isomorphic to 3tfp’4(X;W) for each p.q.
§8. KAHLER manifolds
333
We assume from this point forward that X is compact. Our main point
now is that the Clifford formalism of Chapter I leads easily to interesting
structure in Clifford cohomology. The simplest operation is that of com-
plex conjugation c: Ct(X) -► C€(%). One easily checks that c ° # о c =
—/ and that c ° Ж ° c = — This leads to the following “Serre Duality”
Theorem:
Theorem 8.	5 (Michelsohn [1]). Complex conjugation in Ct(X) induces iso-
morphisms
Wp'q(X) z ^~p'q(X)
for all p,q. More generally, there are isomorphisms
Wp'q(X\W) Ж"Р’”Ч(Х;1У*)
for any holomorphic vector bundle W over X.
Let ★:Ct(X) -> Ct(X) denote the C-linear antiautomorphism given
by the transpose. Recall that for generators v19... ,vpe TxX, we have
★(i?i • • • vp) = vp • • • Up
Theorem 8.	6 (Michelsohn [1]). The transpose antiautomorphism induces
isomorphisms
★ -----------►№’"«(*)
for all p,q.
Clifford multiplication leads to the following interesting duality. Let (•, )
denote the usual L2-inner product on sections.
Theorem 8.	7 (Michelsohn [1]). The C-bilinear pairing fl defined on
H™(X) x H₽’"V) by
= (ф • 1)
is non-degenerate.
In all of the above we have considered Ct(X) as a left module over itself.
Considering Ct(X) as a right module produces a different Dirac operator
(as in II.5) and a corresponding decomposition £T =	+ 0*. These
are related to and by the antiautomorphism ★:
= ★ о Q) о ★ and	= ★ ° Q) о ★.
This leads to cohomology groups	and there are iso-
morphisms
Wp'q(X\W) #fp'~q(X,wy
334	IV. APPLICATIONS
for any holomorphic vector bundle IV. The operators & and ® enter
naturally when considering the $I2(C)-structures
Proposition 8.8. Let W be a holomorphic vector bundle over X. Then the
following relations hold on Ct(X) ® IV:
= 0	9)<£ + && = о
9)<£ +	=	=
[X®] = 0	[Ж,^] = &
[Ж,А] = 4A
Theorem 8.9 (Michelsohn [1]). The cohomology groups Jtfp,q(X;W) admit
an intrinsic filtration
{0} c с C ... c JTP’Q(X;IV)
where
^kq = {[ф] 6	: ^k(p = o}.
Furthermore, the representation of the Lie algebra	preserves the
subspaces
JP'4(X-W) = ker(A) n ker(A') n rC£™(X;IV) = H™(X;IV) n НРЧХ;ИТ
In the basic case when IV is trivial it is shown that A = AA, and so we
have the following:
Theorem 8.10. The Lie algebra	acting on TCt(X) preserves the
subspace ker(A) = H*’*(X). Hence, there is a canonical s\2(C)-structure on
the Clifford cohomology of X.
The induced operators on cohomology are of the form
^ИР.Ч(Х) = 4
: HP’*(X)----►	+2(X)	<£: H™(X)------►	" 2(X).
Definition 8.11. A class <peHp,fl(X) is called primitive if = 0.
Theorem 8.12 (Michelsohn [1]). For all q >0 and all p, the qth power
&Ч:НР'Я(Х) -► Hp,-fl(X) is an isomorphism. Every (p e Hp,fl(X) can be writ-
ten uniquely as
<p = Z &кч>к
k^O
where cpk е Нрл + 2к(Х) is primitive.
§9. PURE SPINORS
335
One might imagine that there is a direct relationship between Clifford
cohomology and the standard Dolbeault cohomology of X. This is indeed
true. There is an explicit element in the Hodge automorphism group
which relates the two. This is computed in Michelsohn [1]. It shows in
particular that
Hr"s’""r~s(X;IV) Н^(Х;ИО Hr(X;Qs(IV)),
and so the Clifford cohomology is independent of the Kahler metric
chosen on X. The role played by Clifford multiplication in H* *(X) does
however depend on the metric.
§9. Pure Spinors, Complex Structures, and Twistors
Thus far we have said relatively little about the role of spinors in local
riemannian geometry although there remains much to be discovered in
this area. There are, however, several places where spinors enter naturally.
For example there is a notion due to Ё. Cartan of spinors of “pure”
type which are related to almost complex structures. These spinors are
also related to the calibrations introduced in Harvey-Lawson [3]. The
interesting fact is that the simplest spinors give rise, under “squaring,”
to the most complicated differential forms.
Whenever there is a parallel spinor on a manifold, there is a reduc-
tion of the holonomy group. Via Bochner’s method, spinors play a central
role in constructing and understanding manifolds with reduced holonomy.
This is particularly true of the exceptional cases of G2 and Spin7
holonomy.
This section and the next one are devoted to a discussion of the topics
just mentioned. We begin with the notion of a pure spinor. Fix R" with its
standard inner product <•,•> and extend this metric C-linearly to C" =
R" C. Let Cln = C£n C be the associated Clifford algebra, and let
$c be a fundamental Ctn-module, i.e., an irreducible complex spinor space.
For each spinor a e $c, we consider the C-linear map
ja: C"----► $c	given by ja(v) = v a.	(9.1)
Generically, this map is injective. However, there are interesting spinors
for which dim(ker > 0, and it is these that we shall now examine. We
begin with the following definition.
Definition 9.1. A complex subspace KcC" is said to be isotropic
(with respect to the bilinear form < v>) if = 0 for all v,w e V.
We define a hermitian inner product (•, ) on C" by setting (u,w) =
<u,w>. Clearly, if V cz C" is an isotropic subspace, then V 1 V in this
336
IV. APPLICATIONS
hermitian inner product. In particular, therefore, we have
2 dimcK n.	(9.2)
Lemma 9.2. For any non-zero spinor a, the subspace ker is isotropic.
Proof. If v • a = w a = 0, then (v • w + w • r) • a = — 2<u,w>a = 0. 
Definition 9.3. A spinor a is pure if ker is a maximal isotropic sub-
space, i.e., if dim(ker = [n/2].
Denote by P$ the subset of pure spinors in $c, and denote by n the set
of maximal isotropic subspaces of C". Both P$ and 3 n are naturally acted
upon by the group Pin„, and the assignment a i—► ker ja gives a Pin„-
equivariant map
K.P$------(9.3)
To see that К is equivariant note that for v e C" and a e Sc, we have
g(v) go = gv g~l go = g-vo for all g e Pin„ с C£*. Hence,
ker jg<T = 0(ker as claimed.
Remark 9.4. Note that in fact we can define a Pin„-in variant “filtra-
tion” of the spinor spaces
$0 ° $1 ° $2 ° • • •	$[и/2] = $c	(9-4)
where $k = {o : dim(ker j'J [n/2] — k} for each k. The subset P$ = $0 -
{0} consists exactly of the pure spinors. The subsets $k are not linear
subspaces. As we shall see, there are orthonormal bases of $c which con-
sist entirely of pure spinors.
At this point our discussion divides as usual into the two cases, n even
and n odd. We shall discuss in detail only the even case. (The reader can
easily carry over the arguments and constructions to the odd case.) From
this point on we shall assume that n = 2m is an even integer, and further-
more that R2w is oriented.
Definition 9.5. An orthogonal almost complex structure on (R2m is an
orthogonal transformation J: R2w -► R2w which satisfies J2 = — Id. For
any such J, an associated unitary basis of (R2m is an ordered orthonormal
basis of the form {eivJei9... ,em,Jem}. Any two unitary bases for a given
J determine the same orientation. This is called the canonical orientation
associated to J.
Let denote the set of all orthogonal almost complex structures on
IR2w. It is easily seen that is a homogeneous space for the group O2m.
It falls into two connected components and <£~ where SO2m/Um
consists of those almost complex structures whose canonical orientation
is positive (i.e., agrees with the given one on IR2w).
§9. PURE SPINORS
337
Associated to any J there is a decomposition
C2w = V(J) ф F(J), where
V(J) = {u g C2w : Jv = - iv} = {vq + iJv0 : vQ g R2w}.	(9.5)
For any v0 g R2w we have that <u0 +	vo + i*^o> =	~
<Ju0,Jr0> + 2i<u0, Ju0> = 0, and so the space K(J) is isotropic. Con-
versely, given any m-dimensional isotropic subspace V c C2w, there is a
unique J g 4>m so that V = V(J). To see this note that if one writes C2w =
R2m ф iR2w, then V(J) is just the graph of J. Hence, there is an O2m-equi-
variant bijection
(9.6)
which associates to J the isotropic subspace V(J). Let ^2m denote the
component corresponding to .
Recall now that with respect to the complex volume element coc —
imei- • • e2m we have a decomposition	into 4-1 and — 1
eigenspaces respectively.
Lemma 9.6. If a g is pure, then either a e or a e
Proof. Let v} = (1/V2)(e, + iff 1 j g m, be a hermitian orthonormal
basis of V = ker j^. Since ± Vj for i j and l\ 1 Vj for all ij, we see that
gp/p... ,em,fm is an orthonormal basis of R2w. Note that (e; + ifj)a = 0
=>	= a for each j. Hence, fte1f r • • • emfma = ±coca = a, and
a g or a g as claimed. 
This lemma gives a decomposition P$ = P$ + II P$~ of the pure spinor
space into positive and negative types. Let P(P$+) denote the projectiviza-
tion of the positive pure spinor space, i.e., Р(Р$+) = P$+/~ where we say
that a ~ a' if a = to' for some t g C. Each of the spaces P(P$±), and
J2m are acted upon by Spin2m, in fact by SO2m.
Proposition 9.7. The maps a i-> K(a) and J i—► V(J) induce SO2m-equi-
variant diffeomorphisms
and
Proof. The second line follows immediately from the first by a change of
orientation. The map V has already been shown to be an equivariant
bijection between homogeneous spaces, and is therefore a diffeomor-
phism. It remains only to show that the equivariant map К is a bijection.
We construct K~1 as follows. Fix V g J2m and let J g be the associated
complex structure. Choose a unitary basis {e^Je^... ,em,Jem} of R2w
and set
= 4(e, - iJej) &} =	+ He)
338	IV. APPLICATIONS
as in §8. Define
(Dj = — efy and = —Ifij	(9.7)
and note that by (8.3):
All of the elements (Oj, a>k for 1 j,k n, commute in C£2m. (9.8)
Furthermore, by (8.3) these elements satisfy the identities
о. + &. = 1	(9.9)
co2 = o)j and coj = co j	(9.10)
cojcdj = (OjO) j = 0	(9.11)
w =	(9-12)
= (£y<p,£j^) = (<p,oty/0 for all <р,ф e $c (9.13)
for each y, where (u,w) denotes any Spin2m-invariant hermitian inner
product in $c.
These identities easily imply the following. Fix any j and let W be a
linear subspace invariant under multiplication by e} and JeP Then there
is a hermitian orthogonal direct sum decomposition
W = Wj ® W]
where
Wj = (Dj • W = ker(^|J and W'j = a>j • W = ker^J J
and where : W -► W is defined by /i£y(w) = £/ w. By (9.12) we see that
pe. maps Wj to W'j isomorphically and so
dim Wj = dim Wj.
We first apply this process with W = $c and j = 1 to obtain a decom-
position $c =	® $1 where = ker is invariant under e2,Je2,... ,em,
Jem. We next set W = and j = 2 to obtain a decomposition =
$2 © $2 where $2 = кег(дЕ2) n ker(/ifl) and dimc$2 = 2m~ 2. Proceeding
inductively we eventually construct
$m = кег(дг,) n ... n ker(№J with dirnc$m = 1.	(9.14)
The complex volume form a)c = imelJel • • emJem has the value +1 on
$m because: £j& = 0 => — iejJejO = a. Therefore, $m <=
We clearly have that У (J) = ker for a e $m. Hence, $m is independent
of the choice of unitary basis and the map V [$m] gives the desired
mapK-1. 
§9. PURE SPINORS
339
Globalizing this result to manifolds gives the following:
Proposition 9.8. Let X be an oriented riemannian manifold of dimension 2m.
Then the orthogonal almost complex structures on X, whose canonical
orientation is positive, are in natural one-to-one correspondence with cross-
sections of the projectivized bundle IP(P$+) of positive pure spinors on X.
In particular, X is Kahler if and only if there is a parallel cross section of
P(P$+).
Note. P(P$+) is an SO2m-bundle and is globally defined whether or not
X is a spin manifold.
Definition 9.9. The bundle x(X) = P(P$+) is called the twistor space of
X.
The total space of t(X) carries a canonical almost complex structure
defined as follows. Note to begin that the fibres of the projection я: т(Х) -►
X are naturally homogeneous complex manifolds (= SO2m/UJ. The rie-
mannian connection of X determines a field of horizontal planes for
я, that is, it determines a canonical decomposition
T(T(X)) = Г © Ж
where iC is the field of tangent planes to the fibres. As noted, 'У has an
almost complex structure integrable on the fibres. The bundle has a
“tautological” almost complex structure defined, via the identification
я*-► Т*Х, to be the structure J itself. (Here we consider t(X) as the
bundle of positively oriented almost complex structures on the tangent
spaces of X.)
In spinor terms the almost complex structure on t(X) can be written
as follows. Note that — n*TX and so	Now
whenever, a vector bundle is pulled over its own projectivization, a line
bundle splits off tautologically. Let Яж denote the tautological line bundle
in я*^с(Х) and let Я^ denote the line bundle in corresponding to
the complex structure on the fibres. Then the line bundle Я = Я^ ® Яж <=
®	<= $c(^ ©	= $c(т(Я0) gives the projective pure
spinor field defining the almost complex structure on t(X).
We now consider the question of integrability. Let J be an almost com-
plex structure defined over an oriented riemannian 2m-manifold X, and let
V g TX ® C be the associated field of totally isotropic m-planes. The
structure J is said to be integrable if it comes from an honest complex
structure on X, i.e., if there is a holomorphically related system of local
complex coordinate charts so that in each coordinate system z19... ,zm,
the field V is given by V = spanc {d/dz19... ,d/dzm}.
340
IV. APPLICATIONS
The fundamental theorem of Newlander and Nirenberg states that J is
integrable if and only if
j = ПЮ	(9-15)
i.e., if and only if for any pair of complex vector fields u,w with values
in V, the Lie bracket [u,w] again has values in V.
Suppose now that a is a (locally defined) pure spinor field whose pro-
jective class defines J. Then since V = ker ja we see that locally
Г(П = {ue Г(ТХ ® C): ira = 0}.
Proposition 9.10. The almost complex structure defined by a pure spinor
field a is integrable if and only if a satisfies the equation
vVwa — w = 0.	(9.16)
for all u,w e T(V) where V = ker ja.
Proof. Fix p,wGr(V) and differentiate the condition wa = 0 with
respect to v. This gives the equation
0 = Vy(wa) = (VyWja + w Vya.
Interchanging v and w and then subtracting shows that
0 = (Vvw - Vwu) • a + w •	- v • Vwa.
Since Vvw — Vwr = [v,w], we conclude that v • Vwa — w •	о
[v,w] • a = 0 о [u,w] g Г(И). 
Note that the equations (9.16) depend only on the projective class of
a, since for any smooth function /, we have (ir Vw — w Vy)(/a) =
f(v- Vw — w VJa. The equations (9.16) also depend only on the conformal
class of the underlying riemannian metric.
Proposition 9.10 has an interesting reformulation in terms of the twistor
space t(X). As mentioned above, t(X) carries a canonical almost complex
structure. Now a C^-map between almost complex manifolds f:(X,Jx) -*
(T,Jy) will be called holomorphic if its differential Д is everywhere J-linear,
i.e., if ° Jx = JY ° /*.
Theorem 9.11. (Michelsohn [3]) Let X be an oriented (even-dimensional)
riemannian manifold with an almost complex structure determined by a pro-
jective spinor field s g Г(т(Х)). Then this almost complex structure is inte-
grable if and only if s is holomorphic.
Note. More succinctly one could say that cross-sections of t(X) induce
almost complex structures, and holomorphic cross-sections induce integrable
ones. However, the condition that a cross-section s be holomorphic is not
§9. PURE SPINORS
341
linear since the complex structure on X depends itself on s. It is rather
of the form: dss = 0, where ds denotes the Cauchy-Riemann operator
associated to s. In this form the equation is reminiscent of many basic
equations in geometry, such as the minimal surface equation &ff — 0,
the Yang-Mills equation AVRV = 0 (cf, Lawson [1], [2]), and the condition
for balanced metrics (d*)wco = 0 (cf. Michelsohn [2]).
Proof of Theorem 9.11. It suffices to work locally, so we may choose a
positive spinor field a g Г(Р$+) with [a] = s. Let V = ker be the field
of (0,l)-subspaces as above. The condition that s = [a] be holomorphic is
equivalent to the condition that
= Xva for all v g Г( V)	(9.17)
where kv is a function depending on v and where V is the riemannian
connection on P$+. We must show that conditions (9.16) and (9.17) are
equivalent. Let {ex,... ,£„} be a local hermitian frame field for V, and
recall from (9.14) that the complex line field s is given by s = ker(/^f).
Now equation (9.16) and the fact that ё] = 0 show that	=
= 0 for all i,j. Therefore we have a g
n ker(/xf.) = s g $+, and since ё-^а e we conclude that
= 0 for all j.	(9.18)
Condition (9.16) says that the C0O(Ar)-bilinear form on Г(У) given by
/?(u,w) = wVyff is symmetric. Hence, (9.18) => ft = 0, and we conclude that
Vptr g Q кег(дг.) = [a] for all v g Г( V). Therefore, (9.16) => (9.18) => (9.17).
On the other hand if (9.17) holds, then = 0 for all u,w g Г(И) and so
(9.16) holds. 
Remark. Theorem 9.11 could be modified by changing the almost com-
plex structure on t(X) to be the one induced by s itself, i.e., by lifting
Js uniformly to all horizontal spaces above each point (instead of twisting
along the fibre). At the point s g t(X) these two complex structures agree.
In low dimensions these constructions are particularly interesting for
the following reason:
Remark 9.12. In dimensions 2m < 6 every non-zero positive (or negative)
spinor is pure, i.e., P$± =	— {0}. This is simply because the group
Spin2m acts transitively on the unit sphere in in these dimensions.
(Transitivity also holds for 2m = 8 if one restricts to the real spinors $±,
but it does not hold for the complex ones.) As a consequence we have
the following fact:
Proposition 9.13. On an oriented manifold of dimension four or six, every
nowhere-vanishing spinor field uniquely determines an almost complex struc-
ture (via (9.3)).
342
IV. APPLICATIONS
Let X be an oriented riemannian 4-manifold. A 2-form (p on X is said
to be self-dual (of anti-self-dual) if *<p = cp (*<p = — (p respectively). Since
the Hodge ^-operator satisfies (*)2 = 1, there is an orthogonal decom-
position
A2(X) = A+ © A_
where A± = {(p e A2(X): *<p = ±<p}. Now the twistor space t(X) =
IP(P$+) = P($c) can be identified with the bundle of unit self-dual 2-forms
by associating to a complex structure J its Kahler form a)j e A+ (given
by y/2a)j(U9W) = <Jl/,IV». It is a 2-sphere bundle over X with a canon-
ical almost complex structure.
The following theorem has played an important role in understanding
the structure of the self-dual Yang-Mills equations over riemannian 4-
manifolds. The seminal ideas for these applications were due to R. Penrose
and R. Ward (see Ward and Wells [1]). A riemannian 4-manifold is called
self-dual if its Weyl conformal tensor W is self-dual, i.e., if = Wwhere
W is considered as a 2-form with values in Hom(TAr,TAr).
Theorem 9.14 (Atiyah-Hitchin-Singer [1]). The canonical almost complex
structure on the twistor space of a riemannian 4-manifold X is integrable
if and only if X is self-dual.
We refer the reader to the original paper for details. The arguments in-
volve the so-called “twistor equation” which is roughly the complement
of the Dirac equation.
One class of manifolds whose twistor space always has an integrable
complex structure is the set of manifolds of constant sectional curvature.
For the case of the sphere S2m9 one has t(S2w) £ */2ш+1 = SO2m+1/U2m
= SO2lB + 2/Uw+1. The map 7t:</2m+1 -► S2m associates to an isotropic
m-plane VczC2m+i the unit vector eeR2m+1 such that C2m+1 =
V © V © Ce. (This determines e up to sign. Since R2m + 1 =
Re{V © V} © Pe, this sign is determined by fixing an orientation on
R2m+1.) In two beautiful papers, E. Calabi [2], [3] used t(S2"*) to classify
the harmonic maps from S2 to Sn. He first showed that a harmonic map
(p: S2 -► S” must lie in a geodesic subsphere of even dimension. He then
showed that any harmonic map <p: S2 -► S2m lifts to a holomorphic map
Ф:52 -♦ /2m+i which is horizontal for the projection n.
We now return to the question of almost complex structures. Suppose
that a g Г($с) is a field of pure spinors over a 2m-manifold X. Then as
we have seen, the projective class [a] g Г(Р(Р$с)) determines an almost
complex structure on X. However, the field itself carries more information.
It determines a trivialization of the canonical bundle A^V associated to
the structure. To see this we need the following lemma. Let $c be the ir-
reducible complex representation for Cf2m. For any a g $c we define the
§9. PURE SPINORS
343
isotropy group of a by
Ga = {g e Spin2m : go = a}.
Lemma 9.15. For any pure spinor о g P$+, one has that
Go = SUW
where SUm <= Spin2m is conjugate to the lifting of the standard embedding
SUW о so2w.
Proof. Set V = ker a C2m as above, and note that if go = o9 then
#(V) = jz, where by definition g(y) = g• v g~1 = Adg(u). Now V =
{x + iJx : x g R2m} where J is the complex structure determined by a. The
action of Spin2m on C2m = R2”* ® iR2m given by Ad, preserves real and
imaginary parts. Therefore, if g g Go, then g(x + iJx) = g(x) + ig(Jx) g V
for all x, and so
g(Jx) = J(g(x)) for all x g IR2m.
Hence, the image of Ga under the covering map Ad :Spin2m -► SO2m is
contained in the subgroup Um = {y g SO2m : yJ = Jy}.
We now show that Ad(GJ SE SUm. Given g e Gff, there exists a unitary
basis {el9Jel9... ,em9Jem} for IR2m which “diagonalizes” g, i.e., so that
g(ek + iJek) = el20k(ek + iJek) for real numbers 0 g Qk < n and 1 ^k^m.
This means that
g = ±n (cos ek ~ sin QkekJek).
к
Now, (ek + iJek)o = 0 => eko = — iJeko => ekJeko = — io for each k.
Hence,
go = ±f] (cos f)k + isin 0k)o = ±eiSSr.
к
Since go = o9 we have 2 £ Qk = 0 (mod 2л) and so detc(g) = el2E0k = 1
as claimed.
The above argument shows that for any g g Ad-1(SUm) we have that
go = ±a. Since SUm is simply-connected, Ad'^SUJ has two connected
components. The identity component fixes o9 the other component con-
tains — 1 and hence cannot fix o. 
Alternatively we could consider the Lie algebra
9, = {<pgA2R2w : <po = 0}
of Ga. For any (p g g^, there is a unitary basis of IR2m so that (p =
E ^kek^ek- Since ekJeko = — io9 we have that
(po= V
and so £ Лк = 0. This condition characterizes sum in um.
344
IV. APPLICATIONS
It is incidentally not true that: Gff s SUm => a is pure. For example, in
dimension eight we could take a = a0 + cr'o where a0 and a'o are pure
and determine almost complex structures Jo and — Jo respectively.
For any 2m-dimensional spin manifold X, Lemma 9.15 implies
Proposition 9.16. Each globally defined pure spinor field a on X determines
a unique reduction of the structure group of X to SUm.
Proposition 9.17. X admits a pure spinor field which is parallel if and only
if X is Kahler and Ricci flat.
Proof. If a is parallel, then the holonomy group of X is contained in
Ad(Gff) = SUm. This means that the almost complex structure is parallel
and the canonical bundle is flat.
Conversely, if X is Kahler and Ricci-flat, then $c = A*(X) and both
and Aq provide parallel pure spinors. 
Corollary 9.18. Suppose dim(X) = 4 or 6. Then X admits a non-zero paral-
lel spinor field if and only if X is Kahler and Ricci-flat.
Proof. If a е Г($с) is parallel, then each component гт+ е Г(^) is parallel,
and as noted in 9.12 each component is pure. 
Using II.8.10, we now conclude the following:
Theorem 9.19. Let X be a compact spin 4-manifold with A(X) 0. Then
any riemannian metric on X with scalar curvature к 0 is Kahler and Ricci-
flat. In fact, any such metric admits a parallel quaternion structure, i.e.,
parallel orthogonal complex structures I and J with IJ — — JI.
Proof. If к 0, then by II.8.8 and П.8.10 and the Index Theorem, we see
that X carries a parallel spinor field a. Each component a+ and is
also parallel. Assume that / 0. Since is an H-line bundle (and the
H-structure is preserved by the connection), we see that a+, ia + ,ja+ and
give a complete parallelization of $<£, that is, if one positive spinor
is parallel, then all positive spinors are parallel. We conclude that X car-
ries a family s P($+) S2 of parallel complex structures, each of which
makes X into a Kahler manifold. 
It should be pointed out that by Yau’s solution of the Calabi Conjec-
ture [2], [3], we know that Ricci-flat Kahler manifolds exist in all
dimensions.
Remark 9.20. A theorem similar to 9.19 was asserted in Hitchin [1]
for all dimensions. It was based on an incorrect calculation that Ga s
SUm for all o. Indeed, one finds counterexamples already in dimension
eight. This will be discussed in the next section.
§10. REDUCED HOLONOMY AND CALIBRATIONS
345
We conclude this section with a remark about what happens in odd
dimensions. Let $c be one of the two irreducible complex spinor spaces
for Cf2m + i and let a e $c be a pure spinor. The totally isotropic subspace
V = ker ja g С2ш+1 has dimension m and we get a hermitian orthogonal
decomposition
C2w+1 = V ® V® Ce
where e e R2m + \ This gives an orthogonal decomposition
R2w+1 =	© Re
where VR ® С = V © V (i.e., VR has a complex structure). Assuming
||e|| = 1, we see that e e V R is determined up to sign. A choice for e cor-
responds to a choice of orientation on R2m+ L In summary then, each pure
spinor a e$c determines a pair (e,J) where e g R2m+1 is a unit vector and
J is an orthogonal almost complex structure on e1. A global pure spinor
field a on a spin (2m + l)-manifold X determines, as above, a reduction
of the structure group to SUm. If Va = 0, then X splits locally as a rieman-
nian product X £ XQ x R where XQ is Kahler and Ricci-flat.
In dimensions three and five every non-zero spinor is pure.
§10. Reduced Holonomy and Calibrations
The holonomy group of a connected complete riemannian n-manifold X
is defined as follows. Fix a point x g X and to each piecewise smooth loop
у based at x, let hy: TxX -> TxX be the orthogonal transformation given
by parallel translation around y. These transformations form a subgroup
3&x g O(TxX) On whose conjugacy class as a subgroup of O„ is inde-
pendent of the choice of base point x. This conjugacy class Ж(Х) is called
the holonomy group of X. Its identity component Ж(Х)° is called the local
holonomy group of X (see Kobayashi-Nomizu [1].) If X = Yk x Zn~k
is a riemannian product, then one easily sees that 3ff(X) = J^(T) x
Ж(И) g Ok x On-k g O„, and if яДХ) = 0, then by a theorem of de Rham
the converse is true. It is therefore sensible to consider manifolds which are
irreducible. This means the universal covering is not a riemannian product.
In 1955 Marcel Berger [1] classified the possible holonomy groups for
irreducible riemannian manifolds. There has been subsequent work refin-
ing Berger’s list (see Bryant [1], [2] for a brief history). We now know
that if X is irreducible and not locally symmetric, then 3&(X)° must be
one of the following:
SOn (the generic case)
Um (n = 2m; Kahler) Spt • Spm (n = 4m; Quaternionic Kahler)
SUm (Kahler and Ric = 0) Spm (Quaternionic Kahler and Ric = 0
346
IV. APPLICATIONS
or must be one of the two exceptional cases:
G2 (n = 7) Spin7 (n = 8)
where by definition SpiSpm = Spj x Spm/Z2 and Z2 is generated by
(-1,-1). Metrics of each of the above types are known to exist.
We have already seen that:
c: Um <=> there exists a parallel projective pure spinor field on X
<tf(X) c: SUm <=> there exists a parallel pure spinor field on X.
In this and the next section we shall show that parallel spinors are also
related to the existence of Spm, G2, and Spin7 holonomy metrics. These
results are due to the authors and Reese Harvey (see Michelsohn [3]).
The fundamental elementary point is the following:
Proposition 10.1. Let X be an n-dimensional (riemannian) spin manifold on
which there exists a globally parallel spinor field a. Then at any point x
the holonomy group satisfies
&x <= GOx	(Ю.1)
where Gax = {g e Spin(7 rX): gox = ax}. In other words,
Jf(X) <= Go
where Ga denotes the conjugacy class of Gffx in Spin„ (which is defined
independently of x because a is parallel).
Conversely, if (10.1) is satisfied for a spinor ax at some point x, then ox
extends to a globally parallel field a on X.
Proof. If Vct = 0, then the holonomy transformation hy e Spin(TxX)
generated by parallel translation around any loopy must preserve a. Hence,
jtfx cz Gax. Conversely, if c: GOx, then parallel translating ax from x
to у is path-independent and defines a globally parallel field. 
We know that on any spin manifold with A 0, any metric with zero
scalar curvature has parallel spinor fields. Therefore in light of Proposition
10.1, it would be interesting to understand the structure of the isotropy
groups G^ c: Spin„. For и 6 we have already seen in §9 that Ga c: S4«/2]>
so the first interesting case occurs when n = 7. Recall that the irreducible
real representation $ of Spin7 has dimension 8. Furthermore, Spin7 acts
transitively on the unit sphere giving a diffeomorphism S7 = Spin7/G2 (see
1.8.2 forward). Hence, all non-zero spinors are essentially equivalent, and
we have the following:
Proposition 10.2. Let $ be the irreducible real spinor space for Spin7. Then
for any non-zero о e $,
Ga = G2
Ad: Spin8
§10. REDUCED HOLONOMY AND CALIBRATIONS	347
Corollary 10.3. A 7-dimensional spin manifold has holonomy g G2 if and
only if it carries a non-trivial parallel spinor field.
The group Spin8 has three irreducible 8-dimensional real representations
A+’Spin8 SO(D
SO(R8)
A :Spin8 -----► SO($ )
related by the triality automorphism (see 1.8). Each is transitive on the
unit sphere and therefore determines a unique conjugacy class of sub-
group. That is, in each representation the isotropy subgroups of any two
non-zero elements are conjugate.
Proposition 10.4. There are three distinct conjugacy classes of subgroups
„ . o ? Spin7+
Spin7 c Spm8
Spin^
defined as the isotropy groups Spin7 = Gv, Spin* = Ga± for non-zero
elements v e R8 and g $±. These three conjugacy classes are cyclically
permuted by the triality outer automorphism of Spin8. Furthermore, the
conjugacy classes Spin* are conjugate in the enlarged group Pin8. In
fact,
Spin7 = e • Spin^ • e~1	(10.2)
for any ее S1 a Pin8 (where S7 a R8 c: Cf8 are the standard embeddings).
The covering homomorphism Ad:Spin8 -► SO8 gives embeddings
Ad: Spiny <—► SO8	(10.3)
which are not conjugate in SO8 but which are conjugate in O8.
Proof. Let Z » 7L2 ® 7L2 denote the center of Spin8. Then the non-
conjugacy of Spin7, Spin7 and Spin7 follows from the fact that they
intersect Z in three distinct subgroups (cf. 1.8). Since the representations
are permuted by triality, so are these subgroups. Given e e S7 a R8, and
a e $+, we have that eo e$~ and clearly that Gea = eGae~l. This proves
(10.2).
To see that (10.3) is an embedding, note that (ker Ad) n Spiny = {!}.
To see that Ad(Spin7) and Ad(Spin<7) are not conjugate in SO8, note
that: #Ad(Spin7 )g~l = Ad(Spin7) for some g e SO8 would imply that
у Spin7 у-1 = Spin^ for у e Spin8 with Ad(y) = g. On the other hand
(10.2) shows that Ad(Spin*) are conjugate by elements Ad(e) (= reflection
in the hyperplane e1) in O8. 
The embeddings (10.3) give a single conjugacy class of the subgroup
Spin7 g O8 and it is exactly this representation which appears in Berger’s
list. Consequently we have
348
IV. APPLICATIONS
Corollary 10.5. An 8-dimensional spin manifold has holonomy <= Spin7 if
and only if it carries a non-trivial parallel spinor field.
Note that if о is a parallel real spinor field on an 8-manifold, then a
decomposes a = a+ + where both cr + and are parallel. If both
and are non-trivial, we get a further reduction of the holonomy group
to Spin7 n Spin<7 = G2. (To see this, note simply that under triality we
have Spin7 n Spin^ = Spin7 n Ad~1(SO7) s Ga for о as in Proposi-
tion 10.2.) Recall that the complex spinor spaces for Spin8 are just the
complexifications of the real ones. Write = $+ ф i$+ and consider an
element = a0 4- iat g Then for g e Spin8, gac = if and only if
ga0 = cr0 and goY = and so
g<7C = GffQ n G<T1
If (7o,ot g $+ are linearly dependent (over R), then G(Tc Spin7. If they are
linearly independent, then Gffc = Spin6 SU4. In this second case Gffc
remains unchanged if we replace ac by ac = a0 4- ia{ where {<?oA}
any orthonormal basis of span^o^). Up to real scalars this orthonor-
mality is equivalent to the fact that <<tc, <tc> = 0. One easily checks that
in this dimension, the pure spinors are exactly those which are isotropic
with respect to the natural real structure on .
Remark. It should be pointed out that R. Bryant [1], [2] has found a
number of beautiful examples of manifolds with G2 and Spin7-holonomy.
We shall now show that the propositions above give a simple neces-
sary and sufficient condition for the existence of a topological G2 or
Spin7-structure. By this we mean the following. Let X be a differentiable
n-manifold and fix a Lie group G c: GL„(R). Then by a topological G-
structure on X we mean a topological reduction of the structure group
of TX from GL„(R) to G. This is defined to be a G-equivariant embedding
PG a PGL(X) of a principal G-bundle into the frame bundle of X. Reducing
the structure group to O„ c: GL„(R) is equivalent to choosing a riemannian
metric on X. If G <= O„ is a closed subgroup, then reducing the structure
group from O„ to G is tautologically equivalent to finding a cross-section
of the bundle
PO(X)/G-----> X
whose fibres are the homogeneous spaces On/G.
Theorem 10.6. Let X be a differentiable 7-manifold. Then X carries a topo-
logical G2-structure if and only if it is a spin manifold.
Proof. If the structure group of TX can be reduced to G2, then because
тг i G2 = 0, X is a spin manifold (see Remark 1.1.10). Conversely, suppose
§10. REDUCED HOLONOMY AND CALIBRATIONS	349
X is spin and let $ be the irreducible real spinor bundle of X. Then since
fibre-dim(^) = 8 > dim(X), there exists a nowhere vanishing cross-section
a of $ (cf. Milnor-Stasheff [1]). By Proposition 10.2 we can identify the
unit sphere bundle in $ with PSpinM/G2- Therefore the normalized sec-
tion a/||a|| gives us a G2-reduction. 
Note that if X is a spin 7-manifold, then the topological G2-structures
on X are in one-to-one correspondence with global spinor fields a e Г($)
such that ||a|| = 1. The story for Spin7-reductions is quite similar.
Theorem 10.7. Let X be a differentiable 8-manifold. Then X carries a to-
pological Spin^-structure if and only if X is spin and either %($ + ) = 0 or
X($~) = 0, where $* denote the irreducible real spinor bundles on X. Equiv-
alently, X carries a Spin^-structure if and only if w^X) = w2(X) = 0 and
for an appropriate choice of orientation on X we have that
Pi(*)2 - 4p2(X) + 8Z(X) = 0.	(10.4)
Note. Reversing the orientation of X leaves the Pontrjagin classes un-
changed but reverses the sign of /(X) in H8(X). Thus, condition (10.4)
could be replaced by the condition that for any orientation on X, one of
the two equations
Pi W2 - 4p2(X) ± 8Z(X) = 0
holds in H8(X).
Note that for manifolds of the type X8 = S1 x У7, equation (10.4) is
automatically satisfied. Furthermore, X8 is spin iff У7 is spin, and Theo-
rem 10.6 applies. There are other special cases of some interest.
Corollary 10.8. Let X be a complex manifold of dimension 4. Then X carries
a topological Spin7 -structure if and only if
Ci[c? - 4c1c2 4- 8c3] = 0
Corollary 10.9. Let M and N be compact spin 4-manifolds. Then the pro-
duct X = M к N carries a topological Spin^-structure if and only if
9sig(M)sig(N) = 4%(M)x(N)
In particular, M x M has such a structure if and only if
3sig(M) = ±2Z(M).
Droof of Theorem 10.7. If X is spin and z($+) — 0, then there is a nowhere
vanishing section а e Г($+) which, via Proposition 10.4, reduces the struc-
ure group of Spin7. In fact, by 10.4 we have the identification
PsPin8W/Spin7+ ^S(F)	(Ю.5)
350	IV. APPLICATIONS
where S($ + ) denotes the unit sphere bundle in $ + . Hence, the Spin7-
reductions are in one-to-one correspondence with spinor fields a e Г($+)
such that ||ff|| = 1. If %($) = 0, the analogous discussion holds.
Suppose conversely that the structure group of TX has a Spin7-
reduction, i.e., suppose there is a Spin7-equivariant embedding PSpin7
POg(X) of a principal Spin7-bundle into the orthogonal frame bundle of
X (for some riemannian metric). Since Spin7 is simply-connected, this
determines a spin structure: PSpin8M -► Лю8М <= po8M> in which PSpin7
lifts to a Spin7-equivariant embedding PSpin7 <= PSpin8(X), which in turn
corresponds by (10.5) to a cross-section of S(J5+). Hence, %($+) = 0.
Note. The reader may find the lack of ambiguity in this last paragraph
somewhat unsettling. If so, let us examine the argument more closely. In
saying that PSpin7 c Л?8М is “Spin7-equivariant” we must specify an
embedding Spin7 c: O8. This choice is important only up to conjugation
in O8. (To see this, fix g e O8 and consider the map g: PO*(X) -► РОб(Х)
which converts the given O8-action to the conjugated “g O8#"1 ’’-action.)
Now the two subgroups Ad(Spin7) are conjugate in O8 but not in SO8.
We choose an embedding whose class in SO8 is Ad(Spin7). Since Spin7
is connected, the embedding PSpin7 Po8M clearly fixes an orientation
on X, i.e., it selects a sub-bundle PSO8(X) c: POs(JV). With this choice of
orientation, PSpin7 lifts to PSpin + <= PSpin8(X). Suppose on the other hand
that we had chosen an embedding Spin7 c SO8 of Ad(Spin7)-type, which
differed from the first choice by an element g e O8 (with det g = — 1). Our
given reduction f: PSpin7 Л>8М must now be conjugated to g ° f о g~1:
PSpin7 °* FO8(X). This new reduction chooses the opposite orientation on
X and lifts to a Spin^ -subbundle of PSpin8(X).
The proof that/(jf1) = pJX)2 - 4p2(X) + 8/(X) is a good exercise in the
algebra of characteristic classes, which we leave to the reader. 
It is an interesting fact that spinors in dimensions seven and eight are
also related to certain exceptional geometries of subvarieties introduced
in Harvey-Lawson [1], [2], [3]. The fundamental idea is this: Let cp be
a differential p-form on a riemannian manifold X normalized so that
(p\P = volF	(10.6)
for every oriented tangent p-plane P on X. This is equivalent to the con-
dition that
fz <P vol(Z)	(10.7)
for every oriented p-dimensional C1 -submanifold Z of finite volume in X.
If such a manifold Z has the property that
J*z Ф = vol(Z)	(10.8)
§10. REDUCED HOLONOMY AND CALIBRATIONS
351
then Z is called a ф-submanifold. These creatures are interesting for the
following reason. Let Z c= X be a compact oriented p-dimensional sub-
manifold, possibly with boundary. We say that Z is homologically volume-
minimizing if vol(Z) vol(Z') for all such Z' with the property that Z — Z'
is a boundary (of some singular (p + l)-chain in X). When Z is not com-
pact, it is called homologically volume-minimizing if this property holds
for every compact subdomain-with-boundary in Z.
Proposition 10.10. Suppose cp satisfies (10.6) and dcp = 0. Then every cp-
submanifold is homologically volume-minimizing in X.
Proof. Let Z be a compact ф-submanifold and Z' a competitor with
Z — Z' = Sc for some singular chain c. Then by (10.7), (10.8) and Stokes’
Theorem, we have
vol(Z) = j*z Ф = j*z. Ф = vol(Z'). 
This proposition shows that every ф-submanifold is in particular a
minimal submanifold, i.e., its mean curvature vector field is identically
zero. It is therefore roughly as regular as X itself. For example, if X is
real analytic, so is any ф-submanifold. The above definition extends im-
mediately from submanifolds to integral currents (see Harvey-Lawson
[3]). The “ф-subvarieties” can then have singularities, although they
remain homologically volume-minimizing.
Definition 10.11. If cp satisfies (10.6) and dcp = 0, then cp is called a
calibration. The pair (Х,ср) is a calibrated manifold and the family of cp-
submanifolds is the associated calibrated geometry.
Calibrations are interesting for the special cases, not the general one.
We know that few topological reductions of the structure group of a mani-
fold are “integrable” in the sense that they are holonomy reductions. Simi-
larly, few calibrations are “integrable” in the sense that the associated
geometry of submanifolds is large. However, many interesting examples
do exist.
Example 10.	12 (Kahler geometry; U„-case). Let X be a Kahler mani-
fold of complex dimension n. Fix p, 1 p n, and consider the 2p-form
where co is the Kahler form of X. Then cpp is a calibration and the asso-
ciated family of фр-submanifolds (фр-subvarieties) consists exactly of the
complex analytic submanifolds (subvarieties) of dimension p in X. Propo-
sition 10.8 says that every complex analytic subvariety is homologically
volume-minimizing in X (a result due to H. Federer [1]).
352
IV. APPLICATIONS
Example 10.	13 (special lagrangian geometry; SU„-case). Let X be a
Ricci-flat Kahler manifold of complex dimension n, and assume that the
canonical line bundle x = A?T*X is trivial. Consider the n-form
A = Re(Q)
where Q is a parallel holomorphic n-form (i.e., section of к). Then A is a
calibration and the associated geometry is called special lagrangian. It is
shown in Harvey-Lawson [3] that the special lagrangian subvarieties are
as plentiful as the complex subvarieties.
There are three rich exceptional geometries which appear in dimensions
seven and eight.
Example 10.	14 (associative and coassociative geometry; G2-case). Con-
sider euclidean space R7 = Im ©, where О denotes the Cayley numbers,
and define a parallel 3-form <p on R7 by setting
<p(x,y,z) = <xy, z>
Then (p is a calibration with a rich geometry of submanifolds. It is called
the associative geometry because the submanifolds are also defined by the
vanishing of the associator [x,y,z] = (xy)z — x(yz). The dual form
ф = *cp
gives a geometry of 4-folds called the coassociative geometry.
These geometries exist (and are rich) on any riemannian 7-manifold
with G2-holonomy. Note that <p and ф are clearly G2-invariant since they
are defined using Cayley multiplication. In fact <p and ф are the only non-
trivial exterior p-forms, 0 < p <7, which are G2-invariant. Any riemannian
7-manifold with G2-holonomy carries parallel forms (p and ф which at
each point are equivalent (over SO7) to those given above. Furthermore,
in any such manifold the geometry of associative 3-folds and coassociative
4-folds is vast. In particular, every analytic submanifold of dimension
two is contained in an associative submanifold (whose germ along the sur-
face is uniquely determined).
Example 10.	15 (Cayley geometry; Spin7-case). Consider euclidean
space R8 = (D and define a parallel 4-form Ф on R8 by setting
<$(x,y,z,w) = <x x у x z, w>
where the triple cross product is defined by x x у x z = |{x(yz) — z(yx)}.
Then Ф is a calibration which, as shown in Harvey-Lawson [3], has a
rich geometry of submanifolds, called the Cayley geometry. It includes as
special cases all complex submanifolds and all special langrangian sub-
manifolds for a 6-dimensional family of complex structures on R8. If we
consider R7 c R8 as ImO = I1 с O, then Ф|И7 = ф and 1 l Ф = cp.
§10. REDUCED HOLONOMY AND CALIBRATIONS
353
The form Ф is Spin7-invariant. It is, in fact, the unique Spin7-invariant
p-form on R8 for 0 < p < 8. Any riemannian 8-manifold with Spin7-holo-
nomy carries a parallel 4-form of this type and has a large geometry of
Cayley submanifolds.
The forms Ф and (p above were first discovered by Bonan [1].
There is a direct and intimate relationship between spinors and cali-
brations. It comes about by what can be roughly termed the process of
“squaring.” We shall examine this process to the point that meets our
needs here. For further discussion the reader is referred to the book of
Reese Harvey [1].
Let n = 2m be an even integer and recall the fundamental algebra
isomorphism
* Homc($c,$c) =	® $*	(10.8)
We introduce on $c a Pin2m-invariant hermitian metric (•,) by which we
can identify with $c. (Jc is the same real vector space as $c but with
the opposite complex structure, i.e., with scalar multiplication by t in $c
replaced by f.) The identification $c® $c = Homc($c,$c) is then
determined by associating to a pair of spinors aba2 e $& the elementary
endomorphism
® ff2(r) = (t, ff2)CT! for T 6 $c.
Inverting (10.8) now gives the following:
Proposition 10.16. There is an isomorphism
$C ® $C	^2m
which associates to a pair of spinors aba2 g $c, the unique element,
cp g Ct2m such that (р-т = (T,a2)ai for all т g $c. Under this isomorphism
we have for each ф g C£2m, an identification of endomorphisms
рф ® Id ~ and Id ® p^ %
where Ьф and denote respectively left and right multiplication by ф on
C£2m, and where ф* = а(ф*\ Via 1.1.3 the isomorphism above determines a
canonical identification
$c®$c A*C2w	(10.9)
which is Pin2m-equivariant, where g g Pin2m acts on $c ® $c by pg ® pg,
and on A*C2m by the complexification of the standard representation of O2m
on forms.
Proof. All assertions but the last are easily checked. For the last, one need
only recall that а(д‘) — g~l for g g Pin2m and therefore pg® pg = pg®
Ptg-iy = Ad^ = the standard representation of g on A*C2w.
354
IV. APPLICATIONS
There is a corresponding real version of this result. For simplicity we
shall consider only the cases where C£„ is a real matrix algebra. Suppose
n = 6 or 8 (mod 8) and let $ denote an irreducible real C£„-module. Then
Clifford multiplication gives an isomorphism
д *. C£„	► HomR($, $).
When n = 7 (mod 8), there are two distinct isomorphisms
---► HomR(^±,^±)
where $+ (and $~) are the irreducible real C£„-modules on which the
volume form co acts by Id (and -Id respectively), and where Ct* =
(1 ± co) • C£„. Inverses to these maps are given explicitly as follows:
Proposition 10.17. Let $ be an irreducible real Ct „-module where n = 6 or 8
(mod 8), and assume $ is provided with a Pinn-invariant inner product
Then assigning to a pair а,т g $ the endomorphism Lff®t() = <-,т)сг deter-
mines an isomorphism
$®$-^Ct„	(10.10)
under which p^ ® p^ % L^® where ф* = %{ф'). Via 1.1.3 this gives
a canonical P‘\nn-equivariant isomorphism
$ ® A*R"	(10.11)
where g g Pin„ acts on $®$ by pg®pg. The isomorphism (10.11)
decomposes into two isomorphisms:
(10.12)
p = 3 or 4(mod 4)
Л2$ —ф APR”	(10.13)
P = 1 or 2 (mod 4)
under the canonical decomposition $ ® $ = (®2$) © (A2$) into symmetric
and skew-symmetric 2-forms respectively.
When n = 7 (mod 8), all analogous statements hold with $ replaced by $+
(or and with Ctn replaced by Ct* (or Ct~ respectively).
Proof. The initial statements are similar to those in Proposition 10.16 and
are proved analogously. For statements (10.12) and (10.13), we note first
that under the isomorphism (10.10), symmetric tensors go to self-adjoint
transformations and skew-symmetric tensors go to skew-adjoint trans-
formations. On the other hand Clifford multiplication by a p-form has
the property that (дф)* = ( —lpp(p+1)д<р. This proves (10.12) and (10.13).
Details of the case where n = 7 (mod 8) are left to the reader. 
§10. REDUCED HOLONOMY AND CALIBRATIONS
355
For any spinor a in $ (or $+ if n = 7 (mod 8)), this proposition gives a
decomposition of the “square”:
ff ® a = фо + <Рз + <?4 + <?7 + Фе + • • • + <Pit- 1 + <l>4t + • • • (Ю.14)
with (pp e APR" for each p. Under special assumptions about a, certain
of these components will vanish or be related to one another by the Hodge
♦-isomorphism. We shall first consider the case where n = 8.
Theorem 10.18. Let $ denote the irreducible real Ct8-module, and let
$ = $+ ® be its standard decomposition into ± 1 -eigenspaces for the
volume form co. Then for any unit-length spinor a g $ +, the decomposition
(10.14) becomes
сг®сг=1+Ф + со
where Ф g A4R8 is the Cayley 4-form defined in 10.13.
Proof. To begin consider о ® a g C£8. Since cocr = a, we have by Proposi-
tion 10.17 that a ® ff = (coa) ® ff = co(a ® a) and similarly that о ® о =
ff ® (off) = (ff ® ff)- a(co') = (a ® ff)o. Now left and right multiplication
are related to the Hodge ♦-operator by (5.35) of Chapter II. This shows
that if we write ff ® ff = £ (pp as in (10.14), then
p(p-1)	p(p+ i)
Z(-i) 2 *<Pp = E</>p = E(-1) 2 *<?₽•
Hence q>p = 0 for p # 0,4,8, and furthermore ♦ф4 = ф4, *cpQ = <p8.
The identity element 1 g C£8 can be written as 1 = £ ffa ® ffa where
{ffi9... ,a16} is any orthonormal basis of $. It follows by choosing ff{ = a,
that <1, ff ® ff) = 1. Therefore ff ® ff = 1 + Ф + co for some Ф g A4R8.
(We are using here the inner product on C£8 for which 1 and eh • • • eip,
1 ij < ... < ip, form an orthonormal basis. This differs from the stan-
dard one on Hom(R16,R16) by a factor of ^.)
Observe now that since the isotropy subgroup of any nonzero element
ff g $+ is Spin7, the image Ad(Spin7) <= SO8 must leave invariant each
component of ff ® ff. In particular Ф is a Spin7 -invariant 4-form. This
identifies it as a multiple of the Cayley 4-form. To determine which multiple,
we proceed as follows. Fix orthonormal vectors e{,... ,e4. Then
(ff ® ff, er ... e4> = (ff ® ff, ... e4-1> = <(e4 • • • eYff) ® ff, 1>
= <e4--e1a,a>^||a||2 = 1.	(10.15)
This proves that the A4-component of ff ® ff is a calibration, i.e., it satisfies
(10.6).
Note that we have equality in (10.15) if and only if e4 • • • e{ff = ff. Set
w =	• • • e4. Since w2 = 1 and w$+ = $\ we can decompose $+ =
$ + ® $- into the ±1 eigenspaces of w. Multiplying by ere5 maps $ +
isomorphically onto $t, so dim$+ =4. The group Spin7 is transitive
356
IV. APPLICATIONS
on the unit sphere in so we may assume a e $Hence equality does
occur in (10.15) for some choices of el9...,e4. This proves that the
A4-component of a ® & is exactly the Cayley 4-form. 
Entirely analogous arguments prove the following:
Theorem 10.19. Let $ be the irreducible real Ct^-module on which the
volume form a> acts by 4-1. Then for any unit spinor a g $, the decomposition
(10.14) becomes
о®а=\+<р + ф + со
where cp g A3R7 is the associative calibration and ф = g A4R7 is the
coassociative calibration.
It is clear that squaring spinors is a natural way to produce interesting
calibrations in all dimensions. (This interesting fact was first noticed and
proved by Dadok and Harvey [1].) The method is efficient both locally
and globally on a manifold. It is also efficient for analysing the condition
dcp = 0 as we shall now see.
Let Ф be a Cayley 4-form and consider the orbit GL8 • Ф cz A4R8. This
orbit is independent of the choice of Ф and of the underlying inner prod-
uct. It consists precisely of all 4-forms whose isotropy group in GL8 is
isomorphic to Spin7. We call the forms in Ca = СЬ8 Ф, the 4-forms of
Cayley type. Note that any such form Ф' determines uniquely an inner
product on К8 which makes Ф' a Cayley calibration.
Theorem 10.20. Let X be a smooth oriented 8-manifold, and suppose Ф is
a smooth 4-form on X which is of Cayley type at each point. Then Ф
is closed if and only if Ф is parallel in the riemannian metric it determines.
Therefore, to find a closed 4-form of Cayley type is to find a metric
with holonomy c= Spin7.
Proof. Introduce the metric for which Ф is a Cayley 4-form. Let a be a local
spinor field so that а®сг=1+Ф + со. Since *Ф = Ф we see that
d<& = 0 о </*Ф — 0 <=> £)Ф = 0 <=> D(o ® а) = 0 where D ~ d 4- d* is the
Dirac operator on C£(X) discussed in Chapter II, §5. Let Ц) denote the
Atiyah-Singer operator on the real spinor bundle $. Theorem 10.19 will be
a direct consequence of the following lemma:
Lemma 10.21. For any о g Г($) with ||a|| = 1, we have
||0(a®< = ||M2 + UM2-
Proof. From 10.17 we see that
D(ff ® ff) = eF4? ® °) = (E ® ° + E
= (0ct) ® a + £ (e/r) ® (Veya).
§11. SPINOR COHOMOLOGY; MANIFOLDS WITH c, = 0	357
Since ||<r||2 = 1, we see that <Va,a> = 0. Furthermore <ela,eJa> =
—	It follows that
||O(tf ® ff)||2 = ||0a||2 + Y
= IIM2 + Z Ml2 = IIM2 + l|v< 
In analogy with the above we define a 3-form (p e A3IR7 to be of associa-
tive type if = {g g GL7: g*cp = cp} is isomorphic to G2. Bryant observed
that the set of such 3-forms is open in A3R7. Each such 3-form determines
a unique inner product on R7 which makes it an associative 3-form.
Arguing as in the proof of 10.20 gives the following:
Theorem 10.22. Let X be a smooth oriented 7-manifold and suppose <p is a
smooth 3-form on X which is of associative type at each point. Then for the
riemannian metric determined by the form cp is closed and coclosed if and
only if it is parallel.
Therefore, to find a metric with holonomy c=G2 it suffices to find a
closed 3-form <p, pointwise of associative type, such that
Ф„ф) = о
where is the Hodge *-operator for the metric determined by <p.
Characterizations of this type were formulated and used in the funda-
mental work of Bryant [1],[2].
§11. Spinor Cohomology and Complex Manifolds with Vanishing
First Chern Class
In this section we shall examine the geometry of Dirac bundles over
complex manifolds. We shall recapture a number of classical results and
prove some new ones. As an application we shall show the existence of
compact manifolds with Sp„-holonomy. The results here are due to the
second author and can be found in Michelsohn [1].
Fix a compact Kahler manifold X of dimension n, and let S' be a
holomorphic bundle of left modules over C£(X) equipped with a hermitian
metric and its associated canonical hermitian connection V. (This is
the riemannian connection determined uniquely by the condition that
a g T(S) is holomorphic if and only if VJya = for all v g TX; (see Wells
[1].) We assume that this metric and connection make S a Dirac bundle
over X.
We define operators and 3) on T(S) by setting
(H.l)
J=1	J=1
358	IV. APPLICATIONS
where {e^... ,en} is any local hermitian frame field defined as in §§8 and
9. The curvature Rs of the canonical hermitian connection is of type (1,1),
i.e., £j = Rft £j = 0 for all ij. From this and the identities (8.3) it fol-
lows that
®2 = 0 and 02 = O	(11.2)
These operators are easily shown to be formal adjoints of one another.
Proposition 11.1. There is a parallel orthogonal direct sum decomposition
of S into holomorphic sub-bundles
S = S° © S1 © ... © Sn
with the property that
Q): T(S1’)---► nS1’+1)	and @: Г($‘’+1)--------► T(Si)
for all i.
Proof. Let euJe19... ,en9Jen be a local unitary frame field on X and define
£j = ?(eJ ~ as above. Consider the family of commuting elements
co19... ,co„, cf,... ,cb„ defined as in (9.7) by setting co, = — sfij and =
— sfij. To each (possibly empty) subset I = {il9... ,ir} c {1,... ,n} with
complementary subset {f,... Jn_r} we associate the element
CD, = CO.- ’ ’ CO. CO;, * • CD:
1	*1	*r Jl	Jn-r
and we denote |/| = r. From the relations (9.9)—(9.11) we can write
1 = п +%) = i nr
J=1	r = 0
where
*r = I CO;	(11.3)
|/|=r
The operators 7tr are independent of the choice of unitary frame field and
are therefore globally defined on X. They have the following basic prop-
erties for all r, s and j:
V7tr = 0	(11.4)
7C2 = Ttr	(11.5)
7tr7ts = 0 if r # S	(11.6)
<7сга, т> = <a, 7сгт> fora, те S	(11.7)
£j7tr-x = 7tr£j and £j7tr+i = Tlr£j.	(11.8)
Properties (11.5)—(11.8) follow directly from (9.8)—(9.13), as does the fact
that nr e C£°’2r-"(X) for each r. Since 0 = V(l) = £ V7tr and V preserves
sections of the subbundle C£p g(X), we then obtain property (11.4). We
§11. SPINOR COHOMOLOGY: MANIFOLDS WITH c. = 0
359
now define
Sr = nr • S r = 0,1,... ,n
From the properties above we see that the л/s form a parallel family of
orthogonal projection operators. Therefore by (11.3) we have the orthog-
onal decomposition S = фг Sr.
We now show that each Sr is holomorphic. Let a e T(S) be a (local)
holomorphic section, and write ar = 7tra. Then for any v e TX we have
that Vjy(c7r) = 7tr(VJt,a) = лДкг) = iar. Hence, each ar is holomorphic, and
one sees immediately that the space of such local cross sections spans the
fibre of Sr at each point. Therefore Sr is a holomorphic sub-bundle.
The fact that ®(F(Sr)) <= F(Sr+1)and ^(F(Sr)) c T(Sr~1) follows directly
from (11.3) and (11.8). 
We now have defined a complex
0-----> r(S°)	ns1) -------------r(S")---------► 0	(11.9)
with adjoint complex
о«— r(s0)	ns1) ----------nsB)<— o.	(ii.io)
This complex is elliptic, i.e., at any non-zero cotangent vector the symbol
sequence is exact. To see this note that the principal symbol of 0 at
£ g TX is given by Clifford multiplication = £c- = |(£ — iJ£)-. Sim-
ilarly we have a^) = £c-. The exactness of (11.9) follows from the iden-
tity $Jc + ^c = -1ИР-
Definition 11.2. The rth cohomology group of the Dirac bundle S is the
quotient
Y _ ker(^lr(sn)
(	> ©(Щ'-1))’
From the ellipticity of the complex (11.9) it follows that each of the
vector spaces Jfr(X,S) is finite dimensional.
Taking S = C£(X) and using both Q) and 9Г we essentially recover the
Clifford cohomology groups of X (see §8).
This viewpoint gives a simple unified approach to the fundamental
vanishing theorems of Kahler geometry. Let us define elliptic self-adjoint
operators V*V and V*V on T(S) by
v*v = - £ v*v = - £ v*.£,
j=l	j=l
witheb ... ,e„ as above. They have the property that f <V*Va,a> = f |Va|2
where V is defined by Vya = |(Vv — iVJy)a. Consequently we have that
ker(V*V) = {holo, sections of S}, ker(V*V) = {antiholo, sections of S}.
360
IV. APPLICATIONS
We now define associated curvature operators
= E ej= E ч• £k• Rt.ix-
j,k=l	j,k=l
Applying the arguments of II.8 and using the fact that R£j£k = Rs£j £k = 0
for all i,j, gives the following (Michelsohn [1]).
Theorem 11.3.
9)3) 4-	= V*V + Яс = V*V + Йс.
Taking the average of these two formulas gives a formula of the type
presented in §8 of Chapter II.
We now consider a holomorphic hermitian line bundle, A, over X.
The curvature Кл of A is an imaginary valued, J-invariant 2-form (i.e.,
(l,l)-form) on X. In terms of a hermitian frame field ... ,e„ as above,
we have
nA _ _____ nA __ nA
for all j,k; that is, the matrix ajk = R^Sk is hermitian symmetric. If this
matrix is positive definite at each point, then A is called a positive hermitian
line bundle. (If ajk is negative definite at each point, then A is called nega-
tive. In this case A* is positive.) It is straightforward to see that any closed
imaginary (l,l)-form representing the first Chern class of A is the curvature
form of a hermitian metric on A. We now have a vanishing theorem of
“Kodaira type.”
Theorem 11.4. Let X be a compact Kahler manifold and let S be a holomor-
phic Dirac bundle as above. Then for any positive line bundle A there exists
an integer N such that for
JT(X, S ® Aw) = 0
for all r > 0.
Proof. The curvature of the bundle S ® Лш is given by
Kv.W ®f) = (Rsv.w°) ®t + mo® (Rv,wt\
Since A is positive we may choose local hermitian frames so that R^j £k =
-z/jk where 0 < zx z2 A„. It follows that
94® ® Л = (e {+ m E ®
\j<k	/	j.k
= ^c(a) ®	— m\£	)	®
\ J	J
= [$Rc(a) +	® t
§11. SPINOR COHOMOLOGY: MANIFOLDS WITH Cj = 0	361
where t = — £	= E Note that for any a e S we have
(та, а) = £ А/ауг, а) = £ А/со?а, а) = £ ^||а)уг||2
Е IMI2 = 2i Е
where w = £ а>7. This element has the fundamental property that
W7lr = Г71г
which follows from the elementary fact that c^coi = coj if i e I and co^ = 0
otherwise. Consequently, if a e Sr, i.e., if nra = a, then w = ra. It follows
that on F(Sr ® Aw)
^0лт^(Я£ + тгЛ)® 1
where z > 0 is the minimum of on X. For r 1 and m sufficiently large
we have > 0, and it follows from 11.3 that 33 4- 33 > 0 on
T(Sr ® Aw). 
There is also a vanishing theorem of “Nakano type” for the Clifford
bundle. We refer to Michelsohn [1] for the proof.
Theorem 11.5. If A is a negative line bundle over a compact Kahler manifold
X9 then
jtp5X,A) = о
for all q < 0.
Let us now suppose that X is a Kahler spin manifold, i.e., that cx(X) = 0
(mod 2), and let $c be the bundle of complex spinors on X with its canonical
riemannian connection. This bundle is holomorphic; in fact, it is equiva-
lent as a complex vector bundle to AJT*X ® <51/2 where <5-1 = A!^T*X is
the canonical bundle (see App. D.). Since X is Kahler, the canonical con-
nection on $c is also the canonical hermitian one. We can therefore apply
Theorem 11.3. To compute the terms and $RC which appear there, we
need to know the curvature tensor of $c. From (4.37) of Chapter II we see
that this can be expressed in terms of the Riemann curvature tensor R of
X by the formula
2и
Ry,W = 4 E	tlj^i ‘ Wj'
i,j=l
where qi9... ,^2и is any real orthonormal basis of the tangent space.
Choosing a unitary basis el9Jel9... 9en9Je„ and writing {e;} as above, we
can reexpress this as
Ry,W = E	* £k* + (Rv,w£p£k)£j ’ Ё?}
j,k = 1
= 2 E	+ E (Rv,w£p£j>
j
362
IV. APPLICATIONS
where we use the fact that e/k 4- ek£j = — djk. It follows that
*c =	(ПП)
= 2 У Rjklm * ek *	* em* + £ ^]kll£j * fik‘
j,kj,m	j.k.l
where Rjk/- denotes {Rihetfihemy. Now the Bianchi identity states that
KjHm = Rjttm + Rikjm = Rjiknr That is, RJklfh is symmetric in к and I. But
еке/ = — ё/ек, and so equation (11.11) becomes
*C = У	’ &k*‘
j,k,l
This relates to the Ricci form of X, which for tangent vectors V and W
is given by
Ric(K,^) = -£ {(Re.vej9 W> + (Rje.yJej9 wy}
j
Thus, we are able to write $RC as
«с = -I Z • ek-.
L j*
Since Ric is hermitian symmetric we can choose our basis so that
Ric(£j,ek) = |2/>k where z7 = Ric^e,) = Ric(Jej,Jej) for j = 1,... ,n are
the eigenvalues. As before, we write —efij = and carry out a similar
analysis for $RC. Thus we obtain the following:
Theorem 11.6. Let X be a Kahler manifold equipped with a spin structure.
Then on the bundle of spinors $c
4-	= V*V 4-~У = V*V -ьУ
4 j	4
where Л19... 9Л„ are the eigenvalues of the Ricci tensor.
Taking the average of these two formulas gives the Lichnerowicz
Theorem (II.8.8). To see this we observe that
V*v + v*v=4v*v and £	+ <5j) = £ = 1-k
1	j	} L
(И.12)
where к = traceR(Ric) is the scalar curvature of X. Theorem 11.6 easily
gives the following:
§11. SPINOR COHOMOLOGY: MANIFOLDS WITH = 0	363
Corollary 11.7. Let A be a hermitian line bundle over a Kahler manifold
X with curvature form R\ and assume that X is spin. If Ric > 2RA, then
A) = 0	(11.13)
for all r > 0. Similarly, if Ric > — 2Ra, then (11.13) holds for all r < n.
Proof. Straightforward computation shows that the terms *RC and Wc in
Theorem 11.3 for the Dirac bundle ® A are exactly given by Clifford
multiplication by the elements
Wc = £ ( -I Ricj,k + R%
j.k \	2	/
/ 1	\	(1L,4)
= £ ( _— Rjk ]^k£j
j.k \ 2	/
Choosing a basis which diagonalizes the hermitian form |Ric — R with
eigenvalues	we find that $RC = £ pjCOj. This proves the first state-
ment. The second is proved similarly. 
Taking the average of the two formulas of Theorem 11.3 in this case
and applying (11.12) gives the formula (cf. Theorem D.12)
4(00 + 00) = V*V +	+ QA	(11.15)
where QA is the Clifford element corresponding to the curvature 2-form
of A.
Theorem 11.8. Let X be a compact Kahler manifold equipped with a spin
structure, and let Abe a hermitian line bundle over X. If the scalar curvature
к satisfies the inequality > pi| + ’ ‘ + PJ at each point, where A19... ,A„
are the eigenvalues of the curvature form of A, then the cohomology groups
Jfr(X, $c ® A) = 0
for all r. In particular, by the Atiyah-Singer Index Theorem,
{ch A • A(X)}[X] = 0.
It is interesting to specialize our discussion to the case where A = <5r/s
where d = AJTX is the anticanonical line bundle of X (the line bundle
generated by the global section ex • • • in C£(X)). An easy adaptation
of the arguments given for D.2 shows the following (see Michelsohn [1]):
Lemma 11.9. Suppose cr(X) = ka where k is odd and a g H2(X;Z) is indivi-
sible (i.e., not a non-trivial integral multiple of any other class in Н2(Х;Ж)).
Then for any odd integer m, there is a bundle of the form $c ® bm/2k globally
364
IV. APPLICATIONS
defined on X where locally $c is the bundle of spinors with its canonical
connection.
For any rational number t, the curvature of d* with its natural metric
is just tR* where for any unit vector e, Rde Je = Ric(e,e) = Ric(Je,Je). If we
choose a unitary basis ei9Jei9... ,en9Jen so that Ric(ej,ek) = АД-к, then
R^Ck = tRic^.eJ = (t/2)Aj6jk, and applying (11.14) gives the following:
Lemma 11.10. On F(SC ® <5'/2)
- - 1 "
0)9) + 99 = V*V + - (1 + t) £ A/w,
4	J=1
= V*V + l(l-t) f
4	J=1
where An .. . ,A„ are the eigenvalues of the Ricci tensor on X.
Remark 11.11. The operators which appear here can be diagonalized as
follows. Suppose that о g $rz ® bt/29 i.e., that = a. From the fact that
7tr = £|Л=Г coy and that
fcoy if i g /	_	(0 if i g I
[0 if l Ф I	[cDj if l Ф /,
we see that on $rc ® bt/2 we have
Ea/°j = E ki<°i and	E
j	uT=r	j	iiT^r
where
Ay =	4- • • • 4- Air and = Ax 4- * * * 4- Аи — Ay (11.16)
when I is the multiindex {in ..., ir}. It follows from (9.9)—(9.13) that multi-
plication by the elements coy constitutes a family of orthogonal projections
onto mutually perpendicular subspaces of $z ® <5'/2. These formulas give
a delicate and precise analysis of what conditions are necessary for the
positivity of £ AjCOj and £ A/ur Note in particular that the numbers
{Ay: |/| = r} are precisely the eigenvalues of Ric acting on ArTX as a
derivation.
We see from Lemma 11.9 that $rc ® Si/2 always exists globally. It cor-
responds naturally to the bundle A° *T*X (see App. D), and there is an
isomorphism Jtfr(X9 $c ® <51/2) Hr(X9&)9 where О is the structure sheaf
of X.
Corollary 11.12. On T($rc ® <51/2),
4-	= V*V 4- У AyCOy = V*V
2 |/|=r
§11. SPINOR COHOMOLOGY: MANIFOLDS WITH c, = 0	365
In particular, if Ric > 0, then Jffr(X, $c ® <51/2) = 0 for all r > 0 and
{ch <51/2A(X)}[X] = Td(X) = 1.
Arguing as above, we can conclude the following:
Theorem 11.1	3. Let A be a hermitian line bundle over X and suppose that
(1 4- t)Ric + 2Ra > 0. If $c ® <51/2 ® A exists globally on X, then
$c ® <51/2 ® A) = 0	(11.17)
for all r > 0. Similarly, if (1 — t)Ric — 2RA > 0, then (11.17) holds for all
r < n.
We now average the formulas of Lemma 11.10 and obtain
Theorem 11.1	4. Let X be a compact Kahler manifold with c^X) = ka,
+ where aeH2(X;Z) is indivisible. Let $c ® Sp/2k be the twisted
bundle of spinors where <51/2k is a local holomorphic 2kth root of the anti-
canonical bundle and where p 4- к = 0 (mod 2). Then on T($c ® bp/2k)
4(@@ + @S>) = V*V + L У [(к + р)Л, + (fc - pM,.]€o,.
|/pr
In particular, if Ric > 0, then
MT{X, ® 5Pt2k) = 0 for all r
whenever |p( < к and p + к is even.
We apply the Atiyah-Singer Theorem to the complex ® bp/2k to get
Theorem 11.1	5. Let X be a compact complex manifold such that cY(X) — ka,
where a is indivisible. If X admits a Ricci-positive Kahler metric,
then the Hilbert polynomial
Pa(t) = {>A(X)}[X]
vanishes for all integers t such that |t| < к and t + к is even.
From Yau [2], [3] we know that X carries a Ricci-positive Kahler
metric if and only if cx(<5) = cx(X) is represented by a closed positive
(l,l)-form.
To illustrate Theorem 11.15 we consider complex projective n-space
P"(C) and let co e J/2(P"(C);Z) = Z denote the standard generator. Then
cr = (n 4- l)co and a direct calculation shows that
This is predicted by Theorem 11.15 since P"(C) carries a Kahler metric
of positive curvature.
366
IV. APPLICATIONS
From the work of Yau [2], [3] we know that the hypersurface Vn(d) cz
i(Q carries a Kahler metric of positive Ricci curvature for d n 4- 1.
Furthermore, for n 2, ^(^"(d)) = (n 4- 2 — d)a> where co is a generator
of H2(K"(d);Z). It follows that the polynomial PJt) on Vn(d) has zeros at
n - d - 2j for j = 0,... ,n - d.
Theorem 11.15 has the following consequence:
Corollary 11.16. Let X be a compact complex manifold of dimension n. If
X admits a Kahler metric of positive Ricci curvature, then к )( cx(X) for
к > n 4- 1.
Proof. The Hilbert polynomial of 11.15 is of degree n and can have
therefore no more than n zeros. 
One case of Theorem 11.14 is of particular interest since it involves only
the scalar curvature к — 2(2 x 4- • • * 4- 2„).
Corollary 11.17. Let X, <5 and к be as in Theorem 11.14. If the scalar
curvature к of X satisfies к > 0, then
W\X, $c ® bp!2k) =	0	for all	p >	- k
JT°(X, $c ® <5₽/2k) =	0	for all	p <	k
where p + k is even. Similarly, if к < 0, then
Жn(X, $c ® bp/2k) =	0	for all	p <	— k
Ж°(Х, $c ® <5p/2k) =	0	for all	p>k
where p + k is even. In particular if к > 0, then all the plurigenera of X
are zero (cf. Yau [1]).
This last statement follows by setting p = k — 2kg for gel+. The
£th plurigenus of X is exactly the dimension of $c ® 8*~9) =
H\X,0(d~9)).
Note that the formula in Theorem 11.14 is particularly simple when
Ric = 0. From Yau’s proof of the profound conjectures of Calabi we know
exactly when such metrics exist.
Theorem 11.18 (Yau [2]). Let X be a compact complex manifold which
admits at least one Kahler metric. Then X carries a Ricci-flat Kahler metric
if and only if cx(X) = 0.
We can now apply the Splitting Theorem of Cheeger and Gromoll [2]
to conclude that any compact Ricci flat Kahler manifold X has a finite
covering X which decomposes into a Kahler product
X = T x Xo
§11. SPINOR COHOMOLOGY. MANIFOLDS WITH ct = 0	367
where T is a flat complex torus and where Xo is a compact Ricci-flat
manifold which is simply-connected.
When Ric = 0, Theorem 11.14 gives us the following simple identity
between operators
&& 4-	= V*V	(11.18)
on $c. In this case there is a canonical connection-preserving isometry
= A0,*(X) = A£TX of graded bundles, which carries 3) to d and
therefore identifies	Hr(X,^) (cf. App. D). Equation (11.18)
therefore implies the following:
Proposition 11.19. On a compact Ricci-flat Kahler manifold, any harmonic
spinor field is parallel. Equivalently, any harmonic (0,reform is parallel.
Suppose now that X is a compact simply-connected Ricci-flat Kahler
manifold of dimension n. Then the line bundles A0 0 and $*c A0 "
are trivial (they have parallel cross-sections). The index of the elliptic
complex (11.9) in this case is just
A(X) = E ( - Ifdim ЖГ(Х,$С) = £ (- l)rdim Hr(X,0) = Td(X)
and dim 0 = dim 1 = 1. Moreover, by the generalized Hodge Decom-
position Theorem (cf. III.5.5) we have Jffr(X,$c) ker(®® + ^®)|Шг} =
kertV*?^^). Consequently, if Td(X) =/=!+( — 1)", then there must exist
non-trivial parallel sections of $rc (i.e., parallel (0,reforms) for some r 0,n.
This implies that the local holonomy group G of X is properly contained
in SU„. If G is a product of two non-trivial groups, then X is a product
manifold. Otherwise G belongs to the list of Berger discussed in §10, and
we conclude that either X is locally symmetric or G = Sp„/2, for n even and
>2. It is elementary that locally symmetric, Ricci-flat manifolds are flat
(and therefore not simply-connected in the compact case). Combining the
three paragraphs above proves the following general structure theorem.
Theorem 11.20 (Michelsohn [1]). Let X be a compact simply-connected
Kahler manifold with c^X) = 0. Then there is a finite covering manifold X
of X which is biholomorphically equivalent to a product of compact Kahler
manifolds with vanishing first Chern class
X = T x Xr x ••• x Xkx Y
where T is a complex torus and each of X b ... ,Xk,Y is simply-connected,
where
0 if dim Xj is odd
2 if dim Xj is even
and where Y admits a metric with Spm-holonomy.
Td(X;) =
368
IV. APPLICATIONS
This decomposition theorem is a wonderfully effective device for de-
tecting manifolds with Spm-holonomy.
Corollary 11.21. Let X be a compact simply-connected Kahler manifold of
dimension 2m with cx(X) = 0. If Td(X) 0 or 2k for some k, then X carries
a metric with Spm-holonomy. Alternatively, if X is not a product and
Td(X) 2, then the same conclusion holds.
Theorem 11.20 with its corollary first appeared in Michelsohn [1] in pre-
publication form. Before printing, it was modified due to the publication
of an announcement of the non-existence of compact manifolds with
Spm-holonomy for m 2. This announcement was subsequently found to
be in error. The first counterexample was produced by A. Fujiki using
the results above. To construct this example we first take the symmetric
square of a Kummer surface. This is the compact analytic space XQ =
K2(4) x K2(4)/Z2 where Z2 is generated by the interchange of factors in
the product. Resolving the singularities along the diagonal gives a compact
complex 4-manifold X. Alternatively, we could construct X by first blow-
ing up the diagonal (i.e., by replacing the diagonal by its projectivized
normal bundle) and then dividing by the natural extension of the “flip”
to this space. The manifold X is a certain component of the Hilbert scheme
ofO-cycles on K2(4). It is straightforward to see that X is simply-connected
and algebraic (hence Kahler) and that cx(X) = 0. However, further com-
putation shows that Td(X) = 3. Consequently the Calabi-Yau metric on
X, given by Theorem 11.18, has Sp2-holonomy. (See Beauville [1] for
further discussion and examples.)
§12. The Positive Mass Conjecture in General Relativity
The classical theory of gravity, as enunciated by Einstein, is a subject
formulated in the language of differential geometry. In many ways it
stands apart from other theories of modern physics. One of its curious
features is that there seems to be no satisfactory way of defining an energy
density for a gravitational field. Nevertheless, there is a concept of the
total energy of a gravitating system, which is defined in terms of the
asymptotic behavior of the field at large distances. It is in many ways
essential to the theory that this energy be ^0 (and = 0 only on flat
Minkowski space).
The proof of this “positive mass conjecture” has a long history during
which many special cases were established. The key case of space-times
admitting a maximal space-like hypersurface was first proved by Schoen
and Yau [3] using minimal surface techniques. Using Jang’s equation
they subsequently adapted their techniques to prove the general result [5].
§12. POSITIVE MASS CONJECTURE
369
Of interest here is the fact that an alternative proof of this theorem can
be given using the Dirac operator on spinors. This proof was found by
E. Witten [1]. Very roughly the idea is as follows: Consider a space-time
X (i.e., a 4-dimensional Lorentzian manifold), with a properly embedded
space-like hypersurface M c= X. We assume that X satisfies Einstein’s
equations
Ric — %дк = T
where the energy-momentum tensor T is positive on time-like vectors. It
is assumed that as one goes to infinity in M, the geometry of M cz X is,
in a very explicit way, asymptotic to the standard geometry R3 cz R3,1 in
Minkowski space. In this context Witten considers the following problem.
Let S denote the spinor bundle of X with its canonical connection V, and
consider the restriction of (S,V) to the hypersurface M. On this bundle
one has a Dirac operator defined by
з
$ = Z
j=i
where ebe2,e3 is any local orthonormal frame field on M. One now applies
Bochner’s method and writes
= V*V + 3	(12.1)
where, since V is the metric connection of X (and not of M), the term 3
involves the second fundamental form of the hypersurface M in X. Com-
putation shows that the positivity of the tensor T on the normal vectors
to M implies the positivity of 3- This in turn shows that on suitable sub-
spaces of T(S) the operator $ is invertible. Since S is asymptotically flat, one
can consider solutions of the equation = 0 which are asymptotically
constant. Witten shows that there are unique solutions for each constant
value at infinity. Let a be such a solution and consider a region Q cz M
with smooth boundary dQ. From (12.1) we have
о = jn {<V*Va,ff> + <3(ff),a>}
= -ВД +	{||V< + <3(a),a>}
where Efi(a) is an integral over dQ. which satisfies
Efi(a)^0	(12.2)
since 3^0. Witten shows that the inequality (12.2) viewed asymptoti-
cally, establishes the positivity of the energy of the system.
For the complete story the reader is referred to the original paper of
Witten [1] and also to the rigorous account of Parker and Taubes [1].
APPENDIX A
Principal G-Bundles
Let X be a paracompact Hausdorff space and G a topological group. A
principal G-bundle over X is essentially a bundle of “affine G-spaces” over
X. To be precise, it is a fibre bundle n: P -► X together with a continuous,
right action of G on P which preserves the fibres and acts simply and
transitively on them. Thus, the fibres are exactly the orbits of G. More-
over, every point in X has a neighborhood U and a homeomorphism
hv:n~l(U) A U x G of the form hv(p) = (я(р),у(р)) and with the prop-
erty that h(pg) = (n(p), y(p)g) for all g e G. Thus, the bundle is locally of
the form
U x G
U
where G acts by multiplication on the right.
Two principal G-bundles n: P -► X and n':P'^>X are said to be
equivalent if there is a homeomorphism H: P -> Pf so that
commutes and so that H(pg) = H(p)g for all g e G. The equivalence classes
of principal G-bundles over X will be denoted by PrinG(X).
Example. Let n: P -> X be a 2-sheeted covering space of X, and let
G = Z2. The group Z2 acts on P by interchanging the sheets. This is
clearly a principal Z2-bundle. In fact it is not difficult to see that
PrinZ2(X) = Cov2(X) for any manifold X.
More generally, any normal covering of a manifold is a principal
Г-bundle where Г is the group of deck transformations of the covering
(with the discrete topology).
Example. Let E be a real и-dimensional vector bundle over X, and let
Pgl(E) be the bundle of bases in E, i.e., the bundle whose fibre at x e X
is the set of all bases for the vector space Ex. This is a principal GL„-
bundle where GL„ is the group of invertible n x n real matrices. The action
370
PRINCIPAL G-BUNDLES
371
of GL„ on PGL(E) is defined as follows. Fix a matrix g = ((au)) in GL„.
Then given a basis p = (v15... 9vn) of Ex at a point x, we set pg =
(t?i,... ,v'n) where
= E vkakj> j=
к
This action is clearly continuous and is simple and transitive on the fibres.
If E is oriented, we may consider the bundle PGL+ (E) of oriented bases
in E. The construction above makes this a principal GL„+-bundle where
GL„+ = {g g GL„: det(^) > 0}.
If E is riemannian, we can consider the bundle PO(E) of orthonormal
bases. This is a principal O„-bundle where O„ is the orthogonal group. If
X is oriented we get the bundle PSO(E) of oriented orthonormal bases,
which is a principal SO„-bundle, where SO„ = {g g O„: det(g) = 1}.
There are natural complex and quaternionic analogues of these
constructions.
Recall that a general fibre bundle В A X with fibre F and
structure group G Homeo(F) is given by the following data. There is
an open cover Ql = {Ua}aeA of X, and over each Ua there is a local
trivialization n~ l(Ua) -4 Ua x F so that pr о ha = n (where pr is pro-
jection onto Ua). The change of trivialization over U„ n Ue is of the form
(Ua nUf)x F - (L/a n Uf) x F where hx °	= (x,g„p(x)f)
and gafi :Uar\Up -► G are continuous functions called the transition func-
tions of the bundle. They satisfy the cocycle condition
= 1	(A.i)
in Ua n Ufi n Uy. The bundle can be reassembled from this data by pasting
together the local products {Ua x F}ae 4 with these homeomorphisms.
Any fibre bundle В with structure group G as above has an associated
principal G-bundle PG(B). It is obtained by simply replacing F by G in the
local products and then pasting together the {Ua x G}aeA by the same
transition functions, where gafi(x) acts on G by multiplication on the left.
Note that we are free to multiply on the right by elements of G. Since
right and left multiplication commute, this multiplication by elements of
G on the right makes sense when we assemble the bundle PG(B). Note,
however, that while each fibre looks like G, there is no preferred element,
i.e., no “identity.”
The original fibre bundle В can be recaptured from PG(B) as an asso-
ciated bundle, that is, В s PG(F) x ф F where (p: G -► Homeo(F) (see II.2).
We now observe that every principal G-bundle over X can be presented
by transition functions gafi: U ar\U p -► G multiplying G on the left as above.
Of course a principal G-bundle is a fibre bundle and therefore is always
given by a family of transition functions Gafi : UanUp -> Homeo(G) over
372
APPENDIX A
some open cover {Ua} of X. Since the bundle is principal, these functions
must commute pointwise with right multiplication by G. Let gap(x) =
Ga^(x)(l). Then Gap(x)(g) = Gap(x)(l)g = длр(х)д and we are reduced to the
simpler case, as claimed.
Thus, we have seen that every principal G-bundle on X is given by a
pair (^Z,{gap}) where ^Z = {Ua}aeA is an open cover of X and where
gap '-Uar\Up—>G are continuous functions satisfying the cocycle condi-
tion (A.l). Such a pair can be thought of as a Cech 1-cocycle with coeffi-
cients in G (or, more precisely, in the sheaf of germs of continuous maps
to G). One can easily check that two such bundles, constructed from
cocycles {gap} and {g'ap} on ^Z are equivalent if and only if there exist
continuous mapsg2'U3^G for each a such that
=	(A.2)
in Ua n Up for all а, /?. (This can be considered a “Cech-coboundary”
condition.)
Therefore, we define two 1-cocycles {gafi} and {g'afi} on ^Z to be equiva-
lent iff there is a “Cech 0-cochain” {ga}aeA such that (A.2) holds. The set
of equivalence classes will be denoted by Hl(W,G). This set naturally
represents the equivalence classes of principal G-bundles on X which can
be trivialized over the open sets of ^Z.
Suppose now that (iHj) is a refinement of <?Z, i.e., 1Г is an open cover
of X and j:	-> ^Z is a map such that V j(V) for all V e Then by
restriction we get a map : H^^Z; G) -►	; G) which can be shown
to be independent of the refinement function j. These maps satisfy the
relation	° for successive refinements	->^Z. Thus
we can take the direct limit
H1(X;G) = limH1(^Z;G).
This limit naturally represents the equivalence classes of principal
G-bundles on X, That is,
PrinG(X) ^H\X;G).
If G is abelian, Hl(X; G) is simply the first Cech cohomology group of X
with coefficients in G.
Example. Let G = Z2. Then Z2-bundles over X are precisely the two-
fold coverings of X, and the correspondence Cov2(X) A H^X; Z2) is
just the isomorphism given in Lemma 1.1.
Remark A.L If X is a C°°-manifold and G is a Lie group, we can
require all the maps gafi9ga9 etc. in the discussion above to be differen-
tiable. The resulting set, denoted H^XjG)^, represents classes of smooth
principal G-bundles over X. The natural map H*(X; G)^ -► Hl(X; G) can
PRINCIPAL G-BUNDLES
373
be shown to be a bijection so we shall in general drop the reference to
smoothness.
An analogous remark can be made concerning complex manifolds X
and complex Lie groups G where the maps gafi and ga are required to be
holomorphic. In this case, however, the corresponding map to
is far from a bijection in general. (An exception is the case where X is
Stein.)
Note that if G is not abelian, Hl(X; G) is not a group. It is merely a set
with a distinguished element given by the trivial G-bundle. Nonetheless, if
1----> К —U G G-------------► 1	(A.3)
is an exact sequence of topological groups, the standard arguments in
Cech cohomology theory (cf. Hirzebruch [1, Ch.I, §2] show that for para-
compact spaces X, there is an exact sequence:
{*}----> H°(X; K) H°(X; G) H°(X; G')
----> Hl(X,K)-±+ Hl(X;G)—^ Hl(X;G')
of pointed sets. H°(X; G) is the set of 0-cocycles and is easily identified
with the space of continuous maps from X to G. The maps i* and Д are
given in the obvious way by coefficient homomorphisms. Note that for a
principal G-bundle P, the corresponding principal G'-bundle j*(P) can be
obtained by dividing P by the action of К on the right.
Remark A.2. If the group К in (A.3) is abelian, then H2(X;K) is defined
and the exact sequence (A.4) can be extended to
...---> H\X; K)----> Hx(X; G)---> Hx(X; G)----> H2(X; К) (A.5)
Example A.3. For the sequence
0---► SO„----► O„-----► Z2---► 0,
the induced map
^:H\X;On)-------------------------->H\X;Z2)
is just the first Stiefel-Whitney class. Clearly wx(P) = 0 if and only if P
comes from an SO„-bundle, i.e., if and only if P is orientable.
Example A.4. For the sequence
0---> Z2-----> Spin. SO.-------► 0,
the induced cobundary map
w2: Hl(X; SO.)---> H\X; Z2)
is the second Stiefel-Whitney class. Clearly w2(P) = 0 if and only if P
comes from a Spin.-bundle, i.e., if and only if P carries a spin structure.
374
APPENDIX A
There is a caution, however. As shown in §1, distinct spin structures may
give abstractly equivalent principal Spin„-bundles.
Note that the definition of a Cech coboundary leads to a nice com-
binatorial definition of w2(P). Let (^Z, {gap}) be a cocycle representing P,
where each set Ua n Up is simply-connected. Lift each map gafi to a map
gafi:Ua n Ufi -► Spin„, and define
^afiy 9afl9py9ya	(A.6)
in Ua n Ufi n Uy. Since £0(wapy) = 1, we see that
waPy:Ua n Ufi nUy^ Z2.
This Z2-cocycle represents w2(P).
Example A.5. Consider the sequence
0-----► Z----► R-----► S1----► 0.
Since all groups here are abelian, the exact sequence in cohomology con-
tinues indefinitely. It is an elementary exercise to show that H\X;R) = 0
for all i > 0 and for any manifold X. (Here R carries the standard, not
the discrete, topology.) We thereby get an isomorphism
Ci: Hl(X- S1) -2-» H\X-, Z)	(A.7)
called the first Chern class. This shows that the equivalence classes of
principal S1 -bundles are in natural 1-to-l correspondence with elements of
H2(X;Z)
Note. For a full and clear discussion of the basic subject of fibre bundles
the reader is referred to the classic book of Steenrod [1].
We conclude this section with some remarks about transferring bundles
from one space to another. Let P Д X be a principal G-bundle
over a space X and let f: Y -► X be a continuous map. Consider the set
of points
/*P	{(y,p) g Y x P : f(y) = я(р)},	(A.8)
in the product Y x P, and note that projection of У x P onto its factors
induces maps on f*P which make the following diagram commute:
/♦p p
п	n	(A.9)
Y ——> X
We now observe that n:f*P—> У is a principal G-bundle over У. It follows
directly from definitions that G acts on /*P and is simple and transitive
PRINCIPAL G-BUNDLES
375
on the fibres of я. The map f commutes with this action. Any set of local
trivializations of P over a family of open sets = {Ua}aeA in X naturally
determine local trivializations of /*P over the family of open sets
/*^Z = {f~iUa}aeA in Y. The transition functions {gafi} for this cover lift
back to transition functions
f*gafi = gafi°f	(A.io)
for /*^Z.
Definition A.6. The principal G-bundle /*P -► Y is called the bundle
induced from P -► X by the mapping /.
It is clear that if two bundles P and P' are equivalent on X, then /*P
and f*P' are equivalent on Y. (Lift the equivalence.) Thus we have a map
/*: PrinG(X)----> PrinG(T).	(A.11)
From equation (A.IO) we see that the cocycle for /*P is just pull-back
via f of the cocycle for P. Thus, (A. 11) corresponds to the usual induced
mapping on Cech cohomology groups:
f*:H\X;G)-------► Hl(Y,G).
This “pull-back” mapping on principal bundles has two important
properties:
Proposition A.7. For continuous maps X Y Z, the induced maps
PrinG(Z) -C PrinF(T) ^PrinG(X) satisfy
W)*=/*°0*	(A. 12)
Proof. That f*(g*P) = (g ° f)*P is obvious from the definition. 
Proposition A.8. Let Ybea compact Hausdorff space. If two maps f0:Y -► X
and fi'.Y-* X are homotopic, then
Л=Л-
We leave the proof as an exercise for the reader.
APPENDIX В
Classifying Spaces and Characteristic Classes
The point of this appendix is to briefly summarize the basic facts from
the theory of classifying spaces and characteristic classes for principal G-
bundles. Details can be found in Milnor-Stasheff [1] and Husemoller [1].
For simplicity we shall assume throughout this appendix that G is a
Lie group. Furthermore all spaces will be assumed to have base points,
and all maps will be base-point-preserving. All spaces will also be assumed
to have the homotopy type of a countable CW-complex.
Definition B.I. A classifying space for the group G is a connected topo-
logical space BG, together with a principal G-bundle EG -► BG, such that
the following is true. For any compact Hausdorff space X, there is a one-
to-one correspondence between the equivalence classes of principal G-
bundles on X and the homotopy classes of maps from X to BG, given
by associating to each map f: X -> BG, the induced bundle f*EG over X
(see the end of App. A).
Thus the induced bundle construction gives a natural bijection
PrinG(X)	[X,BG].	(B.I)
Note that as a special case we have
Prin^S") 7i„(BG).	(B.2)
Given two different classifying spaces for G, say B0G and BrG, there
must exist mappings fQ:BQG —^BlG and f^IfG —> BQG such that
f?Ei+lG £ Efi for i e Z2. This implies that (fi+i ° f^Efi Efi, and
therefore f +1 ° /• is homotopic to the identity on Bfi for i g Z2. Thus B0G
and BXG are homotopy equivalent, i.e., BG is well defined up to homotopy
type.
The bundle EG -> BG is called the universal principal G-bundle. This
bundle has the following homotopy characterization (see Steenrod [1] for
example).
Theorem B.2. Let E -► В be a principal G-bundle with the property that
the total space of E is contractible. Then (В, E) is a classifying space for G.
The theory has definite interest due to the following:
Theorem B.3. For any Lie group there exists a classifying space.
376
CHARACTERISTIC CLASSES
377
The general construction of classifying spaces for principal bundles is due
to Milnor [1,2] and proceeds as follows. Given a group G, we consider
the n-fold join G* ••• *G. (The join of two spaces X and У is obtained
from X x Y x [0,1] by collapsing {x} x Y x {0} to a point for each x
and also collapsing X x {y} x {1} to a point for each y.)
[0,1]
XxYx[Q,l]
X* Y
X x Ух [0,1]------► X* Y
The space G* ••• ♦ G is (n — l)-connected. It also has a free G-action ob-
tained by multiplying simultaneously on the right in each of the factors.
We pass to the limit as n —► oo, and define EG = G ♦ G ♦ G ♦ • • •. This
is infinitely-connected and has a free right G-action. We set BG = EG/G
with n :EG -► BG the quotient map. From Theorem B.2, this bundle is
classifying.
Of course this construction of Milnor can (and has) been carried out
in enormous generality.
It should be noted that since EG is conuactible, the long exact sequence
of homotopy groups for a fibration implies that
^-JQ^BG)	(B.3)
for all n 1. Thus, the homotopy groups of BG present nothing new.
However, the homology of the space is quite interesting.
We begin the discussion with the following. Consider the singular
cohomology H*(BG; A) with coefficients in a ring A.
Definition B.4. Each non-zero class in H*(BG; A) is a universal charac-
teristic class for principal G-bundles.
Fix a class c e Hk(BG', A). Then for any principal G-bundle P -► X there
is a map fP: X —> BG so that P = f*EG. We define the c-characteristic
378
APPENDIX В
class of P to be the class
с(Р) = /*(с)еЯк(Х;Л).	(B.4)
Since fP is well defined up to homotopy, the class c(P) is uniquely defined.
Observe that any such characteristic class transforms naturally in the
following sense. Given a principal G-bundle P -► X and a continuous map
F: Y -> X, the c-characteristic classes satisfy:
c(F*P) = F*c(P)	(B.5)
To see this we simply note that fF*P = fP° F, and so c(F*P) = /J»P(c) =
F*/?(c) = F*c(P).
Given a continuous homomorphism (p: H -+ G of Lie groups, there is
a corresponding continuous map
B(p:BH------► BG	(B.6)
which classifies the principal G-bundle over BH associated to the repre-
sentation (p. That is,
(Bcp)*(EG) = EH
where EH x ф G denotes the quotient of EH x G by the action of H given
by setting Фь(р, g) = (ph~ \ hg) for p e EH, geG and he H. This associa-
tion also transforms naturally. That is, for any composite homomorphism
ЯЛ К Л G, we have
В(|Д ° (p) = В(ф) ° B((p).
If the homomorphism <p: Я -► G is a homotopy equivalence, then so
is the mapping B(p:BH -► BG. This is easily seen by applying the 5-
lemma to the induced map on the long exact homotopy sequences of the
fibrations
Я-----► EH-----► BH and G-------------► EG----► BG.
(Recall that BH and BG are assumed to be homotopy equivalent to
countable CW-complexes.) This gives the following result:
Proposition B.5. Let G be a connected Lie group and K^G a maximal
compact subgroup. Then BG BK, that is, the classifying spaces of G and
К are homotopy equivalent.
Proof It follows from the Iwasawa decomposition of G (cf. Helgason [1])
that the inclusion К G is a homotopy equivalence. 
It follows, for example, that BO„ BGL„(R), BSO„	BGLn+ (R),
BUn BGLn(C), BSUn BSL„(C), BSp„ BSp„(C), etc.
CHARACTERISTIC CLASSES
379
Note that, in general, each mapping of the type B<p: BH -► BG induces
a transformation (B<p)*: H*(BG\ A) -> H*(BH; A) of universal characteris-
tic classes. Any non-zero element which goes to zero under this transforma-
tion is a universal obstruction to the reduction of the structure group
from G to H. That is, if c is such a class and if P -+ X is a principal
G-bundle with c(P) # 0, then P cannot be written as an associated bundle
P = P' x ф G (as above) for any principal H-bundle P' -► X.
The cases of basic interest here are those of the compact classical
groups. For these cases there are useful alternative constructions of the
classifying spaces.
For К = R,C or H, let GK(n,N) denote the Grassmann manifold of all
n-dimensional К-linear subspaces of KN. There are natural inclusions
GK(n, N) c= GK(n, N + N') induced by the “first-coordinate” inclusions
KN c KN+Nz. Taking the direct limit, we get
Gk(„5oo)=-^GVN),
the Grassmannian of К-planes in К °0.
There is a canonical К-vector bundle E„ -> GK(n9 oo) whose fibre at a
plane P g GK(n, oo) consists of all the vectors in that plane. We shall show
that
BOn	GR(n, oo)
BUn	Gc(n, oo)	(B.7)
BSp„	GM(n, oo).
The principal fibrations in each case are obtained by taking the appro-
priate bundle of bases (orthonormal, unitary, or symplectic) in the ca-
nonical bundle E„. Thereby, E„ becomes the universal (real, complex or
quaternionic) л-plane bundle over the classifying space (BOn, BUn, or BSpn
respectively).
We begin our proof of these claims with the real case. For this we note
that for any N, we have GR(n, N) ON/(ON_„ x O„). The bundle of ortho-
normal frames for the canonical n-plane bundle E„ -► GR(n,N) is just the
Stiefel manifold ON/ON_„. Hence, the bundle of orthonormal frames for
E„ -> GR(n, oo) is just the direct limit over N of the principal O„-bundles
^n/^N-п	► ON/(ON-n X
It is an elementary exercise, using the fibrations Ok —► Ok + x -► Sk, to see
that ON/ON_„ is (N — n — 2)-connected. Hence the infinite Stiefel mani-
fold limN ON/ON_n is contractible, and the limiting fibration must repre-
sent the universal bundle EO„ -> BO„ by Theorem B.2.
380
APPENDIX В
The arguments for BU„ and BSp„ are entirely analogous. Here one
uses the facts that Gc(n,N) = UN/(UN_n x Un) and Gw(n,N)^.
SpN/(SpN_„ x Sp„). We omit the details.
The reader has undoubtedly noticed that BO„ can be considered as the
classifying space for real n-dimensional vector bundles. That is, for a com-
pact Hausdorff space X, the equivalence classes of such bundles corre-
spond one-to-one with elements of [X, BO„] by associating to /: X -► BO„
the vector bundle /*En. This follows easily from the discussion above by
passing to the bundles of orthonormal bases. The corresponding remarks
hold for BU„ and BSPn.
It should be noted that the fact that GK(n, oo) is a classifying space for
n-dimensional vector bundles (over K) can be proved by a direct geometric
construction (see Milnor-Stasheff [1]).
We now examine the cohomology of the basic classifying spaces. These
spaces are infinite-dimensional and have non-zero classes in arbitrarily
high dimensions. Nevertheless, as rings under the cup-product multiplica-
tion, they are quite understandable. Basic references for all the following
results are Milnor-Stasheff [1], Steenrod-Epstein [1], and Borel [2]. We
begin with Z2-cohomology.
Theorem B.7. The cohomology ring H*(BO„;Z2) is a Z2-polynomial ring
Z2[wi,w2. • • •,%]
on canonical generators wk e Hk(BO„ ;Z2)for k = \,...,n.
Note. The classes ... ,w„ are characterized inductively by the following
fact. The kernel of the homomorphism H*(BOn;Z2) ——» H*(BO„_1;Z2)
induced by the standard inclusion О,^ —O„, is the principal ideal
<wn>. Analogous statements characterize the canonical generators in the
subsequent theorems.
The class wk is called the universal fcth Stiefel-Whitney class. Thus to any
n-dimensional real vector bundle E -+ X classified by a map fE: X -► BOn,
we have the associated kth Stiefel-Whitney class of E, wk(E) = An
important concept is that of the total Stiefel-Whitney class w = 1 + Wj +
... + w„. It satisfies the Whitney product formula for the sum of two
bundles
w(E Ф E') = w(E) и w(E')	(B.8)
which translates in longhand to the equations
w»(E © E') = £ Wi(E) и wk_,(E')
1 = 0
CHARACTERISTIC CLASSES
381
for к = 1,2,.... This can be considered as a statement about the induced
map on cohomology induced by B(O„ x -► ВОи+и> under the usual
“diagonal block” inclusion Ои x On. с Ои+и>.
Given a smooth manifold X, we define the kth Stiefel-Whitney class
of X to be wk(TX). It is an important fact that the Stiefel-Whitney classes
of a compact smooth manifold are invariants of the homotopy type of the
manifold. That is, given a homotopy equivalence f: Y -+ X between com-
pact smooth manifolds, we have that f*wk(X) = wk(Y) (see Milnor-
Stasheff [1], p. 131.) This will not be true of many other characteristic
classes of manifolds.
Consider now the space BSOn. This can be realized as above as
the Grassmannian of oriented n-planes in R00. There are natural maps
BSO„ -► BO„ -► BZ2 = K(Z2,1). This sequence is a fibration, and we have
the following:
Theorem B.8. The cohomology ring H*(BSO„;Z2) is a Z2-po/ynomiaZ ring
Z2[w2,... ,w,]
where wk denotes the lift of the universal kth Stiefel-Whitney class by the
map BSO„ -► BO„.
Finally we consider the space BSpin„ which sits in a fibration BSpin„ —►
BSO„ -> K(Z2,2).
Remark B.9. The cohomology ring H*(BSpinn;Z2) is quite complex
and since its precise structure is not used here, we refer the interested
reader to the basic paper of D. Quillen [1]. (The “stable” groups
H*(BSpin;Z2) were first computed by E. Thomas [1].)
The Z2-cohomology of the spaces BU„ and BSp„ is merely the mod 2
reduction of the integral cohomology, and we have the following:
Theorem B.10	. The cohomology ring H*(BU„;Z) is a ^.-polynomial ring
Ztcj.cz,... ,c„]
on canonical generators ck e H2k(BU„; Z) for к = 1,... ,n.
The class ck is called the universal kth Chern class. Thus to any n-
dimensional complex vector bundle E -► X classified by a map fE :X -►
BUn, we have the associated kth Chern class ck(E) = f*(ck)- There is the
concept of the total Chern class c = 1 + ct + c2 + ... + c„, and again
there is a product formula
c(E ф E') = c(E) u c(E').	(B.9)
Given any complex manifold X, the tangent bundle is a complex vector
bundle. We call ck(X) = ck(TX) the kth Chern class of X.
382
APPENDIX В
There is now a ring isomorphism H*(BUn; Z2) = Z2[ci,... ,c„] where ck
is of degree 2k for each k. The class ck is the mod 2 reduction of ck. The
natural inclusion Un cz O2„ induces a map BU„ -> ВО2и. It can be shown
that under this map w2k ck and, of course, w2k + 1 -► 0. Hence, for any
complex vector bundle E,
w2k(E) = ck(E)(mod 2) and	w2k+x(E) = 0
for all k.
The picture is much the same for quaternion bundles.
Theorem B.11	. The cohomology ring H*(BSp„;Z) is a ^.-polynomial ring
. ,O
on canonical generators ak g H4k(BSpn;Z) for к = 1,... ,n.
Under the natural map BSp„ -► BU2„ we have that c2k ak and
c2k+ 1	0-
The integral cohomology of BO„ is a more complicated story due to the
presence of 2-torsion. However, away from the prime 2, things become
nice.
Theorem B.12	. Let A he any integral domain containing | (e.g., Z[|] or Q).
Then H*(BOn;A) is the polynomial algebra
A[Pi,  • . ,P[n/2]]
on canonical generators pk g H4k(BO„;A) for к = 1,... ,[и/2].
Choose A — Q. Then the class pk is called the universal fcth rational
Pontrjagin class. Given a real vector bundle E -► X classified by fE:X -►
BO„, we define the kth rational Pontrjagin class of E to be pk(E) = /J(pk).
There is a total rational Pontrjagin class p = 1 + pj + ... + р(и/2] and a
product formula
p(E ф E') = p(E) и p(E')	(B.10)
in H*(X;Q\ There are rational Pontrjagin classes of a smooth manifold
which, in the compact case, are homeomorphism invariants.
To understand the integral case we must examine the Bockstein homo-
morphism. The coefficient sequence 0—>Z—>Z—>Z2—>0 gives rise to a
long exact sequence
...----» Hk(X; Z) Hk(X; Z)------» Hk(X; Z2)	Hk+1(X-, Z)---»• • •
where fi is called the Bockstein. Note that the kernel of ft is the set of
classes in H*(X;Z2) which are mod 2 reductions of integral classes.
CHARACTERISTIC CLASSES
383
Theorem B.13	. The integral cohomology H*(BOn;Z) is an additive direct
sum
Z[₽i, • • • ,p[B/2)] © Imaged)
where pt 6 H4k(BO„;Z) becomes the element pk of Theorem B.12 under
tensor product with A, and where ft is the Bockstein homomorphism.
The class pk is called the universal fcth integral Pontrjagin class. These
classes have the following property. Consider the homomorphism O„ -►
Un induced by complexification. The Chem classes lift back over the in-
duced map BO„ -► BUn. The classes c2k+1 go to zero, and
pk = (-l)kc2k.	(B.ll)
It is conventional to define the Pontrjagin classes of a real vector bundle
E by this formula, i.e., by setting
pk(E) = (-l)kc2k(E® С).	(B.12)
Observe now that the subset /?(H*(BO„;Z2)) <= H*(BO„;Z) consists en-
tirely of 2-torsion elements. Important among these are the classes
(B.13)
which measure whether the (k — l)st Stiefel-Whitney class is the mod 2
reduction of an integral class. Of course all the Wks vanish when pulled
back to BUn or BSp„ under the maps BUn -► BO2n and BSp„ -► BO4„.
We now examine BSO„.
Theorem B.14	. Let A he an integral domain containing |. Then for each n,
there are ring homomorphisms
H*(BSO2B+1;A) = A[p1,...,pB]
H*(BSO2b;A) = A[pb ... ,pB,z]/<z2 - p„>
where pk e H4k(BSO+; A) and where % e H"(BSO2n; A).
The elements pk are the images of the Л-Pontrjagin classes (of Theorem
B.12) under the map BSO„ -► BO„. The class x is called the universal Euler
class. It can be defined as usual for any oriented vector bundle E -+ X by
setting x(E) = /e(z). (For odd-dimensional bundles it is defined to be zero.)
There is a product formula
X(E ® E') = Z(E) и z(E').	(B.14)
There is a natural definition of z as an integral cohomology class, (see
Milnor-Stasheff [1]). Under the map BU„ -> BSO2„, the class z pulls back
to c„. Under mod 2 reduction, z becomes wn in H"(BSO2„;Z2).
The analogue of Theorem B.13 holds for H*(BSOn;Z).
APPENDIX С
Orientation Classes and Thom Isomorphisms
in K-Theory
The point of this appendix is to examine the notion of an orientation class
for К-theory, KO-theory, or KB-theory on a vector bundle. Such classes
exist only on bundles with an appropriate structure, and when they exist
they determine a “Thom isomorphism” for the given theory. These can be
compared with the standard Thom isomorphism for cohomology via the
Chern character. For more elaboration of the matters discussed here the
reader is referred to the book of Karoubi [2].
Let я:: E -+ X denote a vector bundle over a locally compact space X.
This bundle may be real or complex, or even Real (when X is provided
with an involution). Let k"* denote any of the theories Kc”*, or
KR~pf. (This last theory is defined only for spaces with an involution. We
shall let the reader make the obvious adaptations of the exposition to fit
this case.)
Proposition C.l. Via the projection n:E -► X, k”*(E) is canonically a
k~*(X)-module.
Proof. Given any two locally compact spaces A and B, the outer tensor
product induces a graded, bi-additive map
k"*(4) x fc"*(B)----► к"*(Л x В)
defined as follows. Fix elements [F0,Fi;a] g k(A x Rm) = k~m(A) and
[Wo,g k(B x R") — k~n(B). Choose extensions of the isomorphisms
a : Vo -+ Vi and t: Wo -► W19 which are defined outside compact subsets,
to all of A x Rm and В x R" respectively. Fix a metric in each of the
bundles, and let a*: Vt -> Vo and t* : W\ -► Wo be the adjoint morphisms
to a and t. Let	denote the outer tensor product, i.e., Ц |x]	=
(Рг* К) ® (pr* Wj) where prx and pr2 are the projections of A x В x R"+w
onto 4 x Rm and В x R" respectively. Then the product of [F0,Vi;a] and
[И^И^т] is defined to be p] where
Vo = (K) S ^o) © S ^i), V. = (V, S Wy Ф (Fo S W^)
384
THOM ISOMORPHISMS IN K-THEORY
385
and where
_ fa S 1 -1 0 t*\
P - V H T ст* g) 1 /’
From the fact that
л	(стст* 0 1 + 1 El т*т	0	\
V	0	a*a 0 1 + 1 0 TT*7
we see that p is an isomorphism at any point where either a or т is an
isomorphism. Hence, p is an isomorphism outside a compact subset of
A x В x R"+m. All choices involved in this definition are unique up to
homotopy, so the product is well defined.
The desired module multiplication is now given by the composition
k’*(X) x k"*(E)---------------► k"*(X x E)-----► k"*(E)
where the second homomorphism is induced by the proper map (я x Id):
E -+ X x E. Verification that this multiplication is associative and dis-
tributive is left to the reader. 
From this point on we shall assume that X is compact.
Definition C.2. A class и e k(E) — kQ(E) is said to be a fc-theory orien-
tation for the bundle E if k“*(E) is a free k“*(X)-module with generator u.
Example C.3. Let Ё = X x Сш Д X be a trivialized hermitian vector
bundle and define
u = [ЛсепСш, AoddCw;e Kcpt(E)	(C.l)
where Cm s я*Е denotes the trivial m-plane bundle on E and where
ax,vM = VA(p — v*Lcp
for (x,u) g X x Cm and g ЛсепСш. By using the identification R2m Cm
and taking the canonical orientation, this element can be rewritten as
u =	, $q j p]	(C.2)
where	®	= л*$с(Е) is the irreducible complex graded C£2m~
module (extended trivially over E) and where
Дх.Лф) = у Ф
is given by Clifford multiplication. The fundamental assertion of the Bott
Periodicity Theorem is that и gives a К-theory orientation on E (see 9.20,
(9.8) and 9.28 of Chapter I).
Example C.4. Let E = X x R8m Д X be a trivialized riemannian vec-
tor bundle, and define
м = [Г,Г;д]вКОср1(Е)	(C.3)
386
APPENDIX С
where $ = $+ ф $ = я*$(Е) is the irreducible real graded C£8m-module
(extended trivially over E) and where
= v<p
is given by Clifford multiplication. The Bott Periodicity Theorem (see
9.22, (9.8) and 9.28 of Chapter I) states that и is a KO-theory orien-
tation for E.
Example C.5. Let E = X x Cw = X x (Rm ф iJT) Д X be a trivial-
ized Real vector bundle over a compact space X with trivial involution. Set
и = [CC, C£‘; R + fL] g KRcpt(E)	(C.4)
where С£ш = C£° ф is extended trivially over E and where
(R + iL)xu+iv((p) = (p-u + iv-q)
at any point (x, и + iv) e X x (Rm ф Шш). From the (l,l)-Periodicity
Theorem of 1.10 we see that и defines a KR-theory orientation on E.
We now return to the general case of a vector bundle over a compact
space X.
Definition C.6. A class и e k(E) is said to have the Bott periodicity
property if и determines a к-theory orientation in any local trivialization
of E over a closed subset Cc%, i.e., if k"*(E|c) is a free k"*(C)-
module generated by u, whenever E\c is trivial.
Theorem C.7 Let тг.Е -+ X be a vector bundle over a compact space X.
Then any class и e k(E) with the Bott periodicity property is a k-theory
orientation for E.
Proof. Let {С7}У=1 be a covering of X by closed subsets such that E\Cj
is trivial for each j. We proceed by induction on N. For N = 1 the state-
ment is obvious. Suppose we have proved the assertion for E restricted
to A = u • • • u CN_! (or to any closed subset of Л). Set В = CN. For
each theory k~* there exists a Mayer-Vietoris sequence with connecting
homomorphisms of degree 1 (see Karoubi [2]). One easily checks that
multiplication by the appropriate restrictions of и gives a morphism of
exact sequences:
• • • - k~\A к~*(А) ф k”’(B) - к~\А n В) - к"‘ + 1(Л kj B) • • •
- k'W ® k'W -	- ri+1(EUB)
By inductioin the vertical arrows are isomorphisms from k~*(Л), к ’*(B)
and к“*(Л n B) (since A n В is a closed subset of Л). By the 5-lemma we
THOM ISOMORPHISMS IN K-THEORY	387
conclude that the vertical arrows are isomorphisms also from к“*(Л и В).
The same argument clearly also applies to any closed subset of A u B.
This completes the proof. 
Theorem C.7 has the following immediate consequences:
Theorem C.8 (The Thom isomorphism in К-theory for Unbundles). Let
n:E -> X be a complex hermitian vector bundle over a compact space X.
Then the class
A_X(E) = [7t*Arn£,7r*AocddE;a] g Kcpt(E)
where ae(<p) = e л <p — e* l <p, is a К-theory orientation for E. In particular,
the map h: K(X) -► Kcpt(E) given by
it(a) = (л* a)  A_ i(E)
is an isomorphism.
Proof. By Example C.3, A.^E^ has the Bott periodicity property. 
Theorem C.9 (The Thom isomorphism in Л/7-theory for Spin8m-bundles).
Let ti .E X be a real 8m-dimensional bundle with a spin structure, over
a compact space X. Consider the class
S(E) = [тг*#+,7г*Г;и] g KOcpt(E)
where $ = $+ ф $~ is the irreducible graded real spinor bundle of E and
where pe(cp) = e-cp is given by Clifford multiplication. Then S(E) is a KO-
theory orientation for E. In particular, the map : KO(X) -► KOcpt(E) given
by
i,(a) = (я*а) • S(E)
is an isomorphism.
Proof. By Example C.4, S(E) has the Bott periodicity property. 
Theorem C.10 (The Thom isomorphism in A7?-theory for Real bundles).
Let Tt:E0 X be a real vector bundle over a compact space, and let
E = Ео ® С = Eo ф iE0 be the associated Real bundle (with trivial invo-
lution on X). Then the class
U(E) = [7t*C£°(E0), t^CHEo); R + iL] g KRcpl(E)
is a KR-theory orientation for E (where (R + iL)(cp) = <pe — ie'cp ate + ie' g
Ео ф iE0 above a point in X). In particular, the map i,: KR(X) -> KRcpl(E)
given by
h(a) = (тг*а) • U(E)
is an isomorphism.
388
APPENDIX С
Proof. By Example C.5, U(E) has the Bott periodicity property. 
Remark C.ll. For a general Real bundle n:E -+ X over a compact
space X with involution, the class A_X(E) given in C.8 defines an element
in KRcpX(E) which is always a KR-theory orientation for E (see Atiyah [2]).
Theorem C.12 (The Thom isomorphisms in К-theory for SO2m-, Spin2m-
and S pi n2 „-bundles). Let тг.Е -+ X be an oriented real vector bundle of
dimension 2m over a compact space X. Then the class
6(E) = [л*СГ(Е), л*СГ(Е);д] g Kcpt(E) ® Q
defined in (III.12.12) is a (K ® Q)-theory orientation for E. If E has a spin
structure, then the class
s(E) = [л*^(Е),7г*^(Е);д] g Kcpt(E)
defined by (II1.12.13) is a К-theory orientation for E. This remains true for
any Spinc-structure on E where $C(E) = $c(E) © $c(E) is the associated
irreducible spinor bundle (see App. D). The associated maps
ii: K(X) ® Q-----► Kcpt(E) ® Q given by i,(a) = (л*а) • 6(E)
and
h: K(X)-----► Kcpt(E) given by i,(a) = (n*a) • s(E)
(in the Spinc-cflse) are isomorphisms.
Proof. Consider first the spin (or Spinc) case. In any local trivialization
of E, say E\c Л C x R2m, the class s(E) becomes the element
W = [^c,^;^]eKcpt(C x R2m)
which is the pull-back to the product of the canonical generator of
Kcpt(R2m) given by the Atiyah-Bott-Shapiro Isomorphism. Hence, s(E) has
the Bott periodicity property for K-theory.
Suppose now that E is not necessarily spin. From the isomorphism:
2m = Homc($c, $c) = $c® we see that in any local trivialization
E|c Д C x R2m, the class 6(E) becomes
W = [$c ®	® $*’•> д]
— [&, $c; д] * [$*]
= 2m[$c,$c > д]
= 2ws(E).
Hence, after tensoring with Q, 6(E) has the Bott periodicity property, and
the proof is complete. 
THOM ISOMORPHISMS IN K-THEORY	389
The Thom isomorphisms given above can be compared to the standard
Thom isomorphism in cohomology via the Chern character. Formulas
for this are given in Chapter III, §12 (see (III. 12.14) and (III. 12.15)).
Remark C. 13. Each of the Thom isomorphisms given above can be
extended to bundles defined over a locally compact space X. The orienta-
tion classes given explicitly in the Theorems C.8-C. 12 do not define classes
in k(E) when X is not compact. Nevertheless, one can easily show that
they do pair with elements in k(X) to give elements in k(E). Basically, these
orientation classes have compact support in “vertical slices” and elements
of k(X) lift to classes with compact support in the “horizontal directions,”
and so the product has compact support on E.
The reader has probably noted that if X is a manifold and E = T*X
is its cotangent bundle, then the orientation classes given in the theorems
above are the principal symbols of certain fundamental differential opera-
tors. In particular, Theorem C.8 shows that on any 8w-dimensional spin
manifold, the principal symbol of the Atiyah-Singer operator is an orien-
tation class for the KO-theory of the cotangent bundle. Similarly, if X is
any even-dimensional spin (or Spinc) manifold, then by Theorem C.12 the
principal symbol of the complex Atiyah-Singer operator gives an orienta-
tion class for the К-theory of the cotangent bundle. Viewed in this way,
it is clear that the Atiyah-Singer operator is a fundamental one. Twisting
this operator with a general coefficient bundle generates the KO-theory
(or К-theory) of T*X freely as a module over KO(X) (or K(X) respec-
tively). Otherwise stated, at the level of principal symbols, every elliptic
operator on X is an Atiyah-Singer operator with coefficients in some
bundle.
The proof of Theorem C.7 carries over directly to give the following
useful case of the Leray-Hirsch Theorem. Suppose E -+ X is a complex
vector bundle over a compact space X, and let p: P(E) -► X be the asso-
ciated projective bundle, i.e., the bundle whose fibre at each point x is the
projective space of all complex lines through the origin in Ex. Note that
H*(P(E);Z) becomes an H*(X; Z)-module under the homomorphism p*.
Consider now the “tautological” complex line bundle -► P(E) whose
fibre at e P(EX) P(E) consists of all vectors v g 4 c= Ex. Set
u = c1(^)gH2(P(E);Z).
Theorem C.14. Suppose E -+ X is a complex vector bundle of rank k. Then
H*(P(E);Z) is a free H*(X;Z)-module with basis l,u,u2,... ,uk-1.
Proof. When E is the trivial bundle, this follows directly from the Kunneth
formula. In general one chooses a covering {CJJLj of X and proceeds
by induction using the Mayer-Vietoris sequence exactly as in the proof
of Theorem C.7. 
APPENDIX D
Spirf-Manifolds
The notion of a Spinc-manifold is an important one, but it was not treated
in the main body of the text to avoid too much congestion in the expo-
sition. It is essentially the “complex analogue” of the notion of a spin
manifold, and much of the discussion of this case can be carried out in
parallel with the spin case.
We begin with the complex version of a question mentioned in the
introduction. Let X be an oriented riemannian manifold, and ask whether
there exists a bundle of irreducible complex modules for the bundle Cl(X).
If X is spin, the answer is certainly yes. (One merely takes the associated
bundle $c = Pspin(^) хдс V where V i$ an irreducible complex Cl„-
module.) However, the existence of a spin-structure is not necessary for
the construction of such bundles. To see this we examine the complex
representations of Spin„ more closely.
Suppose Ac: Spin„ -► UN is a complex spinor representation, i.e., one
coming from an irreducible Ct„-module. Let z:Uj UN denote the cen-
ter, i.e., the scalar multiples of the identity. Then we get a homomorphism
Ac x z:Spin„ x - U„ which clearly has the element (—1,-1) in its
kernel. Dividing by this element gives the group
Spin;; = Spin„ xZ2 Up	(D.l)
Note that there is a short exact sequence
0----> Z2----> Spin; SO„ X Uj-----------► 1	(D.2)
where the subgroup Z2 <= SpinS is generated by the element [(—1,1)] =
[(1,— 1)]. Now we have
SpinS c ® C
as a multiplicative subgroup of the group of units. (In fact it is obtained
by “tensoring” Spin„ with the unit complex numbers.) To construct the
bundle of irreducible complex modules over X we proceed as in the spin
case. With the sequence (D.2) in mind, we ask: Does there exist a principal
SpinS-bundle over X which admits a SpinS-equivariant bundle mapping:
Pspi„. pso(x) x PVl	(D.3)
for some principal Unbundle PVl over X? By “equivariant” we mean that
SPINC-MANIFOLDS
391
£(pg) ~ t(p)t(g) for all p e PSpinc and all g e Spin^. We analyse this in the
spirit of Appendix A. Consider the exact sequence
/r(Y;Spin;,) Н'(Х;8О„) @ tf’GVjUj) W2+t‘> H\X-;z2) (D.4)
determined by the coefficient sequence (D.2). The coboundary map asso-
ciates to a pair (Pso,Pu,) the element w2(Pso) + ^(Pu,) where сл is the
mod 2 reduction of the first Chern class of PD1 (see Example A.5). Under
the isomorphism (A.7) this coboundary map can be rewritten as
H\X-, SO„) © H\X- Z) H2(X; Z2)	(D.5)
where p:H2(X;Z) -► //2(X;Z2) is mod 2 reduction. Consequently, given
the frame bundle PSOW, we can find the bundle (D.3) provided that
w2(Pso(X)) = P(M) f°r some и e H2(X;Z), i.e., provided that w2(X) is the
mod 2 reduction of an integral class.
Of course this argument carries over to any principal SO„-bundle.
Definition D.l. Let PSOn be a principal SO„-bundle over X. К Spinc-
structure on Pso consists of a principal Unbundle PV1 and also a principal
Spinjj-bundle PSpinc with a Spinjj-equivariant bundle map
Pspin* ► Pson X
The class ceH2(X;Z) corresponding to PD1 under the isomorphism
H2(X;Z) Prin^X) (cf. A.5), is called the canonical class of the Spinc-
structure.
The argument given above proves the following:
Theorem D.2. A principal SOn-bundle P carries a Spincn-structure if and
only if w2(P) is the mod 2 reduction of an integral class, that is, if and only
if
W3(P) = 0.
(See B.13 for a discussion of the classes JVk.)
It should be noted that Theorem D.2 could be established in the spirit
of Theorem 1.4ff by considering the appropriate 2-fold coverings of the
spaces Рюп x PD1.
Definition D.3. An oriented riemannian manifold with a Spinc-struc-
ture on its tangent (frame) bundle is called a Spinc-manifold.
The theorem above has the following immediate corollary:
Corollary D.4. An orientable manifold X carries a Spinc-structure (i.e.,
X can be made into a Spinc-manifold), if and only if the second Stiefel-
Whitney class w2(X) is the mod 2 reduction of an integral class, i.e., if and
only if W3(X) = 0.
392	APPENDIX D
If a manifold carries one Spinf-structure, it often carries many. They
are parameterized by the elements in 2//2(X;Z) ® H1(X;Z2).
There are two important cases where a Spinc-structure exists and is
canonically determined:
Example D.5. Any bundle with a spin structure carries a canonically
determined Spinc-structure. The Spinc-bundle is obtained as PSpinc =
PSpin х21 where Z2 acts diagonally by (-1,-1) and where of
course denotes the trivial circle bundle. We see then that any spin manifold
is canonically a Spinc-manifold.
Example D.6. Any complex vector bundle E carries a canonically deter-
mined Spinc-structure. To see this, note first that w2(E) = cfE) (mod 2),
and so Theorem D.l implies that E carries Spinc-structures. To see that
there is a canonical one we proceed as follows. Introduce a hermitian
metric in E and let Pvn(E) be the principal Unbundle of unitary frames.
There is a canonical homomorphism
; :u„<—> Spinc2B	(D.6)
which we shall construct explicitly below. It is a lifting of the homomor-
phism U„ -► SO2„ x Ux given by g (i(g), det(#)) where i: <-► SO2„ is
the standard inclusion. Thus, we have a commutative diagram
.	. Spinc2„
SO2„ x Ux
The canonical principal Spinc bundle for E is now constructed as the
associated bundle.
PsP^(E)^PVn(E)XjS^2n	(D.8)
Note, incidentally, that the associated Unbundle in this case is
PufE^PuW*^.	(D.9)
This is the principal bundle of the complex line bundle A"E whose first
Chem class is c^E), i.e., Ci(A"E) = cx(E).
One consequence of this discussion is that every complex manifold, in
fact, every almost complex manifold, is canonically a Spinc-manifold.
Before moving on we shall provide the details of the homomorphism
(D.6). It can be shown to exist by proving the existence of the lifting (D.7)
using covering space theory. However, it can be given explicitly as follows.
Let g g U„ and choose a unitary basis {en ... ,en} of C" in which g has the
form: g = diag{e101,... Let {e^Je^,... ,en,Jen} be the canonically
SPINC-MANIFOLDS
393
associated orthonormal basis of R2" = C". Then
Ав) = fl (cos + sin	x e2X9k (D.10)
k=l \	2	/
in Spinc2„ = Spin2„ xZ2 S1.
We have seen that spin manifolds and complex manifolds are all Spinc-
manifolds. In fact, it requires some searching about to find an orientable
manifold which is not Spinc. Note, for example, that IP4" + 1(IR), although
neither spin nor complex, is Spinc.
It was observed by Landweber and Stong that perhaps the simplest
manifold which is not Spinc is the orientable 5-manifold SU3/SO3 whose
only non-zero mod 2 cohomology classes are 1, w2, w3 and w2 • w3. Since
Sqr(w2) = w3 0, one concludes that W3 0.
Example D.7. (Universal non-Spinc-manifolds) We shall construct here
a family of open simply-connected n-manifolds U"(p) which are not Spinc
for each n 9. These examples are universal in the sense that every simply
connected non-Spinc-manifold of dimension n 9, carries some U"(p) as
an open submanifold.
We fix a positive integer p and consider the cell complex
Z = S2^ D3
where the map dD3 = S2 S2 is of degree 2P. We can embed this com-
plex into R6. In fact there is a natural PL-embedding given as follows.
Consider S2 дД3, where Д3 denotes the standard 3-simplex, and embed
the mapping cone Сф = S2 S2 -► S2 ♦ S2 « S5 in the obvious way. Here
denotes the join. This clearly extends to an embedding S2 D3 <-►
S2 ♦ D3 « D6.
Let be a regular neighborhood of E in R6. This is an open parallel-
izable manifold. We now claim that there is a 3-dimensional real vector
bundle E -+ which restricts to be the non-trivial bundle on S2 с: E.
Since retracts onto E, we need only find a map F:E -► BSO3, which
when restricted to S2 represents the non-zero element in 7t2(BSO3). How-
ever, since 7t2(BSO3) tc^SO^ Z2, any map f:S2-* BSO3 extends to
E. Therefore this bundle exists.
We define the manifold U"(p) as follows. Consider E embedded in the
total space of E as a subset of the zero section; i.e., E cz c= E. For each
n 9, let U"(p) be a regular neighborhood of E x {0} cz E x R"“9.
Theorem D.8. The manifolds U"(p), p 1, are not Spinc. Furthermore, any
simply-connected manifold Xn of dimension n 9 is not Spinc if and only if
for some p 1, there exists an embedding
Un(p) Xn
as an open submanifold.
394
APPENDIX D
Proof. Fix p 1 and n 9 and set U = U"(p). Let S2 с U be the non-
trivial 2-sphere. Then TU|S2 Е|$2 ® (trivial), and so w2(U)[S2] =
w2(E)[S2]	0 by construction. Since H2(U;/) = 0 and tc^U) = 0, we see
that w2(U) is not the mod 2 reduction of an integral class. (Note that U is
homotopy equivalent to E.) Hence, U is not Spinc.
Of course any open submanifold of a Spinc-manifold is Spinc. Hence,
we need only show that a simply-connected n-manifold Xn which is not
Spinc must contain some IT(p) as an open submanifold. Since n^X = 0,
we have an isomorphism л2Х Л H2(X\ Z). Since X is not Spinc, the homo-
morphism w2 e //2(X;Z2) Hom(7t2X,Z2) is not the mod 2 reduction of
an element in H2(X\T) Hom(7t2X,Z). Hence, w2 is non-zero on some
torsion class a of order 2P in ti2X. Now a is represented by а тар /: S2 -► X
which extends to а тар F : U"(p) -► X (since 2pa = 0 in n2X). The induced
bundle F*TX is clearly equivalent to E ® (trivial) = TU"(p). Hence, by
Smale-Hirsch immersion theory, F is homotopic to an immersion U"(p)
Xn. The immersion can be made an embedding by putting E into general
position and shrinking U"(p) to a small neighborhood of E. This completes
the proof. 
We did not need to invoke Smale-Hirsch Theory here. The embedding
U"(p) <-► Xn which is homotopic to F can be built explicitly. A tubular
neighborhood of S2 c= U"(p) is diffeomorphic to a tubular neighborhood
of S2 c Xn9 since the normal bundles are equivalent. Completing the em-
bedding to a neighborhood of D3 cz U"(p) is straightforward.
For compact examples we add a boundary to U"(p) and consider the
double
xn(p) = й"(р) ид йп(р)
where U"(p) = U"(p) u dU"(p).
Let us return our attention to manifolds which are Spinc.
There is an alternative approach to this concept which comes from
considering vector bundles.
Definition D.9. Let X be a Spinc-manifold of dimension n. By a com-
plex spinor bundle for X we mean a vector bundle S associated to a repre-
sentation of Spinc by Clifford multiplication, i.e.,
S = PSpi„c(X) хд V
where V is a complex Ct„-module and A:Spin£ -► GL(F) is given by
restriction of the ^„-representation to Spinjj cz Cl„ ® C. If the represen-
tation of Cl„ is irreducible, we say that S is fundamental.
These spinor bundles are bundles of complex modules over Cl(X).
When n is even there exists only one fundamental spinor bundle for X,
denoted S(X). It splits into a direct sum
S(X) = S + (X)® S’(X)	(D.ll)
SPINC-MANIFOLDS
395
where S±(X) = (1 ± coc)S(X) and where coc = in/2ex • • • e„ is, as usual, the
volume form. Multiplication by a non-zero tangent vector maps S±(X) to
S+(X) and is invertible.
When n is odd, there are two irreducible complex representations of Ct„.
However they are equivalent when restricted to Spinjj. Hence, there is only
one fundamental spinor bundle S(X) for any Spinc-manifold X.
For a spin manifold X, the bundle S(X) is just the usual complex spinor
bundle.
For a complex manifold X with its canonical Spinc-structure, we have
that
S(X) = AJTX A0’*
that is, S(X) is simply the direct sum of the complex exterior powers of
the tangent bundle, considered as a complex bundle (which, in turn, is
isomorphic to the direct sum of the bundles of (0,q)-forms, 0 q n).
This follows directly from the interpretation of the complex spinor repre-
sentation given in 1.5 (see 1.5.25 and the attending discussion). In partic-
ular, from this we see that the Clifford multiplication Ct(X) ®R S(X) -►
S(X) is generated as follows. For each tangent vector v9 consider the linear
map A*TxX A£TxX given by
pv(cp) = v^(p-v4(p
where u* is the 1-form corresponding to v under the hermitian metric on
X. Repeating this operation gives
/U/m>)= -IMlV
and so by the universal property of Clifford algebras, the map p extends
to a representation of Ct(X). Since each pv is complex linear the repre-
sentation is complex.
It is instructive to examine these fundamental spinor bundles as one
varies the Spinc-structure on a given manifold. To do this it is good to
have clearly in mind the elementary isomorphisms:
УесЦ(Х) Prinbl(X) tf2(X;Z)	(D.12)
where Vectj(X) denotes the set of equivalence classes of complex line
bundles on X and where the map to H2 is given by the first Chern class
(see Example A.5).
Example D. 10. Let X be a spin manifold with its canonical Spinc-
structure. We can change this structure by changing the Uj-bundle as
ollows. Pick an element ocg//2(X;Z), and let PDi(a) g Prinsi(X) and
le e Vecti(X) be the corresponding elements under (D.12). Then we de-
ine the ath Spinc-structure by
PSpU«) = J’spinW xZ2 Ри,(а)	(D.13)
396
APPENDIX D
Note that this has a Spinjj-equivariant map
PspUa)------► Pso(*) x Pbl(2a)	(D.14)
where PD1(2a) = PD1(a)/Z2 is the “square” of Ри1(а). (In particular Я2а = Я2.)
We now let Sa(X) be the fundamental spinor bundle for the ath Spinc-
structure on X. Then it is easy to see that
Sa(X) = S(X) ® Яа.	(D.15)
Thus, changing the Spinc-structure on X by an element ae//2(X;Z)
amounts to twisting the fundamental spinor bundle by the complex line
bundle Aa.
This is a general fact. The group H2(X;Z) acts on the set of Spiif-
structures on a Spinc-manifold X, by tensoring the fundamental spinor
bundle with complex line bundles (modulo this action, the Spinc-structures
correspond one-to-one with elements of H\X; Z2).) Notice that tensoring
with line bundles preserves the fibre-dimension and thus takes one bundle
of irreducible modules over Cl(X) to another.
A good example of this phenomenon is the following. Suppose X is a
complex manifold which is also spin. This means that w2(X) = cx(X) = 0
(mod 2). Now the class — cx(X) e H2(X; Z) corresponds to the complex line
bundle к = /^T*X, called the canonical bundle. Since — c^X) =	is
even, there exist square roots к1/2 for к. The different square roots are
parameterized by elements of Hl(X;Z2), corresponding to choices of spin
structure. Now the canonical spinor bundle S(X) = A*TX for X as a
Spinc-manifold, and the canonical spinor bundle $C(X) for X as a spin
manifold are related by the equation
$C(X) = S(X) ® k1/2.	(D.16)
Much of this business of Spinc-manifolds becomes transparent when
viewed in terms of the fundamental spinor bundles. For example, let us
return to the original problem of trying to construct a fundamental spinor
bundle (i.e., a bundle of irreducible complex modules for Ct(X)) over a
manifold X. Locally of course we can always do it. Let {Ua}aeA be a cov-
ering of X so that Uai n • • • n Uak is contractible for all an ... ,a„. On
each Ua, bundles can be trivialized and we can find a fundamental com-
plex spinor bundle of the form Ua x V where V is an irreducible Cln ®
C-module. Now in passing from Ua to Up we look for transition functions
gap- ua n Up -► Spin„ so that £ ° gap = gafi: Ua n Up -► SO„ are the cor-
responding transition functions for PsoW- Now the existence of a com-
patible set of choices is equivalent to the vanishing of the Cech cocycle
= g^g^gya- Ua n Up nUy--------► Z2 = ker(£) (D.17)
in H2(X;Z2).
SPINC-MANIFOLDS
397
Suppose now that the class [w] g H2(X;Z2) is the mod 2 reduction of
an integral class g H2(X;Z), and let Я be the complex line bundle cor-
responding to W. Consider the problem of finding a square root of Я, i.e.,
a line bundle Я1/2 with (Я1/2)2 = Я. Let yap: Ua n Up -► S1 be the transition
functions for Я. Since Uar\ Up is contractible we can extract a square root
yap — 7ap2‘- Ua Up -+ S1. However, the compatibility is just the Cech
cocycle
= 7ap7fiy7y*- Ua n Up n U7--► Z2 = ker(a) (D.18)
where
0----> Z2----> S1 S1--------> 0
and a(z) = z2. The class [w'] g H2(X;Z2) is just the coboundary of
Я g H\X\ S1) under the associated long exact sequence in cohomology. In
fact, consider the following commutative diagram:
H\X;Sl)	H^XiS1)	H2(X;Z2)
»	»	II
H2(X;Z) -2— H2(X;Z) —№(X;Z2)
It is clear that our obstructions agree, i.e., [w'] = р(сг(Я)) =	= [w].
And so [w] + [w'] = 0 in the wonderful world of Z2.
This means essentially that while we cannot construct the spinor bundle
and we cannot construct Я1/2, we can construct their product. Adjusting by
coboundaries we can choose gap and yap so that wap7 = w'ap7. Thus the tran-
sition functions
Gap : Ua n Up---► Spin„ xZ2 S1
defined by Gafi = gap x yafi have wafi7 = GapGp7Gya = 0, and thus determine
a global bundle we may think of as
S(X) = S0(X) ® Я1/2	(D.19)
where S0(X) is the fundamental spinor bundle for the possibly non-existent
spin structure on X, and where Я1/2 is the possibly non-existent square root
of Я with сх(Я) = w2(X) (mod 2).
Since S(X) is a bundle of modules over Ct(X) we need only introduce
an adapted metric and connection to make it a Dirac bundle. We do this
as follows. Fix any line bundle Я as above and choose a unitary connection
on Я. This induces a connection on the bundle Я1/2 when it exists. Locally
both S0(X) and Я1/2 exist and S0(X) carries a canonical (riemannian) con-
nection. We give the bundle S(X) = S0(X) ® Я1/2 the tensor product con-
nection. It is not difficult to see that this connection is well defined globally.
398	APPENDIX D
It is easy to check, using the arguments of §4, that this connection makes
S(X) into a Dirac bundle.
Notice that the line bundle Я corresponds to the principal Unbundle
PVl associated to the Spinc-structure, i.e., PV1(A) = PSpin4^) *p Ux where
p:Spin^ -► Ut is induced from the projection Spin„ x Ut -► U? Let us
summarize:
Proposition D.ll. Let X be a Spinc-manifold with associated complex line
bundle Л. Then to each U^connection co on Л there is a canonically asso-
ciated connection on PSpine. 15 Just the 4ft °f product of the canonical
riemannian connection with co via the covering map PSpinc -► Pso x Ри/Д
This connection makes any complex spinor bundle for X into a Dirac bundle.
Associated to any unitary connection on Л there is the curvature 2-form
Q. This is a closed 2-form, and the class [(2л) “1 Q] g H2(X; R) represents
ct(A) or more precisely the image of under the map H2(X;Z) ->
H2(X; R). We call this the real Chern class of L
The curvature form is defined as follows. For a local trivialization of
P^/Я), the connection 1-form becomes simply a real-valued 1-form co.
Then
Q = dco.
If we change trivializations by a transition function g: V n V -► Ut, then
the new connection 1-form is co' = co 4- g~1 dg. Locally we have g = e10 for
some function в. The equation above becomes co' = co 4- i dd. It follows
that dco' = dco, that is, the curvature 2-form Q is well defined globally.
Suppose now that and co2 are two different connections on Л with as-
sociated curvature forms Qi and П2. Then it is easy to see that co t — co2 = a
for a globally defined 1-form a. Hence, — П2 = da, and we see that
the de Rham class [Q] is independent of choice of connection.
We are now prepared to derive the Bochner-type identity for these
spinor bundles.
Theorem D.ll. Let X be a Spinc-manifold with associated line bundle Л,
and fix a connection on Л with curvature 2-form Q. Let S be a bundle of
complex spinors on X with the canonical connection, and let D be the Dirac
operator on S. Then
D2 = V*V 4-1 к 4-1П
where к denotes the scalar curvature of X and where Q denotes Clifford
multiplication by the 2-form Q.
Proof. The computation is local, so we can assume S(X) = S0(Aj ® A1/2
where S0W is a spinor bundle for the local spin structure. We may now
SPINC-MANIFOLDS
399
apply Theorem II.8.17, and it is sufficient to compute the term 91д1/2. Recall
that
91л‘/2(<т ® z) = £ (ejeka) ® R^^(z)
J<k
= ,E ® №(ej>ek)iz)
= I i E	® z
= i ( E ^j,ek)ej л ek I <r ® z
= |(Q • ст) ® z 
We used here the fact that the curvature of the induced connection on
Я1/2 is We also use the fact that R„iW(Z) = Cl(v,w)iz. Note that Clifford
multiplication by iCl is a hermitian symmetric operation.
Given this theorem, the following lemma is relevant:
Lemma D. 13. Let Я be a complex line bundle on a manifold X. Then any
closed 2-form Cl which represents the real Chern class of Я can be realized
as the curvature form of a disconnection on X.
Note D.14. This means, in particular, that if сх(Я) is a torsion class in
H2(X;Z), then Я admits a flat connection.
Proof. Choose any Ux-connection coo on у and let Qo be its curvature
form. Since Q and Qo represent the same de Rham class, there is a 1-form
a so that da = Cl — Qo. Let co = co0 + a. Then the curvature of co is
Cl0 + da = Q. 
Suppose now that X has even dimension and consider the Dirac operator
D+:TS+(X)----------------------------► TS"(X)
for the splitting (D.ll). Using (D.19) and the Atiyah-Singer Formula, we
obtain the following:
Theorem D.15. Let X be a compact Spinc-manifold of even dimension, and
let D denote the Dirac operator of the fundamental complex spinor bundle.
Then
ind(Z)+) = {>- A(X)} [X]	(D.20)
where c is the canonical class of the Spinc-structure, i.e., с = сг(Я) for the
associated complex line bundle Я.
400
APPENDIX D
Combining this with D. 13 and D. 14 gives the following:
Corollary D.16. Let X be a compact Spinc-manifold such that w2(X) is the
mod 2 reduction of a torsion class. If X carries a metric of positive scalar
curvature, then A(X) = 0.
Corollary D.17. Let X be a compact Spinc-manifold with canonical class c e
H2(X;Z). Let Q be any 2-form representing the de Rham class of c in
H2(X;Z) ® R. Then there is a riemannian metric on X whose scalar curva-
ture satisfies
k>2||Q||
only if{e*‘X(X)}[X]=0.
Here the norm || || is taken in the same metric. It is given by ||Q|| =
Y |Aj| where Q = £ Aje2j-1 л e2j for the diagonalizing orthonormal frame
e1?e2,... at the point.
It is a nice exercise to apply this theorem to P"(C) with the standard
Fubini-Study metric, and then to verify directly the vanishing of the
“Hilbert polynomial” at certain integers.
It is interesting to examine the Dirac operator on a complex hermitian
manifold X, considered as a Spinc-manifold. Recall that S(X) = A0’*. The
splitting S(Aj = S+(X) ® S“(Aj corresponds to the even-odd decomposi-
tion A0’* = AO even ® A0,odd. тье j)irac operator can be identified with
8 + 8*: A°’even---► A°’odd
where 5* is the hermitian adjoint of the operator 8. The index of this
operator is the Todd genus of X.
If ct(X) = 0 (mod 2), we can choose a spin structure on X, i.e., we choose
a square root k1/2 of the line bundle к = An,°. Then the associated Dirac
operator can be identified with
d + d*:A°’even ® k1/2----► A°’odd ® к112.
Here 8 is the standard operator on (0,*)-forms with values in the holo-
morphic line bundle к1/2 (cf. (D.16)).
Observe that Theorem D.15 has the following immediate corollary
which was first proved by Atiyah and Hirzebruch [1].
Corollary D.18. Let X be a compact Spinc-manifold. Then for any class
c g H2(X,Z) such that c = w2(X) (mod 2), the (rational) number
{>A(X)}[X]
is an integer.
SPINC-MANIFOLDS
401
As a final remark we point out that Spinc-bundles are bundles which
have natural К-theory orientation classes, just as spin bundles have natural
KO-theory orientation classes. This makes them sometimes useful in
general theories.
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Index
Л-genus, 231, 283, 284
Л-degree of /, 309
Л-enlargeable, 309
Л-genus, 138, 161, 233, 256, 280, 292
higher------, 310
of hypersurfaces, 138
Л-operator, 137
j/-class, 92, 100, 152, 162, 163, 276, 292,
298, 313
Adams operation, 236
Adams Theorem, 46
Adams-Milnor Theorem, 93
adjoint
—	operator, 187, 194
—	representation, 12
algebra
associated graded —,10
Clifford —, 7, 27, 29
exterior —,10, 24, 38
filtered —,10
matrix — isomorphisms, 26
Z2-graded —, 9
almost complex structure, 336
integrable--------, 339
analytic index, 135, 141, 142, 192, 194,
201, 254
— Clifford —, 141, 142, 216, 217
— G—,212
—	mod 2 —, 147
—	of a family, 208
—	of Mishchenko-Fomenko, 312
anti-self-dual, 342
associated
—	bundle, 93, 371
—	graded algebra, 10
associative
—	calibration, 352
—	geometry, 352
associator, 352
asymptotic principal symbol, 245
Atiyah-Bott-Shapiro isomorphisms
—	for K-theory, 69
—	for KK-theory, 75
Atiyah-Milnor-Singer invariant, 144, 276
Atiyah-Segal-Singer Fixed Point Formula,
261
Atiyah-Singer Index Theorem, 244
cohomological form, 254, 255, 258
— for families, 269
— for real families, 272
— for Ctk-linear operators, 273
Atiyah-Singer operator, 121, 137, 256,
267, 274
Ct„-linear, 140
conformal invariance of—, 133
twisted—, 139
automorphism
a—, 9, 95, 127
triality —, 55
bad end, 324
Bianchi identity, 112, 362
big covering spaces, 302
Bochner
— method, 153, 335
— formulas, 155-157
— Theorem, 157
Bochner-Lichnerowicz formulas, 160, 164,
398
Bockstein homomorphism, 382
Bott periodicity
unitary case, 62, 63, 67, 225
orthogonal case, 63, 67, 222
(1,1) case, 72, 73
— property, 386
bundle
associated —, 93, 371
induced —, 375
— of bases, 94, 370
— of orthonormal bases, 111, 371
principal —, 370
virtual —, 59
418
INDEX
C*-algebra of a discrete group, 311
calibration, calibrated manifold,
calibrated geometry, 351
associative, 352
Cayley, 352
coassociative, 352
special lagrangian, 352
canonical
— bundle of complex manifold, 361, 396
— bundle over Grasssmannians, 379
— class of a Spinc-structure, 391
— orientation of a complex structure,
336
— riemannian connection, 112
Cartan-Dieudonne Theorem, 17
Cayley numbers, 51
character, 212
— ring, 212
characteristic class
Chern, 227, 281, 374, 381, 398
Pontrjagin (integral), 228, 282, 283, 382
Stiefel-Whitney, 79, 81, 83, 84, 373, 380
universal, 377
Chern class
divisibility of-----,281
first-----, 374
Ath-------,381
total-----, 277, 381
vanishing first-----, 367
Chern character, 234, 235, 237, 238
reduced ——, 316
-----commutativity defect, 241
Ctk-Dirac bundle, 140
Cfk-bundle, 140, 214
Dirac —,140
riemannian —,215
Z2-graded —,215
Ctk-linear operator
Atiyah-Singer, 140, 144-153
differential, 215
Fredholm, 217
Z2-graded, 215
classical pseudodifferential operator, 244
classification of
-----Clifford algebras, 27, 29
-------- irreducible representations of
Clifford algebras, 32, 33
classifying space, 376
Clifford algebra 7
-----classification, 27, 29
— — periodicity, 27
-----representations, 30, 37
Clifford modules, 30
Grothendieck group of-------, 33, 40, 73
Real------, 73
Z2-graded---------, 39, 73
Clifford
— bundle, 95
— bundle of a manifold, 111
— cohomology, 332
— diamond, 331
— difference element, 236
— group, 36
— index, 216, 217
— multiplication, 30
coassociative geometry, 352
cobordism, 90, 92, 100
cocycle condition, 371
cohomological formulas for the index, 254,
255, 258, 399
cohomology
Clifford, 332
with compact supports, 239
Dolbeault, 335
spinor, 357, 359
cokernel, 192
compact operator, 192
compact support
bundle with-------, 313
cochain with------, 239
cohomology with — —, 238
К-theory with-----,66
compactly enlargeable, 303
complex
— spinor bundle, 96, 394
— spinor representation, 36
— volume element, 34, 99, 135
conformal invariance of the Dirac
operator, 133
conjugate bundle, 227
connected sum, 91
connection, 101
canonical riemannian —,112
— 1-form, 101, 103
— laplacian, 154
— on spinor bundles, 110
projected —, 282
tensor product —,122
constant at infinity, 303
contracting
e — maps, 302, 317
contraction, 24
convolution product, 173
covariant derivative, 102
INDEX
419
— — as derivation, 107-8
-----on spinor bundles, 110
riemannian--------, 103
second-------, 154
curvature, 101
— 2-form, 101, 398
— identities, 112, 126
— of sphere, 162
— on spinor bundles, 110
— operator, 158
positive scalar—, 160-165, 297-326
Ricci —, 156
scalar —, 160
— transformation, 106
de Rham
— complex, 125
— cohomology, 125
degree
— of a map (constant at infinity), 303
pure —, 8
derivative
covariant —,102
exterior—, 123
invariant second—, 154
diffeomorphism
spin structure-preserving—, 86
difference element, 100
Clifford-----, 236
spinor-------, 238
differential operator 167
Cfk-antilinear----, 216
Cfk-linear--------, 215
Cfk-linear Z2-graded--------,215
elliptic-----, 113, 167, 188
G-invariant-------, 211
See also: operator and elliptic operator
dimension
geometric — of a bundle, 47
— of some classical groups, 56
Dirac bundle, 114
C£k-------, 140
rth cohomology group of---------, 359
Dirac laplacian, 113
Dirac operator, 112
Atiyah-Singer-----, 121, 137, 256,
267, 274
Ctk-linear--------, 140
conformal invariance of-----, 133
essential self-adjointness of---, 117
kernel of----, 116, 117
-----on Clifford bundle, 123
principal symbol of------, 169
self-adjointness of------, 114
eigenspace, 196
eigenvalues, 196
growth of—, 196
Einstein’s equations, 369
elliptic complex, 332
elliptic estimates, 194
elliptic operators, 113, 168, 188, 191
continuous family of------, 204, 206
homotopy of--------, 204
product family of--------, 205
— pseudodifferential —,191
enlargeable, 3O2ff
4—, 309
compactly —, 303
weakly —,318
fi dense, 196
e-contracting, 302
------on 2-forms, 317
fi-hyperspherical, 317
e-local operator, 182
essential self-adjointness of Dirac operator,
117
essentially positive self-adjoint operator,
219
Euller characteristic operator, 136
Euler class, 227, 383
even part of Clifford algebra, 9
exceptional isomorphisms of Lie groups,
50, 56, 58
excision property, 248
exotic spheres, 93, 162, 292
exponential mapping, 13
exterior
— algebra, 10, 24, 38
— derivative, 123
family of
------differential operators, 205
— — elliptic operators, 206, 208
------real operators, 210, 270, 272
— — riemannian manifolds, 269
fibre
integration over the —, 239
filtered algebra, 10
fixed-point
— formula, 267
— set, 261
formal adjoint, 187, 194
formal development, 179
420
INDEX
Fourier transform 171
frame bundle, 111
Fredholm operators, 192, 201
and K-theory, 208, 211, 222
Cfk-antilinear, 217
C(^-linear, 217
families of----, 208
G-invariant-----, 212
G2-structures, 347, 348, 357
G-signature theorem, 264
G-bundle, 259
G-index, 212, 259
— theorem, 260
G-manifold, 259
G-spin theorem, 267
G-structure
topological —, 348
Gallot-Meyer Theorem, 158
geometric dimension, 47
good presentation of bundles, 171
graded Hilbert module, 217
Grothendieck group of Clifford modules,
33, 40, 73
group
— actions, 29
Clifford —, 36
isotropy —, 43
К—, 59
— of units in a Clifford algebra, 12
orthogonal —, 13
Pin —, 14
special orthogonal —, 17
Spin —, 14
Gysin homomorphism, 239
harmonic
— p-forms, 125, 195
— space, 332
— spinor, 160, 161, 367
heat kernel, 198, 277
trace of — , 199
hermitian line bundle
positive —, 360
higher 4-genus associated to u, 310
Hilbert polynomial, 365
Hilbert module
graded------,217
Hirzebruch
— L-sequence, 232
— Signature Theorem, 233, 256
— Riemann-Roch Theorem, 258
Hodge Decomposition Theorem, 195, 332
Hodge laplacian, 123
holonomy group, 345
Sp„—, 357, 367, 368
homotopic elliptic operators, 204, 214
homotopy sphere, 93, 162, 292
hyperspherical, 317
hypersurfaces Vn(d) 88, 90
Л-genus of — —, 138
immersions into R", 283, 286
incompressible surface, 326
index (analytic), 141, 142, 208, 312
Clifford—,216, 217
G—, 212
mod 2 —, 147
— of a Cfk-linear operator, 141, 142
— of an elliptic operator, 135, 194, 254
— of a family, 208
— of a Fredholm operator, 192, 201
— on an odd-dimensional manifold, 257
index (topological), 244
G—, 259
— of a family of operators, 269
— of a real family of operators, 272
topological invariance of —,201
induced bundle, 375
infinitely smoothing operator, 179, 186
integrable almost complex structure, 339
integrality theorems, 280, 400
integration over the fibre, 239
interior product, 24
interpolation of Sobolev norms, 175
invariant second derivative, 154
invariant
Atiyah-Milnor-Singer—, 144
irreducible
—	bundle of Ct(E)-modules, 97
— manifold, 345
—	representation 31
isotropic subspace, 335
isotropy group, 43
—	of a spinor, 343
join, 377
к th power operation, 235
K-theory, 59
equivariant —, 259
higher groups, 61
multiplication in —, 59, 66
—	of a point, 62, 63, 149
real —, 60
INDEX
421
Real —, 70
reduced —, 60
relative —, 61
—	with compact supports, 66
КЗ surface, 90
Kahler manifold, 330
—	with Cj = 0, 367
kernel, 192
Kervaire semicharacteristic, 143, 290
Kervaire sphere, 162
Kodaira-type vanishing theorem, 360
Kuiper’s Theorem, 208
Kummer surface, 90
L2-section, 116
L-genus, 233
L-sequence, 232
L-operator, 24, 95, 128
lagrangian
special —, 352
Laplace-Beltrami operator, 168
laplacian
Hodge —, 123
connection—, 154
Lefschetz number, 212
Leray-Hirsch Theorem, 389
Lichnerowicz Theorem, 160, 362
Lie algebra of Spin,,, 40
Lie groups (exceptional isomorphisms), 50,
56, 58
line bundle
tautological-----,47
local trivialization, 371
local holonomy group, 345
loop space, 222
Minkowski space, 369
mod-2 index, 147
modules
Clifford—, 30
decompositions of------, 35
Z2-graded —, 39
Grothendieck group of Clifford —, 33,
40, 73
multiplication
Clifford —, 30
multiplicative property, 249
for sphere bundles, 252
multiplicative sequence, 229
A-, 231
Hirzebruch L-, 232
Todd, 230
Nakano type vanishing theorem, 360
negative hermitian line bundle, 360
non-negative curvature operator, 158
norm, 15
uniform Ck—, 172
Novikov Conjecture, 312
odd part of Clifford algebra, 9
operators
adjoint —, 187, 194
Atiyah-Singer—. 121, 137, 274
bounded—, 192
Ctn-linear Atiyah-Singer —, 140
compact — ,192
differential—, 167
elliptic—, 113, 168, 188
elliptic pseudodifferential —,191
essentially positive—, 219
Euler characteristic —,136
families of —, 204, 206
Fredholm—, 192, 201,217
G—,211
heat —,198
Laplace-Beltrami—, 168
positive self-adjoint —,198
pseudodifferential —, 178, 179, 186
signature—, 136
smoothing —, 179, 186
orientation
^-theory —, 385
oriented
— bundle, 78
— volume element, 21
orthogonal
— almost complex structure, 336
— group, 13
parallel, 139
parametrix, 117, 190, 191
Pauli matrices, 120
periodicity
— in Clifford algebras, 27
Bott —, 62, 63, 67, 72, 73, 222, 225
Pin group, 14
plurigenus, 366
Poincare duality, 239
Pontrjagin classes, 228, 382, 383
positive
— curvature operator, 158
— hermitian line bundle, 360
— self-adjoint operator, 198
422
INDEX
positive scalar curvature, 160-165, 297-326
— and higher 4-genera, 311
obstructions to —, 301
— on bad ends, 324
— on compact manifolds, 299
— on complete manifolds, 306, 310, 319,
322, 324
— enlargeable manifolds, 306, 310
— 3-manifolds, 324
— 4-manifolds, 344
— weakly enlargeable manifolds, 319
— under surgery, 299
the space of metrics with —, 326-330
power operation, 235
primitive cohomology class, 334
principal bundle, 370
----of tangent frames, 94, 111
— — of bases, 370
----of orthonormal frames, 78, 371
—	spin —, 80
universal-----, 376
principal symbol, 113, 168, 243
asymptotic----, 245
----of a pseudodifferential operator, 186
regularly G-homotopic-------, 214
regularly homotopy of-------, 204
product
convolution —, 173
interior —, 24
—	in K-theory, 59
smash —,61
warped —, 321
projectivization of a bundle, 225
pseudodifferential operators, 178, 179, 186
classical-----, 244
elliptic------, 191
formal development of-------, 179
principal symbol of------, 186, 188
symbol of-----, 177, 187
pure spinor, 336
Radon-Hurwitz Theorem, 45
rank of a spinor, 43
real
—	spinor bundle, 96
—	spinor representation, 35
— K-ring, 60
Real
— K-group, 70
—	module, 73
—	space, 70
— vector bundle, 70
— Z2-graded module, 73
reduced Chern character, 316
reduced K-ring, 60
reducible representation, 31
refinement of a covering, 372
reflection, 17
regularly homotopic principal symbols,
204, 214
relative K-groups, 61
Relative Index Theorem, 315
Rellich Lemma, 173
representations (See also “modules”.)
adjoint —, 12
complex spinor —, 36
± decomposition of—,35
irreducible —,31
— of a Clifford algebra, 30, 37
quaternionic —, 30
real spinor —,35
— ring, 211
twisted adjoint —, 14
unitarity of — of Clifford algebras, 35
weights of a —,44
Ricci
— curvature, 156, 308, 367
— form, 362
Riemann-Roch-Hirzebruch formula, 258
riemannian
canonical — connection, 112
— covariant derivative, 103
— curvature identities, 112, 126
— curvature of the sphere, 162
Fundamental Theorem of — Geometry,
112
— structure, 78
ring of virtual representations, 211
Rochlin’s Theorem, 89
^-actions, 291, 292, 295
S3-actions, 291, 296
scalar curvature, 160
positive------, 160-165, 297-326
(See also “positive scalar curvature”.)
Schwartz space, 171
second fundamental form, 282
second covariant derivative, 154
self-adjoint operator, 195
essentially positive-----, 219
positive------, 199
self-dual 2-form, 342
INDEX
423
Serre duality, 333
signature
G—Theorem, 264
— of a manifold, 137, 266, 284, 288, 290,
295, 349
— of a spin manifold, 89, 280
— operator, 136
— Theorem, 233, 256
twisted — operator, 139
smash product, 61
smoothing operator, 179, 186
Sobolev
basic — k-norm, 170
— s-norm, 171
— space, 170, 171
— Embedding Theorem, 172, 176
special lagrangian geometry, 352
special orthogonal group, 17
Spinf-(manifold), (structure), 391
spin
— cobordism, 90, 92, 100
— field, 18
— group, 14
— group as a covering group, 19, 20
— manifold, 85, 88, 89, 90
— structure, 80, 81, 82, 84
— structure preserving, 86, 266
spinor bundle, 96
canonical complex-------of a spin
manifold, 121, 137
canonical complex-------of a Spinc
manifold, 394
complex and quaternionic structure
on-------, 98
-----decompositions, 99
Z2-graded-------,96, 98
spinor
— cohomology, 357, 359
— difference element, 238
harmonic—, 160
pure —, 336
— representation, 35, 36
Splitting Principal, 226, 237
Stiefel-Whitney class, 380
first----, 79, 83, 84, 373
second------,81, 83, 84, 373
kth------, 380
total----, 380
structure
riemannian —, 78
spin —, 80
support of a pseudodifferential operator,
183
surgery, 299
suspension, 61
symbol class, 169, 243
Real------,271
symbol
asymptotic principal —, 245
— of a pseudodifferential operator, 177,
187
principal—, 113, 168, 243
principal — of a pseudodifferential
operator, 186, 188
tangent frame bundle, 111
tautological line bundle, 47
tensor product
-----connection, 122
Z2-graded-------, 11
Thom class, 152
Thom isomorphism, 239, 240, 243, 271,
387-89
-----in K-theory, 243
Todd sequence and genus, 230, 367-8
topological G-structure, 348
topological index, 244, 269
— G—, 259
-----of a family of Cfk-linear operators,
273
-----of a family of complex operators,
269
----of a family of real operators, 272
torsion tensor, 112
total characteristic class, 230
A-, 231
Л-, 231, 237, 238, 242, 254
Chern, 227, 381
Hirzebruch L-, 232, 237, 242
Pontrjagin, 228, 382
Todd, 230, 242
trace of the heat kernel, 199
transition functions, 371
transpose antiautomorphism, 12
triality automorphism, 55
tubular neighborhood, 299
twisted
— adjoint representation, 14, 15
— Atiyah-Singer operator, 139
— signature operator, 139
twistor space, 339, 342
424
INDEX
uniform C^norm, 314
uniform C*-norm, 172
unitarity of Clifford modules, 37
unitary basis, 336
units
multiplicative group of —, 12
universal
characteristic classes, 377ff
n-plane bundle, 379
principal G-bundle, 376
vanishing theorem, 155, 157, 158, 160, 164,
360, 361, 363, 365, 366, 400
vector fields
-----on the sphere, 45, 46
— — on manifolds, 288, 289
virtual bundles, 59
volume element, 21, 33, 128
complex-------, 34, 99, 135
warped product, 321
weakly enlargeable, 318
wedge product of spaces, 61
weights of a representation, 44
Workhorse Theorem, 180
Z2-graded
— algebra, 9
— bundle, 96
Grothendieck group of — modules, 40, 73
— module, 39, 73
— tensor product, 11
Notation Index
A, A,, A, 231
As 267
X 92,100,152,276
A(X), A(X), A(X), A(X)
AfJtX) 310
A(X;E) 139
л?(Х), sf(X,E) 276
A-deg(/) 309
ad, Ad 12
Ad 12
a 9, 95, 127
233, 138
bf(A) 144
BG 376
BO„, BSO„, BU„, etc. 378
C (the complex numbers)
C(n) 26
*2	336
c,(E)	82, 374
c(E), ct(E) 227,381
ch, ch* 234
chc,ch, 260
ch, 285
сЙ 316
dx(M)	12
ct(K<rt	7
Ct°(y,q), Ct'(V,q) 9	
Ctx(K,<?)	12
c£„, ce*, cer,s 20	
ct„ 27	
Ct.*, Ct*	34
Ct(E),Ct°(E), Ct'(E) 95	
ce^E)	107
C€(X)	111
ce(X)	330
ce^x)	128, 136
cep’W	331
coker( )	192
Cov2(X)	78
Z(E) 227,383	
d, d* 123
D (Dirac operator) 112, 113
D°, D1 135,136
D+, D~ 136,137
D	123
0	121
0°, 01, 0+, 0'	138
Ф _ 140
0, &	331,357
333
Da 167
(Poincare duality) 239
deg(/) 303
Diff(X), Diff(E; X\ Diff(E,F; X) 205
div 115
d (anticanonical bundle) 363
8(E)	242, 388
A (Hodge Laplacian,
Laplace-Beltrami) 123, 168
A„, A„+, a; 35
AJ, ДГ 36
[E,F;<r]	64,67
[Eo..... . ,<rn]	64,67
EG 376
6j, «j	331
F„	261
5,201
5х	210
5r	210
5C	212
5*, 5*	217
5Л 5*	219
G„ ft,	343
Ъ И8
Г(Е)	102
Гср1	115
0-0 (the quaternions) 25
O-O(n)	26
426
NOTATION INDEX
H,HP 125
Hcpt 239
H\X\G) 372
<г 331
Ж(Х) (the holonomy group of X) 345
W\X\ W\X\ W)
H'V; И9 332
JH%,S) (spinor cohomology) 359
i, 387, 388
*(0o,0i),	329
Л,	336, 337
ind	135, 194, 208
ind*	141,206
indc, ind, 212
J 330
/	332
k* 14
k~*(E)	384
ker 116,192
K(X) 59
K(X,Y) 61
K(X) 60
К~\Х), K-i(X), К~1(Х,У)	61
К -»(pt)	62
^ср^),кс’Р;(Х),кср;(х,у)	66,67
К0(Х)	60
К0~1(Х), К0~\Х), К0~\Х,У)	61, 62
KOcpt(X), ко-;(Х), кОсЖУ)	66,67
KR(X), KR~'(X), KR-\XY\
KR'XX'Y) 70, 71
KKcpt(X), KF'pt 72, 73
KC(X) 259
к 160
£, L 24,95,128
L, £*, L, £*	232
ЦХ), ЦХ), L(X), ЦХ) 233
L, 263
L0 265
L2 116
£*(£)	170
£И(Х), £„(Х,У)	64
L„(X)cpt, £„(Х,У)ср(	67
SPn (in K-theory) 64
^(HbH2)	201
203
(in operator theory) 223
ЦЕ) 235
A_,	240,387
ArK, Л* И 10
A*(X)	123
Ж., <	33
ЭДИ, ЭДС	40
Ш1КГ.4,Ш1ЯГЛ, ЯИГ5, <0lr,s 73
MSpin*	91
ц 67, 69, 74, 75, 96, 97
О (the octonions) 51
ОИ,ОГЛ 19
O(V,q) 9
OpM(E,F), Ор"(<Г,^)	205
to (connection) 101
to (volume) 21,128
toc	34, 135
toy, to, 338, 358
to,	358
П (curvature) 101
Qjpin	90
92
ОД*	222
p(£), py(£)	228, 382
PGl(£)	370
Pgl-(£)	82, 371
PO(E) 78, 371
Pso(f), PsPin(£) 80, 371
Pgl(AT) 94
ЛюРО, Pspin(AT) Hl
PSpinc(X) 370
P(K^), P(V,q\ Pin(Kq) 13, 14
Pin,,, Pinr s 19
Princ(X) 370
P$, Р$*, PfP^) 336, 337
0>(X)	326
P(£)	225, 389
HR), HQ, И*)	47
nr 358
W£)	236
TCOM(£,F) 244
TDOM(£,F) 186
TDOM 178
^DOKw	183
Q (the rational numbers)
Q[[x]]~	228
q,,s 19
R (the real numbers)
R(n) 26
Ryw	Ю5, 106
«G	no
R	158
«	155
SR£ 164
NOTATION INDEX
427
«с,	360
R(G) 211
Ric 156
pn	39; c€(p„),	Л>„, (хУа, 94, 95
Pv	16
171
$	139
J, Jc 121
«с» $c	137
[S+,S ;//],	69, 70
S® E	122
s(E)	238
s(£)	242, 388
S(£), Sc(£)	96
S°(£), Sl(E) 99, 135
S±(£), S£(E)	98, 99, 129, 136
sig(X) 137
sig(X;£) 139
sig(X,gF) 263
sig( X\ E,g)	265
SL„(R), SL„(C), SL„(H) 55
SL„(R)	56
SL*(K)	58
SO(K<?), SP(K<?)	17
SO„, SOr s 19
so„, spin,	41
sp„, su„	49
Spin(Kqi) 14
Spin„, Spinr s 19
Spin?s 56
Spin0’ 390
sym (symbol of an operator)
SymM 177
<r( )	186
<7( )	243
a<( )	113,168
)	245
о-y (elementary symmetric functions) 227
ста (Pauli matrices) 120
T" 306
TX, T*X 94
8
X(£)	241
Tdc, Td(X), Tdk 230
top-ind 244, 269, 272
top-indG 259
t(X) 339
0	93
49
Kx 16
V”{d)	88
И(Х)	58
Wj(£)	79,	373
w2(£)	81,	373
wk(£), wk(X) 85, 380
Wk(£)	383
£o	20
Ho	41
®	11,40
0м	168
xp	371
# (connected sum) 91
★	333
lx]	384
L	24
( ),	239, 241, 387, 388
( у (in theory of multiplicative
sequences) 229
О (Fourier transform) 171
(v л w)(x)	41
[v,w]	40
u*v 173
II Ik	170, 171
II He-	172
II Нк,с> 194
II 194
ERRATA
1.	In Proposition 1.3, assume к has characteristic zero.
2.	On page 11, line 24, assume that	and J?are filtered.
3.	On page 43, line 10, the Lie group is also connected.
4.	On page 192, line 22, change “Sobolev Embedding” to
“Rellich.”
5.	On page 257, lines 2 and 3 of Remark 13.11, add the following
omitted line: “Any elliptic operator P can be converted to a
pseudodifferential operator of degree zero.”
6.	On page 278, line 33, “... compact connected spin manifolds.”
7.	In Definition 3.2 of Chapter IV, the action should be effective
on each connected component of X.