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The topology of classical groups and related topics
Notes on mathematics and its applications
General editors: Jacob T. Schwartz, Courant Institute of Mathematical
Sciences, and Maurice Levy, Universite de Paris
E. Artin ALGEBRAIC NUMBERS AND ALGEBRAIC FUNCTIONS
R. P. Boas COLLECTED WORKS OF HIDEHIKO YAMABE
R. A. Bonic LINEAR FUNCTIONAL ANALYSIS
M. Davis A FIRST COURSE IN FUNCTIONAL ANALYSIS
M. Davis LECTURES ON MODERN MATHEMATICS
J. Eells, Jr. SINGULARITIES OF SMOOTH MAPS
K. O. Friedrichs ADVANCED ORDINARY DIFFERENTIAL EQUATIONS
K. O. Friedrichs SPECIAL TOPICS IN FLUID DYNAMICS
A. Guichardet SPECIAL TOPICS IN TOPOLOGICAL ALGEBRAS
M. Hausner andJ. T. Schwartz LIE GROUPS; LIE ALGEBRAS
S. Y. Husseini THE TOPOLOGY OF CLASSICAL GROUPS AND RELATED
TOPICS
P. Hilton HOMOTOPY THEORY AND DUALITY
F. John LECTURES ON ADVANCED NUMERICAL ANALYSIS
A. M. Krall STABILITY TECHNIQUES FOR CONTINUOUS LINEAR
SYSTEMS
P. Lelong PLURISUBHARMONIC FUNCTIONS AND POSITIVE
DIFFERENTIAL FORMS
H. Mullish AN INTRODUCTION TO COMPUTER PROGRAMMING
F. Rellich PERTURBATION THEORY OF EIGENVALUE PROBLEMS
J. T. Schwartz DIFFERENTIAL GEOMETRY AND TOPOLOGY
J. T. Schwartz NONLINEAR FUNCTIONAL ANALYSIS
J. T. Schwartz W-* ALGEBRAS
G. Sorani AN INTRODUCTION TO REAL AND COMPLEX MANIFOLDS
J. L. Soule LINEAR OPERATORS IN HILBERT SPACE
J. J. Stoker NONLINEAR ELASTICITY
Additional volumes in preparation
The topology of classical groups
and related topics
S.Y. HUSSEINI
Department of Mathematics
The University of Wisconsin
GORDON AND BREACH
Science Publishers
NEW YORK
LONDON
PARI S
Copyright @ 1969 by GORDON AND BREACH, SCIENCE PUBLISHERS, ] NC.
150 Fifth Avenue, New York, N. Y. 10011
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All rights reserved. No part of this book may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photo-
copying, recording, or by any information storage and retrieval system,
without permission in writing from the publishers. Printed in east Germany
Foreword
THESE NOTES are based on lectures which I gave in the spring of 1963 (and
again in the fall of 1968) to University of Wisconsin graduate students who
have previously gone through a basic course in algebraic topology. A first
version of the notes, written by R.Piccinini (to whom many thanks), was
circulated in mimeographed form in 1964. This version has been extensively
revised and expanded.
The four chapters are generally independent of one another. The exercises
which follow most of the sections are an important part of these notes: some
of them are quite easy, but, generally speaking, all are intended to augment
the material covered in the main body of the text; hints (Which often amount
to rough outlines of the proofs) are given whenever necessary.
A few of the sections are intended as brief summaries of basic material and
techniques which are needed but not necessarily covered in a first year
course. It is hoped that these summaries will serve as a guide to what is the
literature is relevant to the needs of the reader. The reader who finds himself
lacking basic tools is particularly urged to work through the excercises at the
ends of these sections rather than immediately consult the literature.
s. Y.HUSSEINI
v
Contents
Foreword
v
CHAPTER 1 Fibration and their classification
.
.
.
1
1 Fibrations 3
2 The RPT -category 12
3 The universal fibration 19
4 Representation of a loopspace
as an RPT -complex 25
5 The classification theorem, I 28
6 The classification theorem, II 32
7 The classification theorem, ITI 39
CHAPTER 2 Cohomology and homology of the classical group .
1 Graded modules and algebras 44
2 Differential graded modules and algebras 45
3 The homology and cohomology spectral sequences
of a fibration 48
4 Computation of the cohomology of the classical
groups 51
5 Hopf-algebras 55
6 The homology and cohomology algebras
of the classical groups as Hopf-algebras 58
7 Generating complexes for the classical groups 61
.
43
CHAPTER 3 The homology and cohomology of the classifying spaces
and loopspaces of the classical lie groups and Bott perio-
dicity . . . . . 68
1 The cohomology algebra H* (BUn; Z) 68
2 Characteristic classes 78
Vll
V1l1
CONTENTS
3 The homology and cohomology of the loopspaces
of the classical groups 87
4 The Bott periodicity theorem for U 90
5 The Grothendieck group K(X) 95
CHAPTER 4 K-Theory . .
1 Preliminaries 103
2 Extra-ordinary cohomology defined
by a spectrum 106
3 K-Theory 116
4 Products in K-theory 118
5 Relation of K-theory to another based
on the rationals 123
.
.
.
. 103
CHAPTER I
Fibrations and their classification
ONE OF THE primary tools of the study of the topology of Lie groups and their
homogeneous spaces is the theory of fibrations. The object of this chapter is
to offer a unified treatment of fiber bundles and more general fibrations,
leading up to a classification theorem which implies the classical classification
theorem for bundles [4, 13] as well as Stasheff's classification theorem for
Hurewicz fibrations [10].
The point of view of the treatment is as follows. Suppose that
p:X B
is a Hurewicz fibration over the connected space B; and let F be the fiber
over a point b e B. Denote by Band QB the space of Moore paths and
loops, respectively, and let
q:fJJB B
be the fibration which sends a path to its end-point. Then, using a lifting
function for p: X B, one finds a fiber-preserving map
o
(f/ B, QB) x F ---+ (X, F)
which covers the identity map B B. Moreover, the map
y(oo): F p-l (end-point of (0)
which takes e to rp (00, e) is a homotopy equivalence and, hence, induces a
map
ip:QB G(F)
which takes 00 to ip( (0), G(F) being the monoid of homotopy equivalence of F.
(If p : X B should be a K-bundle, where K is a group of homeomorphisms
of F, then ip can be factored through K.) This suggests the following pro-
gram: first, to describe a process that will replace the fiber QB of q : f/ B B
by F to obtain p : X B; and, second, to classify the fibrations p : X B by
1
2
THE TOPOLOGY OF CLASSICAL GROUPS
classifying the maps ip of QB into the submonoids of G(F). (See [3], [7], [11],
[12] for points of view which resemble in part that of the above program.)
The map y: QB G(F) is not a homomorphism but has nice multiplica-
tive properties which tell a great deal about the fibrations p: X B, as
Stasheff showed in [11] and [12]. The map
:&JBxF X
is just a rule which allows us to translate the fiber F along paths in B. Hence
it corresponds to parallel translation defined by a connection in a smooth
bundle, while the map
y: B G(F)
corresponds to the holonomy group.
Just as it is convenient in the case of manifolds to work with piecewise
geodesic paths and loops instead of all paths and loops, it is useful (and
sometimes necessary) to work with combinatorial models for path- and
loopspaces in the case of CW -complexes. Mter a brief account of the neces-
sary concepts and constructions in the theory of fibrations and bundles, we
introduce the category of complexes of the reduced product type (RPT-
complexes) in which the comtinatorial models are to be constructed. (For a
different account, see [5], [6].) An RPT-complex M is essentially a special
complex with an associative multiplication and such that the multiplication
is non-degenerate in the sense that the product of two cells is again a cell.
The appropriate action of M on a complex N is one in which the action is
non-degenerate. One is able to prove in this case that the orbit-space Nil M
is also a special complex and that the natural projection N Nil M is a
principal quasi-fibration. (This corresponds to a principal G-bundle.) If M
acts on a space F (not necessarily non-degenerately), then one can construct
a quasi-fibration
NXMF NIIM
with fiber F. The construction is completely analogous to that in bundle
theory.
In 3 we describe how a given RPT-complex M determines a principal
quasi-fibration
M 81M fJlM,
where 81M is a right M-complex and 81M is contractible. (This is analogous
to a universal bundle for a group G.) A slight modification of this construction
FIBRATIONS AND THEIR CLASSIFICATION
3
allows us to construct an n-universal bundle for a group G of the homotopy-
type of a special complex.
In 4 we show that if B is a simply connected special complex, then there
is an RPT-complex M and a homomorphism
ip:M DB
which can be extended to a fiber-preserving map
q;: M -+ B
of a universal M-fibration to B, which covers a map homotopic to the
identity map B B. These are the desired combinatorial models.
In 5 we show that if p : X B is a fibration, then there is a fiber-preserv-
Ing map
: ( M, M) x F (X, F)
covering a homotopy equivalence B B such that the induced map
y:M G(F)
is a homomorphism. This implies that induces an equivalence of fibra-
tions
fJlMXMF X,
where 81M x M F is the fibration obtained from fJlM B by replacing M
by F. If B is a finite complex, and p : X B is a bundle, one shows that the
equivalence 81M x M F X is one of bundles.
In the remaining sections we classify the fibrations p : X B by classify-
ing the homomorphisms'Y : M G(F). The usual classification theorems, in
terms of homotopy classes of maps [B, BG(F)] (or [B, B K ] in the case of
bundles) are easily deduced.
1 Fibrations
A map p : X B of topological spaces is said to satisfy the covering homo-
topy property (CHP for short) for a space K if, and only if, for all maps
g: KxI Band g: Kx {O} X such that glKx {O} = pg, there is a map
G: K x I X such that pG = g and GIK x {O} = g. In other words, a
homotopy in B which can be lifted at one end can be lifted all the way. A
map p : X B which satisfies the CHP for all spaces K is called a Hurewicz
4
THE TOPOLOGY OF CLASSICAL GROUPS
fibration. A map p : X B which satisfies the CHP for finite polyhedra only
is called a Serre fibration. In either case, X is called the total space of the
fibration; B, the basespace; and Fb = p-l(b), the fiber at b E B.
Now suppose that B is path-connected. By considering a path in B as a
deformation, one can easily show, by appealing to the CHP, that the spaces
F b , for two different b's, are of the same homotopy type. In this case we
usually choose a fixed Fb as a representative F and call it the fiber of the given
fibration p : X B.
When is a given map P : X B a Hurewicz fibration? A very useful criterion
is a rule which allows us to assign, in a continuous manner, to a path in B a
path in X with a pre-assigned initial point. Such a rule is called a lifting
function. To formulate this concept precisely, consider the space
B = {(ro, x)lx E X, ro(O) = p(x)} c B I X X,
where B I is the space of paths in B with the compact-open topology. Denote
by
p: Xl B
the map which sends ro to the pair (pro, ro(O)). A lifting function for p : X B
is, by definition, a map
A: B XI
such that PA = identity.
THEOREM (1.1) A map
p:X B
is a Hurewicz fibration if, and only if, it admits a lifting function A.
Proof Consider the map
g: B xI B
which sends (ro, x, t) to ro(t). Note that the map
g: B x{O} X,
which sends(ro, x, 0) to x, is a lifting of glB x {O}. Hence the CHPimplies that
there is a map
G: B xl X
FIBRATIONS AND THEIR CLASSIFICATION
5
such that pG = g. Observe now that the adjoint map
A:B XI,
which sends (co, x) to the path t G (co, x, t), is a lifting function.
Q.E.D.
One constructs a lifting function by collating a collection of local lifting
functions. This is essentially the gist of the following theorem.
THEOREM (1.2) A map
p:X B,
with B paracompact, admits a lifting function if, and only if, there is a cover
{U } of B by open sets U(I. such that, for all ()("
I -I U . -I U U
pp (I..p (I. (I.
has a lifting function.
To prove the theorem one assumes (without any loss of generality since B is
paracompact) that the cover {U(I.} is locally finite and that for each ()(, there is
amap
frx.:B I
such that U(I. = {b E BIF(I.(b) O}. Given this situation, one can lift the local
lifting functions together in a standard manner. For details, see [9].
Two important corollaries follow immediately. First we need to introduce
two new concepts. We shall say that a map p: X B is a local Hurewicz
fibration if, and only if, there is an open cover {U(I.} of B such that, for each
Uo., the map
plp-l U(I.: p-l U(I. U(I.
is a Hurewicz fibration. We shall say that p: X B is locally trivial if, and
only if, there is a cover {U(I.} of B by open sets U(I. such that, for each U(I.'
there is a homeomorphism
({J(I.: U(I. x F p-l U(I.
which is fiber preserving and covers the identity map of U(I..
COROLLARY (1.3) Suppose that
p:X B
is a map and that B is paracompact. Then p is a Hurewicz fibration if, and only
if, p is a local Hurewicz fibration.
6
THE TOPOLOGY OF CLASSICAL GROUPS
COROLLARY (1.4) Suppose that
p:X B
is a locally trivial map with B paracompact. Then p is a Hurewicz fibration.
Suppose that
p:X B
is a fibration with fiber F. A locally trivial cover for this fibration is defined
by a pair
0/./ = {U lX , lX E J} and tP = {qJlX' lX E J},
where tft is an open cover for B and each qJlX is a fiber-preserving homeomor-
phism
qJlX: U lX x F p - 1 U lX
which covers the identity map of Ua,. By definition, let
qJa"x = qJa,lx x F and lfJlXfJ = qJ -1 a"x qJfJ,x,
and let G denote a group of homeomorphisms of F. Then we say that a locally
trivial cover (0/./, tP) defines a G-bundle structure on
p:X B
if, and only if, the homeomorphism
qJa,fJ(x): F F
is in G and the map
qJa,fJ: Ua, n UfJ G,
which takes x to qJa,fJ(x), is continuous. A fiber bundle is a fibration with a
maximal bundle structure. We shall denote the bundle by the quintuple
E = (X,p, B, F, G).
Observe that the map
CfJa"x: F p-l(X)
is a homeomorphism. Moreover, the set of homeomorphisms
tP; = {qJa,lqJa,:F p-1X}
FIBRATIONS AND THEIR CLASSIFICATION
7
is such that
i) q;fY..:X; e q) ;
ii) q) G c q) ; and
iii) given q;, q;' e q) , there is a unique g e G such that q;g = q;'.
Two bundles E = (X,p, B, F, G) and E' = (X',p', B, F, G), with the same
fiber F, group G, and basespace B, are said to be equivalent if, and only if,
there is a fiber-preserving homeomorphism
f: X X
which covers the identity map of B and is such that
'
fq):x; = q):x;, all x e B.
A very important G-bundle is one such as E = (X, p, B, F, G), in which the
fiber is the group G itself and the action of G is by translation on the left.
Such a bundle is called a principal G-bundle. One can easily show that G acts
freely on the right of X and that the orbit space is B.
Now let Fbe a space on which G acts on the left as a group of home om or-
phisms. Consider the cartesian product X x F, and make G act on X x F ac-
cording to the rule
g(x,e) = (xg,g-le), xeX, eeF, ge G.
I
Denote the orbit space by Xx G F. Note that the natural projection XxF
B induces a projection
p' : X x G F B.
Moreover, one can easily show that the map p' is a fiber bundle map with
fiber F and group G. Denote this G-bundle by
Ex GF = (Xx G F,p', B, F, G).
We shall say that E is the principal bundle associated with Ex G F.
Conversely, any bundle E' = (X',p', B, F, G) is obtainable up to equi-
valence from an associated principal G-bundle in the manner described
above. To see this, let {Ua" (X. e J} be a locally trivial cover with coordinate
functions
CPa.: Ua, x F --. p'-l Ua,.
Consider next the disjoint union U Ua, x G. Define an equivalence relation on
(I
8
THE TOPOLOGY OF CLASSICAL GROUPS
U Uq. xF by the condition that (x, g) E Uq. x G be equivalent to (y, g')
a
E UfJ x G if, and only if, x = y and g' = ([Jq.fJ(x) · g.
Denote the quotient space by X. Observe that there is a natural projection
p : X B which is a fiber bundle map for the principal bundle
E = (X,p, B, G, G)
and that the given G-bundle ' is equivalent to x G F.
A very important class of fiber bundles is that in which the fiber is a real
Rn, complex en, or quaternionic Hn vector space and the group is the cor-
responding general linear group GL(Rn), GL( en), or GL(Hn). Such bundles
are called real, complex, or quaternionic vector bundles.
It is possible to carry out certain operations on vector bundles similar to
those on vector spaces. Let us assume from here on that the vector bundles
are complex. A similar description holds for real and quaternionic bundles.
The Whitney sum E E' of two vector bundles E = (X,p, B, cm, GL(C m ))
and E' = (X',p', B, cn, GL(C n ) over the same basespace B is the vector
bundles whose total space is
X E9 = {(x, u, u')lx E B, u E p-1b, u' E p'-lb}.
This set can be topologized so that the natural projection
P E9 ': X E9 ' B,
which sends (b, u, u') to b', is the projection of a fiber bundle
E E' = (X E9 " P E9 " B, c n + m , GL (c n + m )).
Similarly the tensor product
E <8> ' = (X " P @ " B, e mn , GL (e mn ))
is the vector bundle whose total space is
X @ , = (b, u <8> u')lb E B, U Ep-1b, u' E p'-lp}.
The exterior product
E A = (E A ,P A ' B, e n (n+l)/2, GL n (n+l)/2(C»)
and the dual
E* = (E *, p[, B, en, GLn(C»)
are defined in a similar manner.
FIBRATIONS AND THEIR CLASSIFICATION
9
A vector bundle
E = (E ,p , B, en, GLn(C»)
admits a Hermitian metric in the sense that each fiber can be given a Hermi-
tian metric as a vector space and the metrics vary continuously as the fiber
moves continuously over the basespace. To see this, form the new bundle
Herm (E) = (E . ,p . , B, e n (n+l)/2, GL n (n+l)/2)
where
E . = {(x, lXx)lx E Band lXx a Hermitian metric for the fiber at x}.
The fiber is contractible, and hence the bundle Herm (E) admits a section,
that is to say, a Hermitian metric in the desired sense. The associated sphere
bundle
S( ) = (Es( ), Ps( ), B, S2n-l, Un),
where Un is the unitary group in n letters, is defined by letting
Es( )= {(x, U)IXEB, UEp-1X of lengthl}.
The associated disk bundle is defined the same way.
Remark (1.5) A map
p:E B
which is a fibration in the sense of Serre has the following property: p induces
an isomorphism
p*: ni (X, F b , e) ni (B, b), i > 0,
(*)
for all b E Band e E Fb (F b being the fiber at b). A map p: E B which has
property (*) is called a quasi-fibration. Very often one is led to maps which
are at best quasi-fibrations.
Problems
The relative covering homotopy property
1 Suppose that p: X -+ B is a Hurewicz fibration, and let g: Kx {OJ U LxI -+ X be a
map where K is a CW-complex and L is a subcomplex. Denote pg by i, and assume that there
is a map G : KxI -+ B such that G IKx {OJ U LxI = g. Show that there is a map G: KxI
-+ X such that G IKx {OJ U LxI = g andpG = G .
1 Husseini (0216)
10
THE TOPOLOGY OF CLASSICAL GROUPS
Quasi-fibrations
2 Given any map p : X -+ B, find a Hurewicz fibration p' : X' B' and homotopy equi-
valences f: X -+ X' and g: B -+ B' such that p'f = gp. In particular, a quasi-fibration is
"equivalent" to a Hurewicz fibration.
3 Give an example of a quasi-fibration p : X -+ B which is not a Hurewicz fibration.
4 Show that for a quasi-fibration p : X B all fibers have the same weak homotopy type
if B is path-connected.
\
Principal fiber bundles
5 A principal fiber bundle = (X,p, B, G, G) is trivial if, and only if, there is a map
s: B -+ X such that ps = identity. (The map s is called a section for .)
6 Show that any bundle = (X, p, B, F, G) is trivial if its basespace is contractible.
A bundle map £x: fJ -+ ' of the principal G-bundle = (X,p, B, G, G) to the principal
G' -bundle ' = (X', p', B', G', G') is given by a pair of maps (f, e) where f takes X -+ X'
and e is a homomorphism of G into G' such that
f(xg) = f(x) e(g), for all x e X and g e G.
A bundle map £x: -+ ' of the bundle = (X,p, B, F, G) to the bundle ' = (X',p',
B', F', G') is given by a triple (f, e, h) where (f, e) is a map -+ ' of the associated principal
bundles = (X, p, B, G, G) and ' = (X', p', B', G', G') and h is a map of F into F' such
that the triple (f, , h) induces a map Xx GF -+ X' x G,F'. (Note that, in particular, h (g. y)
= (g) . y for all g E G and y E F.
7 Let = (x,p, B, F, G) and ' = (X',p', B', F', G') be two bundles. By definition, let
XHom( , ') = {(b,b',£x)lb eB,b' eB',£x: ').
Give X Hom( . ') a natural topology so that the projection
PHom( , '): XHom( , ') -+ BxB'
which takes (b, b', £x) to (b, b') is continuous and is the projection for a suitable fiber bundle
Horn ( , '). Show that if B = point, F = F', and the maps (f, , h) = £x: -+ ' are
those for which h is a homeomorphism, then Horn ( , ') isjust the associatedprincipal bundle
of '.
The classical groups and some of their fibrations
Let F denote the real numbers R, the complex numbers C, or the quaternions H. Assume
that the vector space of n-tuples Fn is given the usual inner product, and let On(F) be the
group of automorphisms of Fn which preserve this inner product. Put e 1 = (1, 0, ..., 0), . . . ,
-; 2 = (0, . .., 0, 1), and imbed On-l (F) in On(F) as the subgroup of On(F) which leaves '8 n
fixed. On(R) is the orthogonal group On, On( C) the unitary group Un, and On(H) the symplec-
tic group SPn. The unit sphere is F n in 8"d-l, where d = 1, 2 and 4 when F = R, C, and H,
respectively.
FIBRATIONS AND THEIR CLASSIFICATION
11
8 Show that the map
p: On(F) -+ snd-l
which takes a to a(i n ) is a principal fiber bundle with group On-l(F).Hence On(F)IOn-l(F)
= snd-l.
[Hint. Construct a section for p-l (sn d - l - {8n}) by sending x e snd-l to the rotation in
the {x, -; n}-plane which takes '8 n to x.]
Let GN,n(F) be the Grassmann manifold of n-planes thro 0 e F"; and ON,n(F), the
Stiefel manifold of orthonormal n-frames in Fn. An element! e ON,n(F) is uniquely deter-
- -
mined by a set of n orthonormal vectors fl, ..., In e Fn . We topologize ON ,n(F) using this
identification. There is a natural map
p : ON,n(F) -+ GN,n(F)
which sends a frame f to the plane it generates. Assume that G N ,n(F) has the quotient
topology .
9 Show that the map 0N(F) ON,n(F) which sends a to the frame {a -n+l'...' a }
induces a homeomorphism
ON(F)ION-n(F) ON,n(F).
Show also that the map 0N(F) -+ GN,n(F) which takes a to the plane generated by {a - n+1t
. . ., ae N} induces a homeomorphism
0N(F)ION-n(F) X On(F) GN,n(F).
Check that p: ON,n(F) -+ GN,n(F) is the map which sends a coset of 0N-n(F) to a coset of
ON-n(F) X On(F), and deduce that p is the projection of a principal On(F)-bundle.
10 By definition, let
EN,n = {(x, u) I x e GN,n(F), U a unit vector in x} .
Show that EN,n is homeomorphic to ON,n(F) and that the map ON,,,(F) -+ GN,n(F) of the pre-
ceding problem is just the projection EN,n GN,n(F) which sends (x, u) to x.
Projective spaces
Observe that ON,l(F) = SNd-l and GN,l(F) = PN-l(F), the (N - I)-projective space (real,
complex, or quatemionic).
11 Show that
Pn(F) = Pn-l(F) U Pn_1D"d,
where Pn-l is the fibration snd-l -+ Pn-l(F) defined in Problem 9 and D nd is the standard
nd-dimensional Euclidean disk.
[Hint. Let [xo, ..., x n ] be the set of(n + I)-tuples in F n + l of the form (cxo, ..., cx n ) for
some nonzero c e F. Observe that [xo, ..., x n ] is the set of direction numbers for a line
through 0 e pn+l. Define Vi = {£x e Pn(F) I Xi(£X) # O} . Show that (i) Vi is homeomorphic to
12
THE TOPOLOGY OF CLASSICAL GROUPS
F(n-l)d and (ii) the mapping cylinder of Pn-l : snd-l -+ Pn-l (F) is homeomorphic to a com-
pact neighborhood U of Pn-l (F) in Pn(F) such that Pn(F) - interior (U) is homeomorphic to
Dnd, the nd-dimensional Euclidean disk.] .
By induction one shows that, as a cell complex,
P n( F) = eO U e d U .. . U end.
2 Tbe RPT -category
Recall that a special complex is a countable CW -complex with a single 0-
dimensional cell which is usually taken to be the basepoint. I. M. James
introduced in [8] the following construction for the study of the loop-spaces
of suspensions. Suppose that X is a special complex, and let e E X be its
single O-dimensional cell. Consider
X OO = U xn,
n l
where xn is the n-fold cartesian product of X with itself and is imbedded in
xn+ 1 as the subcomplex consisting of all (n + I)-tuples whose last entry is e.
We shall say that two tuples
(J) = (Xl' ... , x m ) and 'YJ = (y 1, ..., Y n)
of X oo are equivalent if, and only if, the tuple obtained from (J) by leaving
out e is the same as the tuple obtained from'YJ by leaving out e. Denote the
quotient space of X by X 00' and that of xn by Xn. Observe that the spaces X n
give an ascending filtration of X 00 ,
X C X 2 c... c Xc... c X
n 00'
and that a subset C c X 00 is closed in X 00 if, and only if, C n X n is closed in
X n , for all n > I. Moreover, the cellular structure of X induces a cellular
structure on X n , all n > I, with which X n becomes a subcomplex of the CW-
complex X n + 1. Note that
Xn+l = X n U Xx... xX
where the (n + I)-fold Cartesian product Xx... xX is joined by the map
{e} xXx... xXU... U Xx... xXx{e} X n
which sends an n-tuple in which at least one entry is e to its representative in
Xn. Observe now that the juxtaposition in X OO induces an associative multipli-
cation
#: Xoo xXoo Xoo,
FIB RATIONS AND THEIR CLASSIFICATION
13
for which e is the two-sided identity. It is easy to check that Xoo has the
. ...
following property. Let (X oo)(r) denote the r-skeleton of X 00' and form the
special complex
[(X oo)(r), (X oo)(S)] = (X oo)(r+s-l) U (X oo)(r) X (X oo)(S),
where (X oo)(r) X (X oo)(S-I) is attached by the map
(X oo)(r) X (X oo)(S-l) U (X (0)(r-1) X (X oo)(S) (X cxJ(r+s-1)
induced by fl. The property is that fl induces an imbedding of [(X oo)(r), (X oo)(S)]
as a subcomplex nto Xoo,for all r, s > O. James called Xoo the reduced pro-
duct complex of X.
It turns out that this construction is one of many. Suppose that M is a
special complex, and let mo be its basepoint. Assume that
'YJ: MxM -+- M
is an associative multiplication with mo as a two-sided identity. We assume
also that 'YJ is cellular. Denote by M(r) the r-skeleton of M, and form the
complex
[M(r), M(r')] = M(r+r' -1) U M(r) X M(r')
by attaching M(r) X M(r') to M(r+r' -1) with the map
M(r) XM(r'-I) U M(r-l) xM(r') M(r+r'-I),
which is induced by the given multiplication 'YJ. We shall say that M is a
complex of the reduced product type (an RPT-complex for short) if, and only
if, 'YJ induces an imbedding of [M(r), M(r')] into M, for all r, r' > O.
How does one construct RPT-complexes? Suppose that M is an RPT-
complex, and let if be a special complex which contains M as a subcomplex.
THEOREM (2.1) M can be uniquely imbedded in an RPT-complex M' which
contains M as an RPT-subcomplex. Moreover, any map
i: M G
of if into an associative H-space G such that ilM is a homomorphism can be
extended uniquely to a homomorphism
f' : M' G
of M' into G. Iff is cellular,f' is cellular also.
14
THE TOPOLOGY OF CLASSICAL GROUPS
Proof By definition, let
M = MUMxMxM,
where M x M x M is attached to if by the map
MxMxM U {mo} xMx{mo} -+- M.
Clearly M is a special complex. By definition, the element [m 1 , ... , XiXi + 1,
mi+2, ..., m r ] is an amalgam of [m 1 , .o., Xi' Xi+l, mi+2, ..., m r ], where Xi
and Xi+l are in M. This notion defines an equivalence relation on (M co).
Let M' be the quotient space. Again M' has a natural ascending filtration
M c (M)2,M C o.. C ( M )n,M c... c" M',
where ( M )n,M is the image of ( M )n. Observe that
( M )n,M = ( M )n-l,M U MxMxMxMx ... xMxMxM,
where the cartesian product M x M x M x M x... x M x M x M in which M
appears n times alternately with M is joined to ( M) n-l,M by the map
MxMxMxMx... xMxMxM U ... U MxMxMx...
xMxMxM -+ ( M )n-l,M
which sends a tuple containing at most n - 1 entries of M - M to its
representative. With the help of this filtration, one can show that the cellular
structure of M induces a cellular structure on M', with which M' becomes an
RPT-complex.
Q.E.D.
We shall call M' the RPT-complex generated by M relative to M, or the
RPT-complex generated by M and M.
Any RPT-complex can be obtained from a suitable Xco by a repeated
application of Theorem (2.1). To see this, let M be an arbitrary RPT-com-
plex, and let X c M be the largest subcomplex of M all of whose cells are
indecomposable. Then X generates a subcomplex of M which is just X co' the
reduced product complex of X described above. Observe also that 'YJ induces
on X co a multiplication which is just the usual juxtaposition of sequences
with e as the identity.
Next put oM = X co' and let 1 Mbe the smallest subcomplex of M generated
by 0 M and the cells of lowest dimension in M - 0 M. Repeating this, we
FIB RATIONS AND THEIR CLASSIFICATION
15
obtain an ascending sequence of RPT-complexes
Xoo = oM c 1M c ... c nM c ... c M
(2.2)
such that nM is the subcomplex of M generated by n-IM and the cells of
lowest dimension in M - n-IM. This representation of M as an ascending
sequence of RPT -complexes is not unique, but it is, as are all representations
of its kind, extremely useful in constructions involving M, inasmuch as it
gives us an inductive procedure for building up M.
An M-complex can be defined in an obvious fashion. Suppose that N is a
special complex which contains M as a subcomplex. Assume that M acts on
N associatively, with the action extending the given multiplication on M.
Denote the action by
cp: NxM N,
and assume that it is cellular. Denote by N(r) the r-skeleton of N. Consider
the complex
[N(r), M(r')] = Nr+r' -1 U N(r) X M(r'),
where N(r) X M(r') is joined to N(r+r' -1) by the map
N(r-l) X M(r') U N(r) X M(r' -1) N(r+r' -1)
induced by the given action cp. We shall say that N is a right M-complex if,
and only if, cp induces an imbedding of [N(r), M(r')] into N, for all r, r' > o.
A left M-complex can be similarly defined.
As an example, let if be any special complex containing M as a subcom-
plex. Form the complex
[ N, M] = N U N xM,
where N x M is attached by the map N U M x M N, which is the identity
on N and the multiplication on M x M. It is clear that [ N, M] is a right M-
complex, with M acting on the right factor.
Now any right M-complex N can be described as an ascending sequence of
right M-complexes of this type. To see this, define first of all oN to be M
itself. Let I N be the smalles subcomplex of N which contains oN and the cells
of minimal dimension in N - oN. Form the complex
IN = [I N, M] = oN U I N xM,
where 1 N x M is attached by the map oN x M oN induced by the given
action cp. Observe that IN = [I N, M] has a natural M-complex structure
16
THE TOPOLOGY OF CLASSICAL GROUPS
which coincides with that inherited from N. Proceeding in this fashion, we
can define a sequence of ascending M complexes
M.= oN c IN c... c nN c... c N
(2.3)
such that
nN = n-IN U n N xM n > 0
, -,
(2.4)
where nN is the smallest subcolnplex of N containing n - 1 N and the cells of
minimal dimension in N - n-l N, and the attaching map is that induced by
the given action cp.
The following discussion will show some of the uses of this representation
of N. First we need the following concept. Observe that an equivalence rela-
tion is generated on N by the condition that two elements n, n' E N be equi-
valent if, and only if,
n = m'n' or n' = mn
,
for some m, m' E M. Denote the orbit space by Nil M and the natural projec-
tion by
p:N NIIM.
One can easily show, using the description of N given in (2.3) and (2.4) above,
that Nil M is a special complex and that p is cellular.
Moreover,p satisfies that property for quasi-fibrations which corresponds
to a principal fiber bundle. The precise notion is given by the following
definition. A quasi-fiber map p : X B is said to be a principal quasi-fibra-
tion for the monoid M if, and only if, there is an associated action
'YJ:XxM X
with the following properties:
i) (x, e) = x, all x E X, e is the O-cell of M;
ii) 'YJ (x xM) C p-lp(X), all e E X; and
iii) the map M p-Ip(e) which takes x to 'YJ (e, x), all x E M, is a weak
homotopy equivalence.
THEOREM (2.4) The projection
p:N-+N/lM
is a principal quasi-fibration for M.
FIBRATIONS AND THEIR CLASSIFICATION
17
The proof of the theorem depends on the following two basic lemmas in
quasi-fibration. First we need the following notion. Suppose that p : X B
is a quasi-fibration. An open set U c B is said to be distinguished if p : p-l U
U is a quasi-fibration.
LEMMA 1 Suppose that p : X is surjective. If B = U U V and U, V, and
U n V are distinguished, then B is also distinguished.
LEMMA 2 Suppose that p: X B is a map; let B' c B be a distinguished
subset, and put X' = p-IB'. Assume that there are deformations Dt: X X
and d t : B B such that
Do = 1, Dt(X') c: X', DI(X) eX',
do = 1, dt(B') c B', dl(B) c B',
pD I = dIP,
and, for all x E Band i 0, the induced homomorphisms
DI* : :Jti (P-IX) :Jti (p-l (dlx)).
Then B is distinguished.
See [2] for details.
Proof of Theorem (2.4) Consider the sequence (2.3). oN = M; hence
oN II M = e, the O-cell of Nil M; and hence,
Po: oN °NIIM
is trivially a quasi-fibration. Suppose that we have proved that the map
Pm: mN mN//M, Pm = plmN,
is a principal quasi-fibration for M. We wish to do the same for the case
m + 1. Note that
m+IN_mN= UDnr xM
where Dnr ranges over the cells of m + I N - m Nand b k denotes the interior of
D k . Moreover, we can write
m+IN//M = U U V,
where
V = Pm+ 1 (D"r(i) x M)
2 Husseini (0216)
18
THE TOPOLOGY OF CLASSICAL GROUPS
(Dnr(!) being the nr-disk of radius !), and
U = Pm+I (m+IN(!) xM),
where
m+IN(!) = mN U U (Dnr - b nr (!)) xM.
Clearly V is a distinguished set. Hence, to finish the proof we need to show
that U is also distinguished, since Un V is distinguished (Lemma 1). But,
according to Lemma 2, we need prove only that the map
JIm: M P I (Pm + 1 (n))
which takes x to n · x, for all n E m+ IN, is a homotopy equivalence. Observe
that
P I(pm(n)) = P 1 (Pm + I(n))
and Pm = Pm+ 1, if n E m N. In this case the induction assumption implies that
Ym is a homotopy equivalence. If n is in m+IN - mN, then
n = (n, x),
for some n E m+I N - mN and x in M. Hence Yn takes x' to ( n, xx'). Since M
is arcwise connected and
P I (Pm + 1 (n)) = n xM,
Yn is a homotopy equivalence. This establishes the induction step and, hence,
the theorem.
Q.E.D.
There is another very useful construction, maqe possible by (2.1), (2.2)
and (2.3), which allows us i a sense to replace the fibers of the quasi-
fibration
p:N N//M
by something else on which M also acts on the left. To describe this construc-
tion, suppose we are given another left M-complex N'. Consider the equi-
valence relation defined on N x N' by the condition that (nm, n') be equi-
valent to (n, mn') for all n E N, n' E N', and m E M. Denote the quotient space
by Nx M N'. It is quite easy to show, by appealing to (2.2) and (2.3), that
N x M N' is a special complex. By an argument similar to that of Theorem
(2.4), one can show that the natural projection
NxMN' N//M
is a quasi-fibration with fiber N'.
FIBRATIONS AND THEIR CLASSIFICATION
19
Problems
Suppose that M is an RPT -complex, and let M be a special complex containing M as a
subcomplex. Then the complex
[M, M, M] =MUMx M xM
obtained by attaching MxMxM to M by the map MxMxM -+ M induced by the
multiplication in M is a two-sided M-complex. Now denote [M, M, M] by M{ and, by
definition, let (MDl,M = M{ and
(MDk,M = (MDk-l,M U M{ x M ... x MM{, k 2:: 2,
where the k-fold product M x M ... X M M is attached to (MDk-l,M by the natural pro-
jection
MxMM{xM ... xMM U ... U M xM ... xMM{xMM-+ (MDk-l,M.
1 Show that (MDk-l,M is a two-sided M-complex and that
(M{)OQ,M = U (M{)k,M
k l
is the RPT-complex generated by M relative to M (cf. Theorem (2.1)).
2 Suppose that G is an associative H-space, and let p : X -+ B be a principal quasi-fibration.
Show that the map
g : X U TX X G -+ B Up TX
induced by projection on the first sector is a principal quasi-fibration. Here TXx G is joined
to X by the action Xx G -+ X of X on G.
3 Using the result of Problem 2, show that an associative H-space G has a principal quasi-
fibration.
p : EG -+ B G ,
where EG is contractible.
[Hint. Define EG as union U En' where E 1 = G and En = En-l U TEn-l X G.]
n l
3 The universal fibration
The object of this section is to show how one can attach to a given RPT-
complex M certain fibrations which playa universal role in the classification
of M-fibrations. First we need the following concept.
A principal quasi-fibration X B is said to be universal if, and only if, X is
contractible. Now suppose that the given RPT-complex M is of the form
Aoo, where A is a special complex. By definition, let
&6(Aoo) = Aoo U TA xAoo,
(3.1)
20
THE TOPOLOGY OF CLASSICAL GROUPS
where T A is the reduced cone on A and the attaching map ex : A x Aoo Aoo
is that given by the multiplication in Aoo. To show that
&6(Aoo) = Aoo UTA x Aoo
is contractible, consider the ascending filtration of BI(Aoo) by the complexes
[TA, Ar] = [TA, A r - 1 ] U TA xAr, r > 1,
where the attaching map T A x Ar-l [T A, A r - 1 ] is that induced by the
right action of Aoo on &6(Aoo). Observe that
[TA, A] = TA U TA xA
and, hence, is contractible. It is quite easy to see that [T A, A r - 1 ] is a defor-
mation retract of [TA, Ar]. But &6(Aoo) is a CW-complex; hence &6(Aoo) is
contractible.
It is clear that Aoo acts on f!A(Aoo) on the right and that
&6(A) = &6(Aoo)// Aoo = S A A,
where S A A is the reduced suspension of A. Therefore, Theorem (2.4) implies
that the natural projection
PAoo: &6(Aoo) &leA)
is a quasi-fibration. Since &l(Aoo) is contractible, it follows that P : f!A(Aoo)
&6(A) is a principal quasi-fibration.
Since an arbitrary RPT -complex M can be thought of as being formed by
starting with an RPT -complex of the form X 00 and then adding one generator
cell at a time, one can show the existence of a universal quasifibration for any
RPT-complex inductively, provided the following has been established.
Suppose the given RPT -complex M contains an RPT -complex M' with
the following properties:
i) there is a universal quasi-fibration
P' : OlI(M') OlI(M')
for M' with OlI(M') being an M'-complex; and
ii) M - M' has only one indecomposable cell e (therefore M is generated
by M' and e r ).
FIBRATIONS AND THEIR CLASSIFICATION
21
Now consider the M-complex dI/(M') x M' M, and note that
(dI/(M') x M,M)!I M = dI/(M')!I M' = dI/(M').
Hence the natural projection
p' : dI/(M') x M,M -+ dI/(M')
is a quasi-fibration. Let Dr denote the standard euclidean r-dimensional disk,
and let
I: aDr -+ M'
be the attaching map for ere (Here aDr denotes the boundary of Dr.) Since
OI/(M') is contractible, there is a map
1+ : D -+ dI/(M')
such thatl+laD ..is the composite of I and the injection M' -+ dI/(M'). Now
let
f- : D'- -+ dI/(M') x M,M
denote the composite of the map which takes D'- onto e and the injection
M -+ dI/(M') x M' M. Observe that 1_ and/+ together define a map
/: aDr+l -+ tJIJ(M') x M,M.
Consider next the complex
dI/(M) = (dI/(M') x M,M) U Dr+l x M,
where the attaching map aDr+ 1 x M -+ dI/(M') x M' M is that which sends
(x, y) to lex) y. To show that dI/(M) is contractible, we use a filtration of
OI/(M) similar to the filtration of &6(AC()) given above. According to Problem 1
of 2,
M = U (M1)k,M',
k l
where M 1 = [M', M' U e r , M'], and (M1)k,M' is the k-fold reduced product
complex of M 1 relative to M'. Hence
dI/(M') x M,M = U dI/(M') X M,(M1)k,M'.
k l
Now consider the subcomplexes
OI/k(M) = (dI/(M') x M,(M1)k,M') U Dr+l x(M1)k-l,M', k > 1.
22
THE TOPOLOGY OF CLASSICAL GROUPS
Clearly O//I(M) = (O/t(M') x M' M l ) U Dr+l is contractible, and O/t k - l (M) is
a deformation retract ofO/tk(M). This is enough to show that O/t(M) is con-
tractible, since
O/t(M) = U O/tk(M).
k 1
Denote the orbit space O/t(M)/I M by O/t(M). Note that
dI1(M) = O/t(M) U Dr+ 1
where the attaching map is the composite p'l. Also note that, according to
2, the natural map
p : O/t(M) O/t(M)
is a quasi-fibration.
Summing up, to construct a universal quasi-fibration for an RPT-com-
plex M, we proceed as follows. First, we find an ascending sequence of
RPT -complexes
OM c: 1M c: ... c: iM c: ... c: M
such that (i) oM = ACX) for a suitable special complex A, and (ii) i M is
generated by i- 1 M and the cells e" i of minimal dimension in M - i-I M.
Then, procee ding as above, we construct for each i M a universal quasi-fibra-
tion
Pi: f!leM) f!l( iM)
(3 .2a)
where f!le M) is the right complex;
(f!le- 1 M) x i-1 M iM ) U U (D"i x iM)j, i 1,
j
(3.2b)
where j ranges over an index set in one-to-one correspondence with the
number of irreducible cells of i M - i-1M; and
f!l(O M) = ACX) UTA x ACX),
where TA is the reduced cone of A.
The following theorem has thus been proved.
THEOREM (3.3) Suppose that M is an arbitrary RPT-complex. Then there is a
universal quasi-fibration
(3.2c)
p : f!l(M) f!l(M).
Moreover, the cells of f!l(M) are in one-to-one correspondence with the inde-
composable cells of M.
FIB RATIONS AND THEIR CLASSIFICATION
3
Problems
Suppose thatp: A -+ Bis a principal quasi-fibration for the RPT-complex M. We shall say
that p : A -+ B is locally trivial if, and only if, for every point b e B there is an open neigh-
borhood U b of b and an injective M-map
({Jb : U b x M -+ p-l(U b ).
1 Prove that an RPT-complex M possesses a universal principal quasi-fibration
p : qj(M) -+ qj (M)
such that qj(M) is a right M-complex, qj(M) = qj(M)IIM, and that, for all finite subcom-
plexes K of qj(M) ,
p :p- 1 K -+ K
is a locally trivial principal quasi-fibration for M.
[Outline of proof Since M can be thought of as the union of an ascending sequence of
RPT -complexes, each one of which is generated by the preceding one and a single indecom-
posable cell, an inductive proof can be constructed, provided that one can show the follow-
ing. Let M be generated by an RPT-subcomplex M' c: M and a cell en. Suppose that
p' : qj' -+ qj '
is a locally trivial universal quasi-fibration for M', with qj' a right M' -complex and
qj = qj' 11M'. To construct a locally trivial universal quasi-fibration for M, consider
p: qj' xM,M -+ qj' .
Exactly as in the proof of Theorem (3.3), find a map
f:iJD n + l = sn qj'xM,M
which is equal to the characteristic map f of en on one hemisphere and to a null-homotopy
of f: sn-l -+ M' c: qj' on the the other. Set g = pi, and consider
qj " = qj' U g SnxI,
the mapping cylinder of g. Denote by r the canonical retraction of qj" onto qj' . By
definition, let
qj" = {(x,a)lr(x) = p(a)}c: qj "xqj'.
Now, using Lemma 2 of 2, one shows that the natural projection
p" : qj" qj "
is a locally trivial principal quasi-fibration for M. Next consider the map
q;: SnxM -+ qj"
which sends (x, m) to (x,f(x) . m), and form
qj(M) = qj" Utp D n +l xM.
24
THE TOPOLOGY OF CLASSICAL GROUPS
To finish the proof, show that tl/I is contractible and that the natural projection
p : tl/I(M) -+ tl/I( M) = tl/I U g D'J+l
is a locally trivial principal quasi-fibration.]
2 Suppose that (X: M -+ G is a homomorphism into a group G. Then
p: tl/IxMG -+ tl/I
is a principal G-bundle. Here tl/I-+ tl/I is a locally trivial quasi-fibration/or M.
Universal G-bundles
Suppose that G is a special complex, and assume that G is a group with multiplication
p.:GxG-+G.
-
3 Show that there is an RPT-complex G such that
-
i) G is a subcomplex ofG, and
ii) there is a homomorphism
-
r:G-+G
-
such that ir id and ri = id, i being the injection of G in G.
[Outline of proof Begin with G ex)' the reduced product complex of G. The map r 1 : G ex) -+ G
which sends [Xl' ..., X m ] to X1X2 ... X m is a retraction. ImbedGex) into an RPT-complex(2) M
which is universal with respect to the property that the map of G x G -+ G ex) which takes
(x, y) to [x, y] is homotopic to the map which takes (x, y) to rl [x, y] = xy. To do this,
form
M 2 (G) = (G)2 U(X G 2 xl,
where the attaching map is defined by the rules
(X (x, 0) = x, x e G 2 ;
(X (x, 1) = rl(x), x E G 2 ;
(X (x, t) = x, X E G, tel.
Then (2) M is the RPT -complex generated by G ex) and M 2 . Now note that the retraction
r2 : Gex) -+ G can be extended to a retraction (2)M -+ G by sending [x, t] e M 2 to r2(x).
Restrict the retraction to the subcomplex (2) M3 of (2) M determined by the image of
(M 2 x G U GxM 2 ). Imbed (2)M 3 in a complex (3)M 3 which allows us to deform (2)M 3 to
G by joining x E (2)M 3 to r2(x). Let (3)M be the RPT-complex generated by (2)M and
(3) M3. In this fashion obtain an ascending sequence of RPT -complexes
(1) M = G c: (2) M c: (3) M c: ... c: (n) M c: ...
00
such that, for each (n) M, there is a retraction (n) M -+ G and (n) M is generated by (n-l) M and
a complex (n) M n which contains (n-l) Mn-l x G U (n-2) M n - 2 x 2M 2 U ... U G x n-l M(n-l)
and which can be deform-retracted onto G. Show that G = (ex) M = U (n) M is of the homo-
-
topy of G and that there is a retraction G -+ G which is also a homomorphism.]
n
FIBRA TIONS AND THEIR CLASSIFICATION
25
4 Let Pn : f!J n -+ f!J n be a universal principal quasi-fibration for (n) M which is also locally
trivial. Show that f!A = U f!J n X (n)M G -+ U f!An is a universal quasi-fibration for G.
n n
5 Show that
f!J n X (n)M G -+ f!J n
is aprincipalG-bundle and that Tti (f!J n X (n)MG) = v, for i < n. ( f!J n is called the n-projective
space of G.)
6 Show that n = CP n or HP n ifG = Sl or S3, respectively. What is f14nx(n)MG in these
cases?
4 Representation of a loopspace as an RPT-complex
Suppose B is a simply connected special complex, and let Q(B) and (B)
be the Moore loop- and path-spaces based at b o E B. Let
It: (B) xQ(B) (B),
denote the usual action of Q(B) on (B). First let us call a homomorphism
q;:A C,
where A and C are associative H-spaces, a monoid-equivalence if, and only if,
the induced homomorphism
q;* : ni(A) ni( C)
is an isomorphism for all i > o.
THEOREM (4.1) There is an RPT-complex M and a monoid-equivalence
q; : M Q(B)
which extends to a map
q; : f!J(M) (B)
of an M-complex to an Q(B)-space. Moreover f!J(M) = f!J(M)// Mis B, and the
induced map
cp : f!J(M) B
is homotopic to the identity. Here f!J(M) is the total space of a universal quasi-
fibration f!J(M) f!J(M) of M.
26
THE TOPOLOGY OF CLASSICAL GROUPS
Proof Suppose first of all that
B=8I\A
where A is a special complex. By definition, let
q;: (T A, A) --+ ({/J(8 1\ A),.Q (8 1\ A))
be a map such that the map TAlA --+ 8 1\ A induced by the composite
TA --+ {/J (8 1\ A) --+ 8 1\ A is the identity. (For example, such a map can be
obtained by thinking of (t, a) ETA = I 1\ A as the path which goes from the
base point ao E A on the equator to the north pole and then along the great
circle towards the south a distance t.) Assume as usual that q; is basepoint
preserving. Now extend uniquely q;IA, first to a homomorphism
q; : ACX) --+.Q (8 1\ A),
and then to a map
q;: PJ(ACX)) = ACX) UTA x ACX) --+ {/J (8 1\ A)
of an ACX)-complex to an.Q (8 1\ A)-space. Observe that the induced map
ip: PJ(ACX))// ACX) = 8 1\ A --+ 8 1\ A
is the identity. Since PJ(ACX)) and {/J (8 1\ A) are contractible, it follows that
q; : ACX) --+ Q (8 1\ A)
is a homotopy equivalence.
In order to prove the theorem inductively for an arbitrary special complex
B, it is enough to show the following. Assume that
B = B' U Dr+l = B' U e"+1
f
and that there is a homotopy equivalence
q;' : PJ(M') (/J(B')
of the M'-complex PJ(M') to the .Q(B')-space (/J(B'), where M' is an RPT-
complex representing Q (B') and PJ(M'), its universal quasi-fibration. Let
1: aDr --+ M'
be a map such that the composite (Dr, aU) --+ (PJM', M') --+ B'is.homotopic
to f. The first map (Dr, aDr) --+ (PJM', M') is any extension of7. Observe that
1 is homotopic to the adjoint off. Now consider the complex M' U:;= IY, ob-
FIBRATIONS AND THEIR CLASSIFICATION
27
tained by adjoining the cell Dr to M' with the map f. According to Theorem
(2.1) of 2, M' Ul IY generates an RPT-complexM which containsM' as an
RPT -subcomplex.
Observe that the given homotopy equivalence can be extended to a homo-
topy equivalence
q/ : r!l(M') x M,M (B, B')
in a fashion similar to the construction of the homotopy equivalence of the
first part of the proof. Here (B, B') denotes the space of paths in B based
at b o and with endpoints in B'. Now consider the map
f+ : (Dr, oIY) (r!l(M'), M')
such thatp,!+ is homotopic to! This mapf+ and the map
f- : (Dr, oDr) (r!l(M') x M,M, M)
with which the indecomposable cell IY of M - M' is attached together
define a map
1: sr r!l(M') x M,M
such that pi is homotopic to f. Observe finally that q/ defines a map
(r!l(M') x M,M) U:; Dr (B),
which in turn can be uniquely extended to a map
q; : r!l(M) (B)
which respects the actions of M and Q(B) and which induces a homotopy
equivalence r!l(M)//M B. Now the pull-back (ip-l)* fJl(M) Bis a quasi-
fibration which is equivalent in the desired sense to B B.
An immediate corollary is the following
COROLLARY Suppose that B is a special complex and that Q(B) is its loop-
space. Then there exists an RPT-complex M and a homomorphism
q; : M Q(B)
which induces a homotopy equivalence. Moreover, the indecomposable cells of
M are in 1-1 correspondence with the cells of B.
The preceding theorem is useful in computing H*(Q(B)). The Adams-
Hilton technique can be deduced from it [1].
28
THE TOPOLOGY OF CLASSICAL GROUPS
Problems
Computation of H * (QS 1\ X)
Suppose that X is a special complex, and let K be a commutative ring with unit such that
H * (X; K) is K-projective (see Chapter 2 for the necessary algebraic notations).
1 Show that H * (QS 1\ X; K) is isomorphic to the tensor algebra T (H * (X; K» generated
by the module of reduced homology H * (X; K).
2 Show that H * (QS"; Z) with n odd, is, as a HopI-algebra, isomorphic to the polynomial
algebra P [x, n - 1] generated by an element x of degree n - 1.
Samelson product
Suppose that A is an associative H-space, and choose representative a: SP -+ A and
b : sq -+ A for lX and fJ. Then the maps
a X b, b X a : SP X sq -+ A
agree on SP V sq. By definition, the Samelson product {lX, fJ} of lX and fJ is the element of
Ttp+qA represented by the map
d(axb, bxa): Sp+q -+ A
equal to a X b on the upper hemisphere and to b X a on the lower hemisphere.
3 Consider the space SP+l X Sq+l. Assume that as cell complexes SP+l = eO U e P + l and
Sq+l = eO U e q + l and that SP+l X Sq+l = eO U e P U e q U e P + q + 2 , where the attaching map
ofe p + q + 2 is the Whitehead product of the identity maps SP -+ SP and sq -+ sq. Let M' be the
RPT-complex (SP V S oo. Then an RPT-complex M representing Q (SP+l X Sq+l) can be
obtained by adjoining a cell e P + q + l to M' by a map f: iJ DP+q+l -+ M'. Show that lis a repre-
sentative of the Samelson product tip, i q }, where ip and iq are the homotopy classes of the
imbeddings SP -+ QSP+l and sq -+ {JSq+l, respectively. Deduce that the adjoint of the White-
head product is the Samelson product.
5 The classification theorem, I
Suppose that B is a simply connected special complex. Then according to the
previous section, there is a commutative diagram
M f!l(M) f!l(M) = B
l 1 1;
Q(B) (B) ) B
where the vertical maps are homotopy equivalences, q; is a homomorphism
and q; a map of an M-complex to an Q(B)-space. Moreover ip is homotopic
to the identity. Broadly speaking, we shall see that all fibrations over Bare
FIBRA TIONS AND THEIR CLASSIFICATION
29
obtained from the universal quasi-fibration for M f!l(M) PA(M) by chang-
ing the fiber in a suitable fashion. So suppose that
p: X B,
is a Hurewicz fibration with fiber F, and let G(F) denote the monoid of
homotopy equivalences of F.
THEOREM (5.1) There is a homomorphism
'Y : M G(F)
which induces homotopy equivalences : f!l(M) x M F X and : 8l( M) B
such that the diagram
f!l(M) x MF X
1 lp
f!l( M) L B
is commutative and (; = ip.
Proof The proof proceeds by induction. Suppose first of all that
B = S 1\ A,
where A is a suitable special complex. Then, according to the preceding sec-
tions,
M = Aoo and f!l(Aoo) = Aoo UTA x Aoo,
where the attaching map is the multiplication A x ACX) Aoo. Observe that
the representation map q;: f!l(Aoo) (S A A) defined in 4 enables us to
think of f!l(Aoo) as paths in B based at b o E B. Now choose for the given
fibration p: X B a lifting function A defined on the set of paths T A
c: f!l(Aoo), and observe that the equation
'Y 0 (x, z) = A ( q;( x), z), X ETA, Z E F,
defines a map
'Yo:TAxF X
such that the map Z 'Yo (x, z), for x E A, is a homotopy equivalence of F.
Denote the map z Yo (x, z) by 'Yo (x) E G(F), and observe that it can be ex-
tended uniquely to a homomorphism
'Y : Aoo G(F).
30
THE TOPOLOGY OF CLASSICAL GROUPS
It is clear now that the maps Yo and y together define a map
' : r!l(AC()) x F X
such that ' (yy', z) = ' (y, y'z), with y, y' E AC() and z E F. Hence ' induces
an equivalence
: r!l(AC()) x AcoF X
such that the induced map <5 : B6'(AC()) S 1\ A is a homotopy equivalence.
In order to construct an inductive proof for the theorem, it is enough to
show the following. Suppose that
B = B' U e r + 1 ,
and let y' : M' G(F) be a homomorphism which induces the homotopy
equivalences
' : (M') x M' F X', X' = p-1B',
-
' : r!l(M') B',
-
with ' = ip', and such that
R(M') x M,F X'
1 1 p'
( M') LB'
is commutative. Here M' is the RPT-complex representing Q(B') and
r!l(M') (M') is its universal quasi-fibration as in 3. We wish to show
that there is a homomorphism y: M G(F) which induces homotopy
equivalences : f18(M) x M F X and : r!l (M) B, where M is the RPT-
complex corresponding to Q(B), such that ylM' = y', 1r!l(M) x M F = ',
and (;( r!l( M') = <5', = ip, and the diagram
f18(M) x MF X
1 lp
f!J( M') L B
is commutative.
Suppose q; : r!l(M) (B) is the map given in 4. cp enables us to think of
f18(M) as a set of paths in B based at b o E B.
Observe now that the rule which associates to (w, y) E r!l(M') xFthe path
t ' (w(t), y) determines forp' : X' B' a lifting function A', defined on the
FIBRATIONS AND THEIR CLASSIFICATION
31
set of paths r!l(M'), with the property that
A' (w, y) (t) = ()' (w(t), y), (w, y) E r!l(M') x F.
We need the following lemma.
LEMMA (5.2) Suppose that (K, L) is a pair of a complex and a subcomplex of
paths in B, and assume that A is a lifting function for p : X B defined on L.
Then there is a liftingfunction A, for p : X B defined on K, which extends A.
Proof By definition, let
K = {(w,y))w(O) = p(y)} c: KxX c: B 1 xX,
and denote by L the subset of K corresponding to L. Consider the commuta-
tive diagram
x
/
/
/
/
/
/
/
L xl-+Kxl
ilL
"
-
g
r B
where g «(w, y), t) = wet), and t gL «w, y), t) is the path A (w, y). The
relative form of the covering homotopy theorem implies that there is a map
g: K x I such that pg = g and g(L x I = gL. Clearly the map which sends
(w, y) to g (w, y, t) is a lifting function A which satisfies the lemma.
Now recall that M is generated by M' and a cell e"+l cOfresponding to
e"+ 1 c: B, and that
BI(M) = (r!l(M') x M,M) U (e r x M).
First, by Lemma (5.2), find a lifting function A l , fOf P : X B defined on
91(M') U e" c: fJl(M') x M,M (e" is the additional generator of M), which
extends A', the lifting function determined by the map ()' : r!l( M') x M' F X'.
Second, using A l , extend ()' to a fiber-preserving map
(r!l(M') U e") xF X.
32
THE TOPOLOGY OF CLASSICAL GROUPS
which is a homotopy equivalence on each fiber. This latter map can be
uniquely extended, in a straightforward fashion, to a fiber-preserving map
l : (£?l(M') x M,M) x M F X,
which is also a homotopy equivalence on each fiber. In particular, we get a
homomorphism
y : M G(F)
such that ylM = y'. Clearly the induced map
-
l : £?l(M') B'
is just '. Now l determines a lifting function A2 for P : X B defined on
91(M') x M' M and extending AI. By Lemma (5.2), Al can be extended to a
lifting function A 2 for p: X B defined on £?l(M') x M' M U e"+ 1. Finally,
A 2 gives first a fiber-preserving map
(£?l(M') x M,M) U e"+I) xF X,
which can then be extended uniquely to an equivalence
: £?l(M) x M F X
as required.
Q.E.D.
Remark Suppose that p : X B is a G-bundle. Then, if we take for the
universal quasi-fibration q : £?l(M) f!I(M) one which is locally trivial when
restricted to every finite subcomplex K of f!I(M), it can be proved, similarly
to Theorem (5.1), that there is a homomorphism
y:M G
which induces homotopy equivalences : fJ4(M) x M F X and : f!I( M) -+ 84
which commute with the projections £?l(M) x M F f!I(M) and X B.
Moreover, is an equivalence of G-bundles on the restrictions of (M) x M F
and (5* X to finite subcomplexes K c £?l( M).
6 The classification theorem, II
Suppose that B is a simply connected special complex. We have seen that any
fibration p : X B is equivalent, in some sense, to a quasi-fibration
Prp : £?lM x M,rp F £?lM
FIBRATIONS AND THEIR CLASSIFICATION
33
for a suitable homomorphism cp: M G(F) = G. However, the choice of cp
is not unique. How are all the choices of cp related? In order to answer this
question we must formulate the appropriate notion of equivalent fibrations
p : X Band p' : X' B' on the one hand, and the corresponding notion of
equivalent homomorphisms cp: M G and cp': M' G on the other.
We shall say that the two quasi-fibrations
p: X Band p': X' B',
where fibers, total spaces, and basespaces are CW -complexes, are equivalent
if, and only if, there is a quasi-fibration q: Y C and fiber-preserving maps
: X Y and ' : X' Y which induce the commutative diagrams
F X B
I g 1 6 I (j and
'i'
K Y C
F' X' B'
1 $' 1 Q' 1 ;f'
K Y C
such that all of the vertical maps are homotopy equivalences. It is easy to
check that this defines an equivalence relation among quasi-fibrations.
Observe that Y C can be taken to be a Hurewicz fibration, since any
quasi-fibration is equivalent in the above sense to a Hurewicz fibration.
Denote by C (B, F) the set of equivalence classes of those quasi-fibrations
whose fibers and basespaces are of the homotopy type of F and B respec-
tively.
Suppose now that cp and cp' are homomorphisms
cp : M G(F) and cp': M' G(F)
of the RPT-co plexes M and M' into G(F), the monoid of homotopy
classes of F. We shall say that cp and cp' are equivalent if, and only if, there is
an RPT-complex N and homomorphisms i: M N, i': M' N, and
'ljJ: N G(F) such that
"Pi = cp and 1Jli' = cp'
and i and i' are monoid equivalences. Denote by {M, G} the set of equi-
valence classes of pairs {M', cp'} where M' is monoid-equivalent to M.
Suppose now that B is a simply-connected special complex, and let M be
the RPT -complex which is monoid-equivalent to Q(B) and generated by cells
in one-one correspondence with the cells of B. A homomorphism cp: M
G(F) determines an action M x F F, which in turn allows us to con-
3 Husseini (0216)
34
THE TOPOLOGY OF CLASSICAL GROUPS
struct the quasi-fibration
Prp: ffiM x M,rp F M
obtained from the universal quasi-fibration for M, ffiM fflM, by replacing
M by F. Denote Prp : ffiM x M,rp F ffiM by X(cp).
THEOREM (6.1) The correspondence q; X(CP) induces a bijection
{x} : {M, G(F)} C (B, F).
Proof To see that [x] is well-defined, let cp : M G and cp' : M' G be
homomorphisms, and suppose that N is an RPT -complex with a homo-
morphism 1Jl : N G. Assume that i: M Nand i' : M' N are monoid
equivalences such that 1Jli = cp and 1Jli' = cp'. Note that i and i' induce fiber-
preserving homotopy equivalences ffI M x M ,rp F ffi N x N, '" F and fJI M'
X M' ,rp' F ffi N x N, '" F, proving that X( cp) and X( cp') are equivalent.
Next we show that {X} is injective. Suppose
prp : M x M,rp F fJ4M and Prp': fflM' x AI' ,rp' F -+ ffiM'
are equivalent. Hence there is a fibration K -+ Y -+ C and fiber-preserving
maps : M x M ,rp F Y and ': 81M' x M' ,rp' F -+ Y which induce the
commutative diagrams
-
F -+ 81M x M,rp F -+ 84M
I I {, I CJ and
K --:)0 Y --:)0 C
F -+ fJ4M' x )f' ,rp' F fflM'
l' l' 1
K ) Y --:,.C
where the vertical maps are homotopy equivalences. Assume that K y
C is so chosen that (5 and (5' imbed ffi M and ffiM ' in Cas subcomplexes
with the basepoint only in common. Let N be the RPT -complex monoid-
equivalent to QC and generated by cells in one-one correspondence with the
cells of C. Note that (5 and 3' induce imbeddings i: M N an4 i' : M' N
which are also monoid equivalences. Now, proceeding as in the proof of
Theorem (5.1), find a homomorphism 1Jl: N G(F) such that 1Jli = cp and
"Pi' = cp'. This proves that (M, cp) and (M', cp') are equivalent and, hence,
{X} is injective.
The proof that {X} is surjective is similar. Suppose F X B is a Hure-
wicz fibration. According to 5 there is a homomorphism cp: M G(F)
FIBRATIONS AND THEIR CLASSIFICATION
35
which induces an equivalence
F PAM x M.rp F PAM
1 6 1° 1 6
F > X > B
where the vertical maps are al homotopy equivalences. To show that any
other homomorphism cP' : M' G(F) which sets up a similar equivalence
' : ffi M x M ,rp F X is equivalent to cp, proceed as in the proof that {X} is
injective. To finish the proof we need to show that if F' X' B' is another
fibration equivalent to F X B, then it gives rise to a homomorphism
cp': M' G(F), so that (M, cp) and (M', cp') are equiyalent. Suppose that
K Y C is a Hurewicz fibration with maps : X Y and ' : X' y
which induce the commutative diagrams
F X B
1 6 1° 1 6
K y C
F' x' B'
and 1 8 ' 1 0' 1 6 '
K y C
in which all the vertical maps are homotopy equivalences. Certainly we can
assume that (5 and (5' imbed Band B' as subcomplexes into C with only the
basepoint in common. Next proceed as in the proof that {X} is injective to get
a homomorphism 1Jl : N G(F) which extends cp and cp'. (Here N is the RPT-
complex monoid-equivalent to .Q( C).) Clearly N is monoid-equivalent to
each of its RPT -subcomplexes M and M'. This establishes the theorem.
What is {M, G} like? Can it be described entirely in terms of M and
G(F) = G?
THEOREM (6.2) Suppose cp and cp' are homomorphisms of M and M', respec-
tively, into G(F). Then (M, cp) and (M', cp') belong to the same class in {M, G}
if, and only if, there is one-parameter family of homomorphisms
CPt : M G(F)
such that CPo = cp and CPl = cp'e, where e is a monoid equivalence M M'.
Proof Suppose that
cp : M G and cp': M' -+ G
are homomorphisms of Minto G, and let
CPt : M G
36
THE TOPOLOGY OF CLASSICAL GROUPS
be a one-parameter family of homomorphisms such that CPo = Cp and CPl
= cp'e, where e : M M' is a monoid equivalence. To show that (M, cp) and
(M', cp') represent the same element in {M, G} we need find a pair (N, 1p) where
N includes M and M' as deformation retracts, and 1p is an extension of cp and
q/. We shall construct (N, 1p) by induction on the dimension of the generators
of M. Assume first of all that e : M M' is cellular. By definition let (k) M'
be the sub-RPT-complex of M' generated by the k-skeleton of M'. Then M'
is filtered by the ascending sequence
(0) M' c: (1) M' c: ... c: (k) M' c: ... c: M'
of RPT-subcomplexes where (O)M' = {e}. Similarly, let (k)M be the sub-
RPT-complex generated by the k-skeleton of M. Then
(0) M c: (1) M c: ... c: (k) M c: ... c: M
is a filtration of M and e takes (k) Minto (k) M'. Suppose that we have been
able, by induction, to construct an RPT-complex (k-1)Nwith the following
properties:
l)k-l there are imbeddings ik_l:(k-1)M (k-1)N and i _1:(k-1)M'
(k-1)Nsuchthatimik_l nimi -1 = {e};
2) k-1 there is a deformation retraction
(k-1)eX t : (k-1) N (k-1) N
such that (k-1) eX l (k-1)N) c: (k-1)M' and (k-l)eXl i k - 1 : (k-1)M (k-1)M'
is just el(k-1) M; and
3) k-l there is a homomorphism
1pk-l: (k-l) N G
h th t (k - l) .
suc a 1pk-1 eXt lk-l = CPt.
We wish to show the same thing for the case k. So consider the complex
(k) N = (k)MU (k-1)NU (k)M',
where (k)Mn(k-l)N=(k-l)M, (k-1)Nn(k)M' = (k-l)M', and (k)M
n (k)M' = {e}.
Now suppose that {e k } are the indecomposable cells of M of dimension k,
and let
f: (Dk, ODk) (M, (k-1)M)
FIBRA TIONS AND THEIR CLASSIFICATION
37
be the characteristic map for eke Let
h: ODk xIU Dk xoI (k) N
be the map defined by the formulas:
h (x, t) = (k-l) ()(,t (ik-lx),
X E Dk X {O} ,
X E D k X {I} ,
(x, t) E oD k xl.
h (x, 0) = f(x),
h (x, 1) = ef(x),
Denote by lk : (k) M (k) N and z : (k) M' (k) N the natural imbeddings. By
definition, let (k) N be the universal RPT -complex generated by (k) N
U h U (Dk X l), so that Zk and z induce imbeddings
ik: (k)M (k)N and i : (k)M' (k)N
of RPT-complexes. Define
1Jlk : kN G
to be the homomorphism such that
1Jlk(X) = 1Jlk-l (x),
XE(k-l)N
,
1Jlk (x, t) = q;t (ikx), (x, t) E Dk X I,
1Jlk(X) = q;(x),
1Jlk(X) = q;'(x),
X E (k) M
,
X E (k) M .
It is not hard to show that the deformation retraction (k-l)()(,t: (k-l) N
(k-l)N can be extended to a deformation retraction (k)()(,t: (k)N (k)N
which satisfies conditions (2)k and (3)k. This establishes the induction step
and finishes the proof under the assumption that e: M M' is cellular. The
general case is taken care of by the following lemma.
LEMMA (6.3) Suppose that
e: M M'
is a monoid equivalence where M and M' are RPT-complexes. Then there is a
one-parameter family
q;t : M M'
of homomorphisms such that q;o = e and q;l is cellular.
38
THE TOPOLOGY OF CLASSICAL GROUPS
Proof Consider the usual filtration
(0) M c: (1) M c: ... c: (k) M c: ... c: M
,
where (k)M is the sub-RPT-complex generated by the k-skeleton of M.
Suppose that we have been able to define a one-parameter family of homo-
morphisms
(k-l)q;t: (k-l) M M'
such that (k-l)q;o = el(k-l)M and (k-l)q;1 is cellular. Suppose that {e k } are
the indecomposable k-cells of M, and denote by
f: (Dk, ODk) (M, (k-l)M)
the characteristic map of eke First, by applying the Homotopy Extension
Property to (k-l)q;tf, find a one-parameter family of maps
(k) q; t: (k-l)M U U e k M'
such that (k)<ptl(k-l) M = (k-l)q;t, (k)<PO = e and (k)<Pl is cellular. This being
done, extend (k)<pt to a homomorphism
(k)q;t: (k) M M'
uch that (k)q;o = el(k) M and (k)q;l is cellular. (The extension (k)q;t of (k)<pt is
unique.)
Now let us prove the necessary part of the theorem. We need to show that
if
q; : M G, q;' : M' G, and 1Jl: N G
are homomorphisms, and if there are imbeddings i: M Nand i' : M' N
of RPT -complexes such that i and i' are monoid equivalences and 1Jli = q;
and q;' = 1Jli', then there is a one-parameter family of homomorphisms
q;t : M G
such that q;o = q; and q;1 = q;'e, where e : M M' is a monoid equivalence.
It is enough to prove the following lemma.
LEMMA (6.4) Suppose that M is a sub-RPT-complex of N, and assume that
the injection i: M N is a monoid equivalence. Then there is a deformation
retraction
(Xt: N N
such that each (Xt is a homomorphism and (Xl(N) c: M.
FIBRATIONS AND THEIR CLASSIFICATION
39
The proof is similar to that of Lemma (6.3). Start with the ascending
filtration
(0) N c: (1) N c: ... c: (k) N c: ... c: N
where (k)N is the sub-RPT-complex generated by the k-skeleton. Then
construct tX t by induction on k, by deforming the indecomposable cells e k
into M and then extending the deformation to a one-parameter family of
homomorphisms as required.
Problems
1 Suppose that M is an RPT-complex and G an associative H-space. Let {M, G}' be the set
of equivalence classes of homomorphisms q; : M -+ G, where two homomorphisms are equi-
valent if, and only if, they are connected by a one-parameter family of homomorphisms of M
into G. Show that {M, G}' [ t?lM , t?l G], where [ B6'M, t?l G] is the set of basepoint-pre-
serving homotopy classes of the classifying space f1lM to the classifying space t?lG .
2 Suppose that G is a group. Show that the set of equivalence classes of G-bundles over the
finite complex K is naturally equal to [K, £?l G].
7 The classification theorem, III
The classification given in Theorem (6.1) does not distinguish between a
Hurewicz fibration p: X B and its pull-back Pf :f*X B, f being a
homotopy equivalence of B. To refine the classification so that o e is able to
distinguish a fibration from its pull-back by a homotopy equivalence of the
basespace one must first modify the preceding notions of equivalence.
We shall say that two quasi-fibrations
P : X Band p': X' B
with the same fiber F, where fibers, total spaces, and loopspaces are CW-
complexes, are equivalent if, and only if, there is a Hurewicz fibration
q: Y BxI
and fiber-preserving homotopy equivalences : X Y and ': X' Y
which induce the commutative diagrams
F X B
1 8 1 6 1 6 and
K Y BxI
F X' B
t t t
K Y' BxI
40
THE TOPOLOGY OF CLASSICAL GROUPS
such that J and J' are homotopy equivalences, while (5 and (5' are the imbed-
dings of B as B x {O} and B x {I}, respectively. It is easy to check that the
preceding notion defines an equivalence relation. Denote by B (B, F) the set
of equivalence classes of quasi-fibrations over B with fiber F. Note that every
A
class in B (B, F) has a Hurewicz fibration for a representative.
Suppose now that cP and q/ are homomorphisms
9' : M G and 9": M G
of the RPT-complex M into the associative H-space G. We shall say that cP
and cP' are equivalent if, and only if, there is a one-parameter family of
homomorphisms
CPt : M G
such that CPo = cp and CPl = cp'. Denote by 7(;0 Hom (M, G) the set of equi-
valence classes of homomorphisms M -+ G.
Now let B be a simply-connected special complex, and let M be the RPT-
complex monoid-equivalent to Q(B) and generated by cells in one-one
correspondence with the cells of B. A homomorphism cP : M -+ G(F), the
monoid of homotopy equivalences of F, determines an action M x F -+ F,
which in turn allows us to construct the quasi-fibration
Prp: ffi(M) x M,rpF -+ B
obtained from the universal quasi-fibration ffiM -+ B = 84M by replacing
M by F. Denpte prp : ffi(M) x M,rp F -+ B by x(cp).
THEOREM (7.1) The correspondence cP -+ x(cp) induces a map
{x} : 7(;0 Hom (M, G(F)) -+ B (B, F),
which is bijective.
The proof is analogous to that of Theorem (6.1) and will therefore be
omitted.
Now suppose cP : M -+ G is a homomorphism. It is quite easy to see that cP
induces a map
O(,(cp): ffiM = B -+ BG
where BG is the basespace of the universal quasi-fibration G.
FIBRA TIONS AND THEIR CLASSIFICATION
41
PROPOSITION (7.2) The correspondence cp tX(cp) induces a map
[tX] : no Hom (M, G) [B, B G ]
which is bijective.
([B, B G ] is the set of homotopy classes of base-preserving maps.) The
proof is through arguments similar to those of 6.
An immediate corollary of (7.1) and (7.2) is the following.
A
COROLLARY (7.3) [X] [tX]-l : [B, B G ] B (B, F) is bijective.
Suppose finally that G is a topological group and let
p:X B
be a G-bundle with fiber F. Denote by B (B, F) the set of equivalence
classes of G-bundles over B with fiber F. (See 1. A notion of equiva-
lence of bundles similar to that for quasi-fibrations can be formulated.
The equivalence classes it defines are the same as those defined by the notion
of equivalence of G-bundles given in 1.) Let no Horn (M, G) be the set of
homotopy classes of homomorphisms M G, the homotopics being through
homomorphisms.
Results corresponding to (7.1), (7.2), and (7.3) can be proved for G-
bundles, provided that B is finite. Moreover, [X] [tX]-l induced by the map
which assigns to a mapf: B BG the pull-backf*X G B, where XG BG'
is a universal G-bundle.
Bibliography
[1] J. F. Adams and P.Hilton, "On the chain algebra of a loopspace", Comm. Math. Helv.
20 (1955), 305-330
[2] A. Dold and R. Thom, "Quasifaserungen und unendliche symmetrische Pro dukte" ,
Ann. of Math. 67 (1958), 239-81
[3] L.Conlon and A.F. Whitman, "A note on holonomy", Proc. Amer. Math. Soc. 16
(1965), 1046-51
[4] D.Husemoller, Fibre Bundles, McGraw-Hill, 1966
[5] S. Y.Husseini, "Constructions of the reduced product type, II", Topology 3 (1965),
59-79
[6] S. Y.Husseini, "When is a complex fibered by a subcomplex?" Trans. Amer. Math.
Soc. 124(1966),249-91
42
THE TOPOLOGY OF CLASSICAL GROUPS
[7] S. W.Kamber and P. Tondeur, "On flat bundles", Bull. Amer. Math. Soc. 72 (1966),
846-49
[8] I. M. James, "Reduced product spaces", Ann. of Math. 62 (1955), 170-97
[9] E.Spanier, Algebraic Topology, McGraw-Hill, 1966
[10] J.D.Stasheff, "A classification theorem for fiber spaces", Topology 2 (1963), 239-46
[11] J.D.Stasheff, '''Parallel' transport in fiber spaces", Bol. Soc. Mat. Mex. 11 (1966),
68-84
[12] J.D.Stasheff, "Associated fiber spaces", Mich. Math. J., 15 (1968), 457-70
[13] N.E.Steenrod, The Topology of Fiber Bundles, Princeton Univ. Press, 1951
CHAPTER 2
Cohomology and homology
of the classical group
SUPPOSE THAT G = Un, SOn, or SPn. The object of this chapter is to
compute the cohomology H*(G) and homology H*(G) with various coef-
ficients. The main tools are the fibrations of G over spheres described in 1
of Chapter I.
Since fibrations are, in a sense, a way of forming new spaces out of the
fiber and the base, the question arises as to how the cohomology and homo-
logy of the total space are related to those of the base and fiber. The simplest
fibration is a product, and in this case the answer is provided by the Kiinneth
formula. We have seen in the prece<:ling chapter that an arbitrary fibration
can be thought of as a tower of twisted products; the answer in this case
is given by a suitable spectral sequence. In order to describe the answer,
we need to recall some algebraic notions. The account given here is, by
necessity, brief. The reader is referred to [3] and [6] for a detailed exposi-
tion.
The computation of H*(G) for the unitary and symplectic groups is done
in 4. The case of SOn is more delicate and is taken up in 7. Then we note
that the multiplication in G induces a diagonal map H*( G) -+ H*( G)
(g)H*( G) in the cohomology with appropriate coefficients and a multiplication
H*(G) (g) H*(G) -+ H*(G) which turns H*(G) into a ring-the so-called
Pontryagin ring. Thus the multiplication G x G -+ G defines the structure of
a Hopf-algebra on H*(G) and H*(G) with appropriate coefficients. There is a
brief recapitulation of the basic concepts of Hopf-algebras. (For a detailed
study of Hopf-algebras, the reader is referred to [2], [3], [5].) The computa-
tions of H*(G) and H*(G) as Hopf-algebras are given in 6 and 7. Also in
7 we give generating complexes for Un, SOn, and SPn, i.e., subcomplexes
of G whose homology and cohomology are generators for the algebras
H*(G) and H*(G). These complexes are quite useful. For example, the com-
puting of the action of the Steenrod algebra on H*( G) is simplified greatly
(see, for example, [7]).
These results on the homology and cohomology of the classical groups are
43
44
THE TOPOLOGY OF CLASSICAL GROUPS
special cases of far more general results on the homology and cohomology of
Lie groups. The general study calJs for different and more sophisticated
techniques. The reader is referred to [I], [2], and [3] for more details.
1 Graded modules and algebras
Let K be a commutative ring with a unit. A graded module A over K is, by
definition, a direct sum L A p of K-modules Ap. A p is called the pth homo-
D
geneous component of A, and its elements are the homogeneous elements of
degree p (deg p, for short). If Ap = {O} when p < 0, then we say that the
grading is positive. A homomorphism of degree k
f{J:A-+B
of the graded modules A and B is a homomorphism which takes Ap into
Bp+k. The tensor product A <8>K B of the graded modules A and B is the
usual tensor product graded by taking the rth homogeneous component
(A (8)KB)r to be equal to L Ap (8)KA q .
p+q=t
A graded algebra over K is a positively graded K-module A = LAp,
with a homomorphism of degree zero (a multiplication) p o
f{J:A <8> A-+A
and a unit 1 e Ao of degree zero. If Ao = K, then A is said to be connected. A
is said to be commutative if
a · b = ( - l)pq b · a, a e Ap, b e Aq
where a · b stands for f{J (a <8> b). A homomorphism of graded algebras
lX:A-+-B
is a homomorphism of graded modules of degree zero which is also a homo-
morphism of algebras. The tensor product A <8> K B of the graded algebras A
and B is the tensor product of A and B as graded modules with the multipli-
cation defined by the rule
,
(a <8> b) · (a' (8) b') = (-l)qp aa' <8> bb', b E Bq, a' e A p '.
Bigraded modules and algebras and the corresponding notions can be defi-
ned similarly.
A filtered K-module A is a module with an ascending sequence of sub-
modules
... c: FA c: F A c: ...
p p+l
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 45
such that FooA = A and F-ooA = {O}. The associated graded module is, by
definition,
EO(A) = L F p AjF p _ 1 A.
p
A grading on filtered module A induces a bigrading on the associated graded
module Eo(A) by putting
E ,q(A) = Fp,q(A)fFp+l,q-1A
where Fp,q(A) = FpA n A p + q , A p + q being the (p + q)th homogeneous com-
ponent of A. p is called the filtration degree . q, the complementa"y degree .
and (p + q), the total degree.
Afiltered graded algebra is a graded algebra A which is filtered as a module
and is-such that
FpA. FqA c Fp+qA.
The associated bigraded module EO(A) inherits from the multiplication a
natural multiplication which makes it a bigraded algebra.
2 Differential graded modules and algebras
We shall consider in this section the algebraic framework for homology only,
the case of cohomology being similar.
Suppose that K is a commutative ring with unit. A differential graded K-
module A (DG-module for short) is a graded module with a homomorphism
d: A -+ A
of degree -1 with d = O. The derived module H(A) of A is Z(A)jB(A),
where Z(A) = ker d and B(A) = image d. The grading on A induces one on
H(A). A filtered DG-module A is a DG-module filtered as a module and such
that d (FP A) c FP A for all p. A filtered DG-module gives rise to spectral
sequence, i.e., a sequence of bigraded DG-modules L E;,q, r > 1, and
p,q
differentials d r : Er -+ Er of bidegree (-r, r - 1) such that H(Er) = Er+l.
If the filtration on A is positive, i.e., FP A = {O}, p < 0, then {Er, dr, r > I}
converges and E oo = EO (H(A)), where H(A) is filtered by the submodule
FPH(A) = image (H(FDA) -+ H(A)).
(See the problems at the end of this section for details.)
46
THE TOPOLOGY OF CLASSICAL GROUPS
A differential graded algebra A (DG-algebra for short) is a graded algebra
and, as a graded module, is a DG-module whose differentiation
d: A -+ A
has the property that
d(a.b) = (da)-b + (-l)Da.(db), aeA p , beA.
A filtered DG-algebra A is a DG-algebra which is filtered as an algebra and
d (FpA) c FpA for all p. A filtered DG-algebra gives rise to a spectral se-
quence {Er(A), dr, r > I} of bigraded DG-algebras.
Problems
Suppose that M is a K-module with differentiation d: M M; i.e., d is a homomorphism
such that d 2 = o. Assume that M has an ascending filtration
...c F"Mc Fp+1Mc .0.
such that F-ooM = {OJ and FooM = M. By definition, let
Z; = {x e FpMfd(x) e Fp-rM}
B; = {dx e FpMlx e Fp+rM} .
Then
. . 0 C Zr C zr-l C . 0 . . . 0 C zr-l C zr co. 0
p p+ l' p-1 p ,
... c B; c B;: c .; 0, and B; c Z;, for all r, s.
Put, for r 2:: 0,
r Z;
E =
P zr-1 ffi B r - 1
p-l Q7 p
E' = " E r
, p L p
p
and let d! : E' E' be the differential induced by d: M M.
1 Show that
a) for each r, d r has degree - rand H(E r ) is naturally isomorphic to E r + l ;
b) E = F"M/Fp-1M; and
c) E; = H (FpIFp-1M) and d l : E -+. E _l is the boundary homomorphism of the triple
(F"M,Fp-1M, F p _ 2 M).
The sequence of graded modules E l , E 2 , . .. and homomorphisms d 1 , d 2 , . .. is the
spectral sequence associated to the filtration ... c F"M c Fp+1M coo 0 .
By definition, let
z; = {x € F"Mldx = O},
B; = {dx eF"Mlx eM}, and
E; = Z;IZ;:l B;.
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 47
2 Suppose that F"M = {O}, if p < O. Show that
EO; E (H(M»,
where EO (H(M») is the graded module associated to the filtration
... c: FpH(M)C. Fp+1H(M)c: ".,
F (M) being image (H (FpM) H(M».
If FpM = {OJ, for p < 0, the sequence {E } converges toE'; in the sense that for any p
there is an integer r(p) such that if r > r(p) then E = E +l = ... = E oo . We say then that
E OO is the limit of {E r }.
Suppose now that M is in addition a DG-module. Put
Z ,q = Z; n Fp+qM,
B;,q = B; n Fp+qM.
3 Show that the grading on M induces a natural bigrading
E r = L E ,q,
p,q
where
Z'
E ' = P.q > 1
, r _ ,
P.q Z,-l ffi B,-l
p-1.q+ 1 W p.q
and d r : E r E r has bidegree (-r, r + 1).
4 Suppose that M is a filtered DG-algebra, and assume that FpM = {O}, for p < O. Show
that the multiplication on M induces for 1 :5 r :5 00 a multiplication on E r which turns Er
into a bigraded DG-algebra. Moreover, the natural isomorphism H(E') E r + l carries the
algebra structure defined on H(E') by that on E r to the algebra structure on E r + l . Also the
isomorphism E oo EO(H(M» is one of bigraded algebras.
There is another method for defining the spectral sequence associated to a filtration.
Suppose that M is a module with differentiation d: M M, and let
...c FpMc Fp+1Mc ...
be an ascending filtration such that FpM = {O}, for p < 0, and F ooM = M. By definition,
put
Ap = H(FpM), A = LAp.
p
The map FpM -+ Fl'+lM induces a homomorphism
i: A A
of degree 1. Moreover, the map FpA FplF p + l induces a map
j: A E l
of degree 0, where El = L E , E = H (FpIF p - l ), is the El term of the spectral sequence
associated to the filtration as in Problem 1.
48
THE TOPOLOGY OF CLASSICAL GROUPS
5 Show that the triangle
A A
a "-.J/' i
E l
is exact where iJ is the boundary homomorphism of the pair (Fp F p - l ). Observe that deg (iJ)
= -1. Moreover jiJ : El 'E l is the differential d l : E 1 £1.
6 Proceed by induction to show that for each r there is an exact triangle
or A i or A
l l
a"-.J/'ii-"
Er+l
such thatji- r iJ : Er+l E r + l is the differential d r + l : E r + l E r + l . Here i r is the rth iterate
of A and irA, the image ofi r .)
7 Suppose, in addition, that M is graded. Then A becomes bigraded if we put
Ap,q = Hp+q(FpM).
Show that bidegree i = (-1, 1), bidegree j = (0,0), and bidegree iJ = (-1,0).
The method of 6 and 7 is the so-called method of exact couples. We leave to the
reader the task of defining spectral sequences for the case of cohomology. Just bear in mind
that the filtrations are descending and that differentiations d: M M have degree 1. The
resulting spectral sequence Er = L E:,q and differentials d r : E, E, are such that d r has
bidegree (r, - r + 1).
3 The homology and cohomology spectral sequences of a fibration
Suppose that p : X -+ B is a Hurewicz fiber space with fiber F over the simply-
connected special complex B. Then, according to 5 of the preceding chapter,
there is a homomorphism r : M -+ G(F) and an equivalence
: PA(M) x MF -+ X
of fibrations which induces a homotopy equivalence :PA (M) -+ B on the
basespaces. (Here M -+ PAM -+ PAM is the universal quasi-fibration associ-
ated with the RPT-complex M representing QB.) Now suppose B(s) is the
s-skeleton of B, and let X(s) be p-l B(s). It is easy to see that induces an
equivalence
(S) : (PA(S) M x (s)MM) x M F -+ X(s)
of fibrations, where (s) M is the RPT -subcomplex of M generated by the cells
corresponding to B(S). In particular, it follows that
X(s) = X(s-l)U U DS xF
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 49
where the attaching map j : aDs x F -+ X(s-l) is such that jID -l x F takes
x xF to F and jID -1 xF takes x xF to the fiber over f(x). [Recall that
we regard aDs = 8 s - 1 as the union of two hemispheres Ds.:- 1 U D - 1. f is
the attaching map aDs -+ B(s-l) for the s-dimensional cells of B.] Thus we
get the filtration
F = X(O) c ... c X(s) c ... c X
of X which corresponds to the filtration of B by its skeletons.
THEOREM (3.1) The filtration of X given above induces a spectral sequence
{Er, r > I} such that
E ,* = H* (B; H* (F; K))
and
E:,* = E ,* (H* (X; K)),
where K is an arbitrary abelian group and EO (H* (X; K)) is the group associated
with the filtration on H* (X; K) defined by image H* (X(S); K) -+ H* (X; K).
The proof of Theorem (3. I) is a great deal easier if F is a CW -complex-
in which case one can use cellular chains. Then
E ,* . C*(B) (g) H* (F; K),
and d 1 : El -+ El can be identified with
Q9 1 : C*(B) (g) H* (F; K) -+ C*(B) Q9 H* (F; K)
where a is the boundary homomorphism for the cellular chains. [One needs
the description of aDS xF -+ X(s-l) given above.]
In the case of comology, the cochain algebra C* (X; K), where K is a ring
with a unit, inherits a filtration as DG-algebra from the filtration of X by
X(s). The terms of the resulting spectral sequence are therefore all bigraded
alge bras.
THEOREM (3.2) The filtration of X induced by the filtration of B by its skele-
tons induces a spectral sequence ofbigraded algebras {Er, r > 2}, such that
E 2 *' * = H* (B; H* (F; K))
and
E '* = £0*'* (H* (X; K)),
where Eo (H* (X, K)) is the graded algebra associated to the filtration in-
duced on X by ker (H* (X; K) -+ H* (X(S); K).
4 Hussein i (0216)
50
THE TOPOLOGY OF CLASSICAL GROUPS
This spectral sequence is most useful when it is trivial. The following
theorem describes a fairly common situation.
THEOREM (3.3) Suppose that the induced homomorphism
i* : H* (X; K) -+ H* (F; K)
is an epimorphism. Here i stands for the injection of F in X as afiber. Then
E 2 = E3 = ... = Eoo.
Problems
The following discussion will be limited to homology. The case of cohomology is similar
and will be left for the reader.
Suppose that p : X B is a Hurewicz fiber space with fiber F. Assume also that B is a
simply-connected special complex. Let B' c: B be a subcomplex of B, and define (B, B')(S)
to be B' U B(s) where B(S) is the s-skeleton of B. Now denote p-l B' by X' and p-l (B, B)(S)
by (X, X')(S). Then we get an ascending filtration
X' c ... c (X, X')(S) C ... c X, s 2:: 0
such that
(X, X,)(O) = X' and (X, X')(OO) = X.
This filtration leads to a spectral sequence {Er, d r } for homology, and a spectral sequence
{Er, d,} for cohomology. For simplicity assume that F is a special complex.
1 Show that
E;.t = Hs «B, B'); Ht (F; A)
and
E;:; = E t (H* (X, X'; A)),
where A is a K-module.
[To prove 1 use the classification theorem of Chapter 1. The computation of the E2
terms depends on identifying d l : E .* C* «B, B'); H* (F; A)) -+ E .* C* «B, B');
H * (F; A). This can be done easily by considering how (X, X')S is obtained from
(X, X')(S-l).]
2 Prove that if Hi (B, B') = {O},for i < b, and Hi(F) = {O},for i < a, then
p* : Hi (X, X') Hl (B, B')
is an isomorphism for i :5 a + b - 1 and onto for i = a + b. The homology groups have
arbitrary coefficients.
3 Suppose that Hi(B) = {O}, for i < a, and Hi(F) = {O}, for i < b. Then the sequence
Ha+b(X) Ha+b(B)"':"--:,. H a + b - l (F) -+ ... Hi(x) P* --:,. Hi(B) ...:..-+ H i - l (F) ...
is exact. Here 1: = iJp;l, where iJ is the boundary homomorphism Hl (B, F) H l - l (F). The
homology groups have arbitrary coefficients.
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 51
The map 1: = Op;l is called the transgression. In general it is defined as a submodule
T* c H*(B) of H*(B) and takes its values in a quotient module H*(F)/V* of H*(F).
4 Show that 1: coincides wit:z
d . E S,o E O,S-l
S. S S.
4 Computation of the cobomology of tbe classical groups
The object of this section is to compute the cohomology of the classical
groups. We shall carry out the computation for the unitary group Un, since
it is easier and therefore serves to illustrate the methods. The computation
for the symplectic groups will be briefly indicated in the problems at the end
of the section, while the computation for the orthogonal group, which is more
delicate, will be taken up in 5.
Before we can state the main result, we must introduce a few concepts.
Suppose A is a connected algebra over a commutative ring with unit K. Let A
be the ideal generated by all elements of positive degree in A. By definition,
the K-module of indecomposable elements of A is q(A) = A/A2.
We shall also need certain algebraic constructions. Suppose that
V = L V 2i + 1
i O
is a graded K-module whose non-trivial homogeneous component all have
odd degree. The tensor algebra, T(V), of V is by definition K + L Tk(V),
k l
where Tk( V) is the k-fold tensor product of V with itself. Let S be the ideal of
r(V) generated by elements of the form a · b - (l)dega o de g b b · a. Then the
exterior algebra of V is by definition
E(V) = T(V)jS.
Note that E(V) is the universal commutative graded algebra generated by V.
Now the cohomology ring H* (Un; Z) with integral coefficients is a con-
nected commutative graded algebra. Denote the indecomposable elements
H*jH*2 = q (H* (Un; Z)) by Ln.
n-l
THEOREM (4.1) Ln is afree graded Z-module L (L n )2i+l, and each (L n )2i+l
i O
is of rank one. Moreover, any lifting in: Ln --+ H* (Un; Z) can be extended to
an isomorphism
Vln: E (Ln) --+ H* (Un; Z)
of graded algebras.
52
THE TOPOLOGY OF CLASSICAL GROUPS
Observe that the theorem implies that H* (Un; Z) is torsion-free and,
hence, that the structure of H* (Un; K), K being any ring, is similar.
Denote the basic vectors of en by 8 1 , ..., 8n. Consider the map
Pn: Un -+ S2n-1
of Un onto the unit sphere in en, which sends a to a(8n). One can easily see
that Un is filtered as a principal bundle over S2n-1, the fiber being U n - 1 (see
Problem 8 of 1 of Chapter 1). The proof of the theorem depends on. an
analysis of the sequence of graded algebras
H* ( S2n-1. Z ) p* -+ H* ( u · Z ) H* ( u · Z )
, n, n-1, ,
where i* is the homomorphism induced by the injection i: U n - 1 -+ Un which
identifies U n - 1 with the subgroup of Un leaving On fixed.
DEFINITION A sequence
... -+ A fP n _> A fPn+l) A . 2 -+ ...
n n+ 1 nor
of graded algebras and homomorphisms is said to be exact if ker f/Jn+1 is the
ideal generated by the elements of positive degree in image f/Jn.
Denote by 1 the graded algebra over K which is isomorphic to K in degree
zero and is trivial otherwise.
LEMMA Let
l-+A-+B-+C l
be an exact sequence of connected graded algebras over K. Then the sequence
q(A) q(B) -+ q( C) -+ 0
is exact.
The proof is immediate and will therefore be omitted.
LEMMA 2 The sequence of graded algebras
p*
1 -+ H* (S2n-1; Z) > H* (Un; Z) -+ H* (U n - 1 ;Z) -+ 1 (FTn)
is exact.
Proof Let us denote by H'*(X) the cohomology algebra with integral co-
efficients. The proof is by induction on n. If n = 1, U 1 S1, since U o is a
single point, and hence (FT 1 ) is trivially exact. Assume now that the sequence
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 53
(FT k ), k > 1, is exact. We wish to show that (FT k + l ) is also exact. Note first
of all that the exactness of (FT k ) and Lemma 1 imply that
q (H*(S2k-I)) q (H*(U k )) q (H*(U k - I )) 0
is exact. Hence a simple inductive argument shows that the graded algebra
H*( Uk) is generated by homogeneous elements Xl , ... , X 2 k-1 such that deg Xi
= i, i = 1, 3, ..., 2k - 1. Consider next the spectral sequence {Er (Pk+l),
r > 2} of the fibrationpk+l: U k + l S2k+l. Observe that
E 2 *'* = H*(S2k+l) <8> H*(U k )
as algebras. Since d 2 (Xi) is ofbidegree (2, i-I) and 2 < 2k + 1, for k > 1,
it follows that d 2 (Xi) = O. Hence d 2 kills all the elements in the fiber and
therefore all of E 2 . Hence E 2 = E3. A similar argument shows that E3 = E4
= Es = ... = Eoo. This is enough to prove the induction step and hence the
lemma.
Q.E.D.
The preceding argument shows that the spectral sequence {Er(p), r > 2}
is trivial. Hence
q (Eo (H*(U n )) = q (H*(S2n-l)) + q (H*(U n - 1 )).
Moreover, as a simple inductive argument shows, H*(U n ) is torsion-free.
LEMMA 3 q (Eo (H*(U n )) q (H*( Un)).
Proof Let {Xl' X3, ..., X 2n -l} be a basis for q (Eo (H*(U n )) and elements
Xl' ..., X2n-1 E H*(U n ) which map onto Xl' ..., X 2n -l. To prove that {Xl' ...,
X 2n -l} generate H*(U n ) it is enough to show that every homogeneous ele-
ment X E Hk( Un) can be written as a polynomial in the X/So Consider therefore
the filtration of Hk( Un),
Hk(U n ) = JO,k JI,k-1 ... Jk,O {O},
which defines Eo (H*(U n )). Suppose X e Jj,k-l - Jj+l,k-j-l. Then X, the
equivalence class of X modules Jj+ l,k-j-l, is equal to apolynomialfl (Xl' ...,
X 2n - l ) in Xl' ..., X 2n - l . Hence X - fl (ql, ..., qn) eJ j + l ,k-j-l. Since the
filtration of Hk(U n ) is finite, we can proceed as above and obtain polynomials
12' ...,h such that
X = 11 (Xl' ..., X 2n - 1 ) + ... + h (Xl' ..., X2n-l).
54
THE TOPOLOGY OF CLASSICAL GROUPS
Hence {Xl' ..., X 2n -l} generate the graded algebra H*(U 2n - I ). The linear
independence {Xl' ..., X2n-l} is easily proved.
Q.E.D.
We conclude from the preceding lemma that Ln = q (H*( Un)) is free and
hence can be lifted by a monomorphism
in: Ln -+ H*(U n ),
and that Ln is graded as desired. Since H*(U n ) is torsion free and commu-
tative, in can be extended uniquely to an epimorphism of graded algebras
1pn : E(Ln) -+ H*( Un) ·
LEMMA 4 1pn is an isomorphism.
Proof Recall that the Poincare series of a graded free module A is by
definition
P (A, t) = L akt k
kE;O
where a k = rank (A k ). Note that
P(A (8) B; t) = P(A; t). P(B; t).
Now
n
P (E( Ln); t) = TI (1 + t 2 k - I) ,
k=l
SInce
n
E(Ln) = (8) E(X2k-I),
k=l
where E(X2k-l) is a k = 1 exterior algebra on our generator of degree 2k - 1,
and
P (A (8) B) = P(A) . P(B).
n
To prove P (H*(U n ); t) = TI (1 + t 2k - I ), argue inductively on n, using
the fact that k= 1
H*(U n ) = H*(U n - l ) (8) H*(S2n+I).
Problems
Complex and quaternionic Stiefel manifolds
1 Consider the fiber bundle map
Pn,k : On,k(F) -+ On, 1 (F)
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 55
which sends a k-frame (fl, ..., fk) to fk. (Here F is either the complex numbers C or the
quaternions H. Observe that the fiber of Pn,k is On-1.,k-l .) Show that the induced sequence of
cohomology algebras
p* i*
1 -+ H* (On,l; Z) n.k H* (On,k; Z) n-l.k-l H* (On-I,k-I; Z)
is exact. Here in-I, k-I is the canonical injection of On-I, k-I in On,k.
2 Deduce from (1) that H* (On,k(F); Z) is, as an algebra, the exterior algebra of a Z-
submodule Ln,k = L (Ln,k)dJ-I, with each (Ln,k)dJ-I being of rank 1. (d = 2 when
j=n-k+1
F = C and 4 when F = H.)
Recall that the multiplication in On,n(F) turns H* (On,n(F); Z) into an algebra.
3 Show that H * (On,n(F); Z), as an algebra, is isomorphic to E(Ln,n) where Ln,'J is a Z-
n
submodule L (Ln,n)dJ-I c: H* (O",n(F); Z) and each (L" n)dj-I is of rank one.
j=l
[Hint. First let i: On,n(F) 02n,2n(F) and i' : On,n(F) -+ 02n,2n(F) be the imbeddings of
On,n(F) as the subgroups that leave fixed the first n basic vectors and last basic vectors
respectively. Then the diagram
On,n(F) x 0n,n(F) i)( il -)- 02n,2n(F) X 02n,2n(F)
1 1
On,n(F) 02n,2n(F)
where the vertical maps denote multiplication in the groups, is commutative up to
homotopy. Since i*: H* (On,n(F); Z) -+ H* (02n,2n(F); Z) is a monomorphism, deduce
that H * (On,n(F); Z) is commutative as an algebra. Next show that H * (On,n(F); Z) = H *
(On-l,n-I(F); Z) @ H* (Sdn-I; Z) as H* (On-I,n-I(F); Z) modules. Now proceed by in-
duction to finish the proof.]
5 Hopf-algebras
The multiplication in the groups Un and SPn did not figure in the description
of their cohomology groups. In fact, in the computations themselves, these
multiplications were not used, except indirectly in defining the fibrations
over the spheres (see Problem 1).
We wish to show that the multiplications in groups lead to an added struc-
ture on the cohomology algebra, but in order to do so we must introduce
some new algebraic concepts.
We shall say that A is an algebra over K, with multiplication cp, unit'f} and
augmentation 8 if, and only if,
1) A is a grade algebra with multiplication cp;
2) there exists a homomorphism of graded K-modules of degree zero
'f}:K A
56
THE TOPOLOGY OF CLASSICAL GROUPS
(where K is the graded module whose Oth homogeneous component is K, and
is trivial otherwise) such that the composites in both ways
7K@A @1
A )'A <8> A A
@K/I@
are each equal to the identity (i.e., 1'}(1) is a two-sided identity of A); and
3) 8 : A K is a homomorphism of graded algebras and 81'} = identity.
As an example, consider the multiplication
cp:GxG G
in a topological group G. Then cp induces a homomorphism of graded mo-
dules
H*(G) <8> H*(G) H* (G x G) H*(G)
where the homology groups have their coefficients in a commutative ring
with unit K. Moreover, one can easily see that if e EGis the identity element,
then the injection 1'} : {e} G and the constant map 8: G {e} induce a
unit 1'} * and augmentation 8*, respectively on H *( G).
Another example is the cohomology algebra of any space H* (X; K), the
coefficients being in the commutative ring with unit K. A unit is provided by
the constant map X {xo}, Xo being a point in X, and the augmentation by
the inclusion {xo} X.
The next concept we need is that of a coalgebra. We shall say that a
graded module A is a coalgebra with comultiplication 1p, unit 8 and augmenta-
tion 1'} if, and only if,
1) 1p : A A <8> A is a homomorphism of graded modules of degree zero;
2) 8 : A K is a homomorphism of graded modules (K is the graded
module concentrated in dimension zero) such that the composites both
ways
2@y.K@ A "{
A A @ A::>,. A
1@£ A @ K/-==
are each equal to the identity; and
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 57
3) 'f} : K A is a homomorphism of co algebras with unit-i.e.,
a) the diagram
A :!!!. -+ A <8> A
7l t t 71 @71
I I
K ..!!.-+ K <8> K
commutes, where 0(1) = 1 <8> 1, and
b) 'f}8 = identity.
The homology groups H *(X) of any topological space X, with suitable co-
efficients so that H*(X) <8>K H*(X) -- -+ H* (Xx X), is acoalgebra, thecomul-
tiplication being induced by the diagonal map X X x X. On the other hand
the cohomology groups of space have, in general, no comultiplication unless
the space carries a multiplication with a two-sided unit.
An algebra A with multiplication cp is said to be associative if the two maps
cp (cp <8> 1) and cp (1 <8> cp) of A <8> A <8> A into A are equal. A coalgebra A
with comultiplication 1p is said to be associative if the two maps (1p <8> 1) 1p
and (1 <8> 1p) 1p of A into A <8> A <8> A are equal.
Let 7: be the map A <8> A A <8> A which takes a <8> b to (-I)pq b <8> a,
where a E Ap and b E Aq. If A is an algebra with multiplication cp, then we
say that A is commutative if, and only if, cp = cp7:. If A is a coalgebra with
comultiplication 1p, then we say that A is commutative if, and only if, 7:1p = 1p.
The primitive elements of A are, by definition, ker ( A :!!!.-+ A <8> .J), where
A = A/'f}(K).
The two concepts of algebra and coalgebra are combined in the concept of
a Hopf-algebra. We shall say that a graded module A over a commutative
ring K is a Hopf-algebra if
1) A is an algebra with multiplication cp, unit 'f}, and augmentation 8;
2) A is a co algebra with comultiplication 1p, augmentation 'f}, and unit 8;
and
3) the co multiplication 1p: A A <8> A is a homomorphism of graded
algebras.
Suppose that G is a topological space with a map
cp:GxG G
such that cp (e, g) = g = cp (g, e), for all g E G, where e is some fixed element
in G. Assume that K is a field. Then H* (G; K) is a Hopf-algebra whose
multiplication is induced by cp, while its comuitiplication 1p is induced by the
58
THE TOPOLOGY OF CLASSICAL GROUPS
diagonal map. The co multiplication is always associative and commutative;
but the multiplication is associative if cp : G x G G is at least homotopy-
associative, and is commutative if cp: G x G G is at least homotopy-
commutative. In general it need not be either.
Problems
Suppose that V is a graded vector space
algebra K + L @ V of V.
k>O k
1 Show that the map : V -+ V Ef> V which sends x to (x, x) induces a comultiplication
T( V) -+ T( V) @ T( V) which turns T( V) into a Hopf-algebra.
2 Assume that V is graded by even degrees, and let P[V] = T(V)II, where I is the ideal
generated by the elements {x @ y - y @ xix, y e V}. (P[V] is the polynomial algebra gener-
ated by v.) Show that : V -+ V Ef> V induces a co multiplication P[V] -+ P[V] @ P[V]
which turns P[ V] into a Hopf-algebra.
3 Assume that V is graded by odd degrees and the characteristic of K =F 2. Let E( V)
= T(V)/I where I is the ideal generated by the set {x @ y + y @ xix, y e V}. (E(V) is the
exterior algebra generated by v.) Show that : V -+ V Ef> V induces a comultiplication
E(V)-+ E(V)@ E(V) which turns E(V) into a Hopf-algebra.
4 Suppose that V is a graded vector space over a field of characteristic 2. Define E( V) to be
T(V)/S, where S is the ideal generated by all elements of the form x @ y + y @ x and
x @ x. Show that : V -+ V Ef> V induces a comultiplication E(V) -+ E(V) @ E(V) which
turns E( V) into a Hopf-algebra.
If V is one dimensional we shall write T(x, m) for T(V), P [x, m] for P[V] and E(x, m)
for E(V) where x e V and m = deg x.
5 Suppose that A is a connected commutative associative Hopf-algebra, over the prime
field Zp. A is monogenic; i.e., it is generated by one element x EA. Prove that
L
i>O
Vi over a field K. Let T( V) be the tensor
{ p [x, m]
A = P [x, m]/(x S )
E (x, m)
m even
s a power of p, m even
m odd,
ifp =F 2, and
A _ { p [x, m]
- P [x, m]/(x S ) s a power of 2
if p = 2.
6 The homology and cohomology algebras of the classical groups as Hopf-
algebras
Again we shall carry out the computations for the unitary groups; the other
cases will be taken up in the exercises.
We shall assume, unless noted otherwise, that the homology and cohomo-
logy algebr have their coefficients in the integers.
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 59
Observe first of all that
H*(U n ) = Hom (H*(U n ); Z),
since H*(U n ) is torsion-free. Hence H*(U n ) is a Hopf-algebra. It is easy to
check that the multiplication in H*(U n ) is dual to the comultiplication in
H*(U n ), while the comultiplication in H*(U n ) is dual to the multiplication in
H*(U n ) (i.e., H*(U n ) and H*(U n ) are dual Hopf-algebras). Hence
pH*(U n ) = Hom (qH*(U n ); Z),
and therefore the non-trivial homogeneous components of pH*(U n ) occur in
dim 1, 3, ..., 2n - 1.
Form the exterior algebra E(pH* (Un)) and note that the map
pH*(U n ) E (pH*(U n )) <8> E (pH* (Un))
which sends x to x <8> 1 + 1 <8> x, determines uniquely a co multiplication
on E (pH*(U n )) which turns it into a Hopf-algebra.
THEOREM (5.1) The natural injection pH*(U n ) H*(U n ) can be extended
uniquely to an isomorphism
ex : E (pH*(U n )) H*(U n )
of Hopf-algebras.
Proof First we note that a straightforward computation shows that the
element
xy - (_l)rs yx, x E Hr(U n ), Y E Hs(U n ),
is primitive, provided x and yare primitive. But there were no non-trivial
primitive elements in even dimensions. Hence the subalgebra of H*(U n ) is
commutative; and therefore the natural injectionpH*(U n ) H*(U n ) can be
extended to a homomorphism of Hopf-algebras
ex: E(pH*(U n )) H*(U n ).
To show that ex is an isomorphism, consider the dual
ex*: H'*(U n ) Hom (E(pH*U n ); Z).
Observe that
Hom(E(pH*U n ) = E(q(H*(U n )).
60
THE TOPOLOGY OF CLASSICAL GROUPS
Hence lX* is an isomorphism according to Theorem (4.1) of 4 of this chap-
ter.
Q.E.D.
Since
q(E* (pH*U n )) = pH*(U n ),
we can conclude immediately the following theorem.
THEOREM (5.2) The natural injection p (H*U n ) H*(U n ) can be uniquely
extended to an isomorphism
E(pH*U n ) H*(U n )
of Hopf-algebras.
Problems
The Hopf-algebras H* (SPn; Z) and H* (SPn; Z)
1 Show that the Z-submodule of primitive elements Ln of H* (SPn; Z) is of the form
n
L (L n )4j-l and each (L n )4j-l is of rank one and that the natural injection Ln -+ H* (SPn; Z)
j=l
can be extended uniquely to an isomorphism
E(Ln) -- H* (SPn; Z)
of Hopf-algebras.
Similarly for H * (SPn; Z).
The Hopf-Borel Theorem on the structure of Hopf-algebras
2 Suppose that A is an associative commutative Hopf-algebra over the prime field Zp.
Assume also that A is finitely generated. Then as an algebra,
A ::::: @ (I) A ,
i
where @ (i) A is the tensor product of monogenic algebra (i) A over Zp (see Problem 4 of ' 5).
i
[In order to construct an inductive proof, it is enough to show that if A' is a sub-Hopf-
algebra of A and B = A//A' is monogenic, then
A = A' @ B
as algebras. Here A//A' = AIA'A.]
a) Show that
A = A' @ B
as A' -modules. [Choose a liftingj: B -+ A. Consider the map q; : A' @ B -+ A which sends
s
a @ b to aj(b). Suppose that q;(z) = 0, where z = L at @ x s - t , at E A'; x is a generator of
;=1
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 61
s
B; and s is the height of x. To show that ai = 0, consider the image of L ad(x)S-t under
i=l
the composite A -+ A .@ A -+ A @ B, where the first map is the comultiplication in A and
the second is that induced by the natural projection A -+ B. Show that ai @ j(X)S-i cannot
cancel and, hence, that ai = 0.]
b) Deduce the theorem from (a) when deg x is odd and p is 0 or odd, or when deg x is even
and s, the heibnt of x, is infinite.
To finish the proof suppose that s, the height of x, is finite and that A' is generated by
elements of deg n = deg x. Choose x' E A such that x' -+ X e B = AlIA'. Step (a) above
implies that x's is a polynomial in the powers of x', with coefficients in A'. Moreover, all
t e coefficients have positive degree. Now show that x's E A', for otherwise the diagonal
expansion of x'S (i.e., the image of x's under the comultiplication in A) would not be right.
Next note that the image of x'S in A' I lA's is primitive and, hence, is zero, since A' I lA's is
generated by elements of deg n and the height of every element in A' I I A's is s. Here
- -
A's is the sub-Hopf-algebra Zp + A's where A's is the s-fold product of A' by itself. There-
fore, there is an element YEA' such that
y S = x S .
Observe now that
(x' - y)S = O.
7 Generating complexes for the classical groups
We have seen in the preceding sections that the homology (and cohomology)
algebras of the classical Lie groups Un and Spn can be described by means of
a "universal construction" on graded modules. For example, H * (Un; Z) is
an exterior algebra on a free abelian group graded by odd degrees correspond-
ing to the spheres S1, S3, ... , S2n-1 that occur in the natural fibrations
U k - 1 -+ Uk -+ S2k-1. So, in some sense, the information about H*(U n ) is
concentrated in the generators.
By definition, a generating complex for SUn is a CW-complex X which can
be mapped into SUn by a map i: X -+ SUn in such a way that the induced
map on the integral homology coalgebras
i*: H*(X) -+ H*(SU n )
can be extended to an isomorphism of Hopf-algebras
E(H*(X)) -+ H* (SUn; Z).
The object of this section is to show that such a generating complex can be
found for SUn. Now consider the commutator map
c: U i X Un -+ SUn
62
THE TOPOLOGY OF CLASSICAL GROUPS
which sends (r, a) to a7: a- 1 7:- 1 . As can be easily verified, c induces a map
c: U1 X(U,,/U1 x U;-l) --+ SU".
But U,,/U 1 x U -l = CP,,-l the complex (n - I)-dimensional space. Thus
we can write
- 1
c:S XCP"_l--+SU".
Note that c takes Sl V CP,,-l to the identity elements of SU". Hence we get
amap
f,,: S A CP,,-l --+ SU"
where S A CP n - 1 is the reduced suspension of CP n - 1 .
Observe that the following diagram
., ,
In+l Pn+l /
S A CP,,-l -04 S A CP" -04 S A CP" S A CP n - 1
ifn lfn+l lhn+l (7.1)
SU" . ) SU,,+l ) sm+1
I n +l P n +l
is commutative. Here i"+l and i +l are the inclusion maps; P"+l is the ca-
nonical fiber map which sends a E SU,,+l to a(8,,+1) E S2n+1, and p + 1 is the
map which identifies S A CP,,-l with the basepoint in S A CP".
PROPOSITION (7.2) The map
-
h"+l: S A CP,,/S A CP n - 1 --+ S2n+l
is a homeomorphism.
Proof Let us prove first of all that h"+1 is injective. So suppose that s,
s' E S A CP"/S A cp,,-1 are such that h"+I(S) = h,,+I(S'). Let (7:, a) and
(7:', a') be elements of U 1 X U,,+1 representing sand s' respectively. The
assumption h"+l(S) = h"+I(s') implies that
a7:G- 1 -r- 1 (s,,+I) = a'7:'a'-17:'-1(8,,+1).
But 7:, 7:' E U 1 . Hence
-1 ( ) , , '-1 ( )
a7:a 8" + 1 = a 7: a 8n + 1 · .. .
(1)
Let VI C C,,+ 1 be the line generated by 81' and let vt be its orthogonal
complement. Write,
a- 1 (C"+I)=tX+P, tXEV 1 , PEVt
and
a'-1(8,,+I) = tX' + P', tX' E VI' P' E vt.
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 63
These last two equations imply that
En+l = a(lX) + a(f3) = a'(lX') + a'(f3') ....
(2)
Thus
G7:a- 1 (En + 1) = O"i (lX) + a(f3)
and
a' i' a' - 1 (8n + 1) = a'i' (lX') + a' (f3') .
2 °t 2 0,
Since i, i' E U 1 , we can write i = e 7U and i' = e 7Ut . Now, using (2) we get
(1 2nit ) ( ) (1 2nit' ) ' ( ' )
- e a lX = - e a lX ....
(3)
Case 1: a(lX) = O.
Since a is invertible, lX = 0 and, hence, a- 1 (En+ 1) = f3. Choose a" E U such
that a"-I(f3) = En+l. Then a"-la- 1 (En+l) = En+l and thus aa" E Un.
Hence (i, aa") E U 1 X Un represents a point in 8 1\ CPn-l and thus is map-
ped into the basepoint of 8 2n + 1 = SU n + 1 /8U n . Since (i, aa") differs from
(i, a) by an element in U , it is in the same class (i.e., (i, a) is a representative
of the basepoint of 8 1\ CP n /8 1\ CP n - 1 ), and therefore (i, a) is mapped
onto the basepoint of 8 2n + 1. By (3) either a'(lX') = 0 or e 2nit ' = 1. If
a'(lX') = 0, then, as before, (-r', a') is a representation of the basepoint of
8 2n + 1 and thus s = s'. If e 2nit ' = 1, then
/"+1 (i', a') = a'i'a'-l i '-1 = 1,
and so (i', a') is mapped by hn+l into the basepoint of 8 2n + 1 ; this shows that
in this case too (i', a') is a representative of the basepoint of 8 1\ Cpn / 8
1\ Cpn - 1 .
Case 2: a(a) =F O.
If i = 1, same as before. Thus assume that i =F 1 and, hence, a'(IX') =F 0 and
i' =F 1. Notice that (3), together with the facts that lX, lX' E VI and a, a'
preserve the length of the vectors, shows that we can write
a(E 1 ) = e 2niU a'(81)
, -1 ( ) 2niu
a a El = e 81.
In other words, a'-l a takes VI into VI and vt into vt and, therefore,
a' -la E U 1 x U . Hence we can assume without loss of generality that a = a'.
Therefore lX = f3', f3 = f3', and i = i'. Thus h n + 1 is injective; (8 A cpn/8
64
THE TOPOLOGY OF CLASSICAL GROUPS
A cpn-l) being compact and connected, image (h n + 1 ) is a connected
subspace of S2n+l homeomorphic to S2n+l-which is enough to imply that
h n + 1 is a homeomorphism.
THEOREM (7.4) The induced homomorphism
(h+l)*: H* (S /\ CPn; Z) --+ H* (SU n + 1 ; Z)
is an isomorphism of H* (S A CP n ) onto p (H* (S A CPn; Z)), the Z-sub:'
module of primitive elements of H* (SU n + 1 ; Z). Moreover, (/"+1)* can be
uniquely extended to an isomorphism
E( H* (S A CP n ; Z)) --+ H* (SU n + 1 ; Z)
of Hopf-algebras.
There is a similar theorem for cohomology and, since H * (SUn + 1; Z) is
free of torsion, for coefficients in Zp, all primes p.
The proof of Theorem (7.4) can be easily constructed by induction, using
the commutative ladder with exact rows
{O} --+ P (H* (S A CP n - 1 ; Z)) --+ P (H* (S A CP n ; Z)) --+ p(H* (S2n+ 1; Z) --+ 0
1 1
o --+ p (H* (SUn; Z)) ) P (H* (SUn + 1; Z)) --+ p (H* (S2n+ 1 ; Z) --+ 0
and noting that H* (S /\ CP n ; Z)) is a coalgebra in which every element of
positive dimension is primitive.
Problems
The real projective space RPn
1 Show that
{ Z, k = 0, k = n, n odd
H* (Rpn; Z) = O Z2, k odd and <n
otherwise
and when n is odd that the homomorphism (Pn)* : Hn (sn; Z) -+ Hn (Rpn; Z), induced by the
twofold cover Pn : sn -+ Rpn, takes a generator of Hn (sn; Z) onto two times a generator of
Hn (RPn; Z).
2 The cohomology ring H* (RP n ; Z2) is generated by an element t E HI (RPn; Z2) such
that t n + 1 = O. (I.e., H* (RPn; Z2) is the truncated polynomial ring P [t, 1]/(t n + 1).)
[Hint. Proceed by induction. Relate Pn: sn -+ Rpn to the reflection map of S".]
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 65
Rpn-l as a generating complex for SOn
3 Consider the map
0 1 X On --+ SOn
which takes (r, 0') to 0'1:0- 1 1:- 1 . Show that it induces a map
f n
Rpn-l ) SOn
which takes Rpn-l into SOn and such that the induced map Rpn-l/RPn-2 -+ sn-l is a homeo-
morphism. Here Rpn-l/RPn-2 is the space obtained from RPn-l by collapsing RPn-2 to
a point.
The Stiefel manifold On,2(R)
Consider the fiber bundle map
Pn,2 : On,2 On,l = sn-l
with fiber On-l,l = sn-2.
4 Show that the induced sequence of algebras
{ I} -+ H* (sn-l ; Zp) -+ H* (On,2 ; Zp) -+ H* (sn-l ; Zp) -+ {I}
is exact if p = 2, or n even and p arbitrary.
5 Deduce that additively
H* (On,2; Zp) = H* (8"-1 X sn-2; Zp),
where p = 2, or when n is even and p is arbitrary.
[The case when n = 3 and p = 2 shows that this equality is not necessarily one of al-
gebras.]
6 Show that, as algebras,
H* (On,2; Z) = H* (sn-l x 8"-2; Z),
if n is even.
7 Show that ifn is odd then, additively,
H* (On,2; Z) = H* (8"-2 U e n - l U s2n-3; Z),
where 8"-2 U e n - l is the space obtained by joining the (n - I)-disk Dn-l to sn-2 by a map
O -l -+ sn-2 of degree 2.
[Hint Use the imbedding Rpn-1/RP n - 3 -+ On,2 to determine the differentials of the spec-
tral sequence of Pn-2 : On,2 -+ sn-l.
The cohomology algebra H* (On,k; Z2)
8 Show that the usual fibration
Pn,k: On,k -+ sn-l, fiber = 0"-1, k-I
which sends a k-frame (VI, ..., Vk) to.Vk induces the exact sequence of algebras
1 --+ H* (8"-1; Z2) --+ H* (On,k; Z2) H* (On-l, k-l; Z2).
5 Husseini (0216)
66
THE TOPOLOGY OF CLASSICAL GROUPS
Hence, additively,
H* ( 0 . Z ) /'OIJ H* (8" -1 X S n-2 X ... X S n-k. Z )
n,k, 2 /'OIJ , 2.
To compute the algebra structure of H* (On,k; Z2) proceed as follows.
9 Show that H* (On; Z2) is commutative.
[Show that the maps IX, p : On X On -+ 02n, where IX (x, y) = xy and p (x, y) = xy' (y' being
the element of On one gets by letting y act on the last n basic vectors), are homotopic.]
10 The map In: Rpn-l -+ SOn induces a monomorphism (!n)*: H* (RPn-l; Z2}
-+ H * (SOn; Z2) which is uniquely extendable to an isomorphism of HopI-algebras
({In: E ( H * (RPn-l; Z2») =--+ H * (SOn; Z2).
[Hint Proceed by induction. Assume fn-l : RPn-2 -+ SOn-l induces a monomorphism
(!n-J* : H * (RPn-2; Z2) -+ H * (SOn-I; Z2)
which in turn extends to an isomorphism
({In-l: E(H* (Rpn-2; Z2» -+ H * (SOn_l; Z2)
of Hopf-algebras. To prove the corresponding statement for n, show first that the square of
every element in image (!,,)* is o. So let t n - k E Hn-k (RPn-l ; Z2) be the generator dual to
t n - k E Hn-k (Rpn-l; Z2). Note that
(fn)* (t n - l ) (fn)* (t n - l ) @ 1 + 1 @ (fn)* (tn-l) + L (fn)* (tn-i) @ (!n)* (t i ).
i>O
The induction assumption implies that (!n)* (t n _ l )2 is primitive. But there are no primitive
elements of H* (SOn; Z2) in dimension 2 (n - 1), since there are no indecomposable
elements of that dimension in H* (SOn; Z2). Hence (!n)* extends to a map
({In : E (Ii * (RPn-l; Z2») -+ H * (SOn; Z2)
of Hopf-algebras. Finally by comparing dimensions one proves the assertion.]
11 By computing the primitive elements of E ( H * (RPn-l; Z2»), show that
H* (SOn; Z2) = P[t l ]/(t: 1 ) @ ... @ P[t2i-l]/(t j_l) @ ...
where deg t2l-l = 2i - 1, and Sl is the smallest power of2 such that Si (2i - 1) 2:: n.
12 Deduce from 11 that H*(On,k; Z2) is isomorphic to the subalgebra of H* (SOn; Z2)
generated by the elements tr 1 , ..., t;T-l' whererl is the smallest power 0[2 such that ri (2i - 1)
2:: n - k.
The cohomology algebra H* (On,k; Zp), p 9= 2
This computation is easier than that in the case p = 2.
COHOMOLOGY AND HOMOLOGY OF THE CLASSICAL GROUP 67
13 Suppose that n is odd. By considering the fibrations
SOn-2 -+ SOn -+ On,2,
show that H* (SOn; Zp) = E (X3, Xl, ..., X2n-3), where deg Xi = i. [E (X3, Xl, .. ., X2n-3)
is the exterior algebra of the graded vector space generated by X3, ..., X2n-3.]
14 By considering the fibrations
SOn -+ SOn+l sn, n odd,
show that
H* (SOn+l; Zp) = E(X3' Xl, ..., X2n-3; X n ),
where deg Xi = i.
15 Deduce from 13 and 14 the structure of H* (On,k; Zp).
Generating complex for SPn+l
Suppose Sp is the subgroup of SPn+l which leaves the first basic vector £1 fixed. Let SPl act
on SPI X SPn+l/Sp according to the rule (x, (a, -i) -+ (xax- 1 , Xr), where y is the image of y
,..,
in SPn+l/Sp . Denote by Ln the orbit space (SPl X SPn+l/Sp )/SP1. Now consider the map
c: SP1.XSPn+l -+ SPn+l
( a, '7;) -+ 't'-l a't' .
,.., ,..,
It induces a map of Ln into SPn+l; and, since Ln ';:) Hpn = SPn+l/SPl X Sp , the n-dimen-
,..,
sional quaternionic projective space, it induces a map.t;, of Ln = LnlHpn into SPn+l.
16 Show that fn induces a monomorphism fn*: H* (Hpn; Z) H* (SPn+l; Z) which is
uniquely extendable to an isomorphism
E (Ii * (Hpn; Z») -+ H * (SPn+l; Z)
of HopI-algebras.
Bibliography
[1] A. Borel, "Sur'1a cohomologie des espaces fibres principaux et des espaces homogenes
de groupes de Lie compacts", Ann. of Math. 57 (1953), 115-207
[2] A. Borel, "Sur l'homologie et la cohomologie des groupes de Lie compacts connexes",
Amer. J. Math. 76 (1954), 273-342
[3] Seminaire H. Cartan 1959-60, Ecole Normale Superieure
[4] H.Cartan and S.Eilenberg, Homological Algebra, Princeton Univ. Press, 1956
[5] J. Milnor and J. Moore, "On the structure of Hopf-algebras", Ann. of Math. 81 (1965),
211-64
[6] E. Spanier, Algebraic Topology, McGraw-Hill, 1966
[7] N.E.Steenrod and D.Epstein, Cohomology Operations, Ann. of Math. Studies No. 50,
Princeton Univ. Press, 1952
CHAPTER 3
The homology and cohomology of the
classifying spaces and loopspaces of the
classical lie groups and Bott periodicity
WE SHALL COMPUTE the cohomolgy and homology groups for the case of the
unitary groups. The real and ,symplectic cases will be briefly indicated in the
exerCIses.
The computations we give describe H* (BUn; Z) as the algebra of sym-
metric functions 0 f a polynomial algebra P [t 1 , ... , t n ], deg t i = 2, and hence
is itself a polynomial algebra P [0"1' ..., O"n], deg t i = 2i. The fact that
H* (BUn; Z) is a polynomial algebra depends only on the fact that H * ( Un; Z)
is an exterior algebra. It is a special case of Borel's theorem [2], [5]. Similar
descriptions for H* (BSO n ) and H* (BSP n ) are outlined in the problems at
the end of the section. These results are of course special cases of theorems
which are proved, with more sophisticated techniques, for a general Lie group.
(See [2] and, for a treatment in the context of homological algebra, [5].)
In 2 we define characteristic classes for bundles following the Borel-
Hirzebruch method [3]. In the exercises we outline some of the other ways of
looking at characteristic classes.
In 3 the homology of the loopspace QSU n is computed as a preliminary
to proving the Bott periodicity theorem for U. The first algebraic-topological
proof was given by Toda in [6] (see also [5]).
In the last section the Orothendieck group K(X) is defined and the Bott
periodicity is formulated as a formula of the Kiinneth type [4]. This formula-
tion is due to Atiyah and is the starting point of his proofs of the periodicity
theorem [1].
1 The cohomology algebra H* (BUn; Z)
Suppose that Tn C Un is the subgroup of diagonal matrices
e21ti61 0
.
0"=
o · e21ti9n
68
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 69
The homomorphism
Ok : Tn -+ S 1 , k = 1, ... , n,
which sends G to its kth diagonal element e21ti9k, induces, at the cohomology
level, the homomorphism
0: : HI (8 1 ; Z) -+ HI (Tn; Z), k = 1, ..., n.
By definition, let
Uk = O:(x),
where x is a generator of HI (SI; Z). The elements Ul, ..., Un are certainly a
basis for HI (Tn; Z). Now suppose that
tX n : E -+ BUn
is a universal bundle for Un. Since Tn is a subgroup of Un, it acts freely on E,
and the natural projection
13n: E -+ E/Tn
is a universal bundle for Tn; moreover, we get the commutative diagram
E E/Tn = BTn
l rn
BUn
Observe that 'Yn is a bundle map with fiber Un/Tn. Now let
N T I1 = {x E Un I xTn x -l c Tn};
i.e., NTn is the normalizer of Tn. It is easy to see that NTn/T = 8n, the full
symmetric group on n letters. We know from the first chapter that an auto-
morphism of a group can be extended to a bundle equivalence of its universal
bundle. Thus the group 8n acts on BTn.
[One can exhibit the actions of 8n = N Tn/Tn on the universal bundles
directly, without appealing to the general considerations of Chapter 1, by
selecting suitable models for the universal bundles. Consider therefore
UN+n/U N " the Stiefel manifold of orthonormal n-frames in CN+n. Here U
is the subgroup of U N+n which leaves the first n basic coordinates fixed. Sup-
pose that Un is imbedded in U N + n as the subgroup leaving the last N basic
coordinates fixed. The inner automorphism of U N + n defined by an element
70
THE TOPOLOGY OF CLASSICAL GROUPS
X E NTn takes the subgroups U , Tn X U , and Un into themselves and, hence,
induces bundle equivalences
UN+n/U; -+ uN+n/u;
1
UN+n/T" X uft -+ UN+,,/T" x u;
and
U N + lI /Tn x u; --+ U N + lI /T" x U;.
! 1
UN+n/U lI X u; --+ uN+n/u" X UN'
Now if we let N --+ 00, we obtain the desired actions.]
What is the effect of the action of 8n on the cohomology of H* (BI n; Z)?
To describe the action we need the following.
PROPOSITION (1.1) The transgression in the universal bundle of Tn
7:: HI (Tn; Z) --+ H 2 (BTn; Z)
is an isomorphism, and the cohomology algebra H* (BTn; Z) is a polynomial
algebra on the elements t 1 = -7:(Ul)'''.' t n = -7:(u n ) of H 2 (BTn; Z).
Proof The first statement is just that
d · E O ,1 E 2,0
2. 2 --+ 2
of the spectral sequence of 13n: E -+ BTn is an isomorphism. The second
statement follows immediately from the fact that BTn is of the homotopy-
type of the n-fold cartesian product Cp oo x ... x Cpoo of the complex
projective space Cpoo.
It is clear now, since 7: is natural, that Sn acts on H* (BTn; Z) as the full
symmetric group of the symbols {t 1 , ..., t n }. The main object of this section
is to prove the following theorem.
THEOREM (1.2) The homomorphism
(Yn)*: H'* (BUn; Z) --+ H'* (BTn; Z)
induced by Yn is an isomorphism of H* (BU; Z) onto the subalgebra of sym-
metricfunctions in H* (EjTn; Z). Hence H* (BUn; Z) is a polynomial algebra
p [Yl, ..., Yn], where deg Yi = 2i.
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 71
Proof Proceed by induction on n. If n = 1, 1'1 is the identity map, and
the theorem follows trivially. Assume now that
H* (BUn-I) f'OoJ !7P [t 1 , ..., tn-I] C H* (E/Tn-l), n > 2.
We need to show that
H* (BUn) f'OoJ !7 P [t 1 , ..., t n ] C H* (E/Tn).
Consider first the fibration
n: E/U n - 1 --+ EU n
with fiber S2n-l = Un/Un-I' which is associated with the usual imbedding
U n - 1 --+ Un.
LEMMA (1.3) The induced homomorphism
: : H* (BUn-I) --+ H* (BUn-I)
has the following properties:
i) n* is an epimorphism;
ii) : is an isomorphism in dimensions 2n - 1;
iii) ker ( : : H2n (B Un) --+ H 2 n (B Un _ 1)) f'OoJ z.
Proof The spectral sequence {Er( n)' r > 2} of the fibration n: BU n - 1
--+ BUn determine the exact sequence
... --+ Hk (BUn-I; Z) --+ Hk (S2n-l ; Z) Hk+ 1 (BUn; Z) --+ ...
--+ H2n-l (BUn-I; Z) -+ H2n-l (S2n-l; Z) 2-> H2n (BUn; Z)
-+ H2n (BUn-I),
since Hi (S2n-l; Z) = {O} for i 2n - 2 and HI (BU 1 ; Z) = {O}. [Cf. 3
of Chapter 2.] One can easily show that : is an epimorphism in dimensions
2n - 2. But by the induction assumption H* (BUn-I; Z) is generated by
elements of degree 2n - 2. Hence * is an epimorphism in all dimensions.
The other two parts are proved using the preceding exact sequence.
Q.E.D.
Consider next the fibration
#n: EjU I x U:- 1 --+ BUn
corresponding to the subgroup U 1 x U -I' where U;-1 is the subgroup of Un
consisting of all these elements which leave the first basic vector 81 fixed.
Note that the fiber of /In is U n /U 1 x U;-1 = CP n - l .
72
THE TOPOLOGY OF CLASSICAL GROUPS
LEMMA (1.3) The spectral sequence {Er{P,n), r > 2} of f1n is trivial.
Proof Observe that H* (cpn-l) = P[t]/(t n ). Since d 2 : E 2 E 2 is a
derivation of algebras and E 2 = H* (BUn) (8) H* (CP-l) as an algebra, it is
enough to show that d 2 (t) = O. But this is so for dimensional reasons. Hence
E3 = E 2 = H* (BUn) (8) H* (cpn-l). Now d 3 (t) = 0 because H3 (BUn)
= {O}, n > 2, as in Lemma (1.2). (The higher differentials vanish for
dimensional reasons.) .
Consider next the ascending sequence of groups
Tl xTn-l c: U 1 x U;-1 c: Un.
From it, we obtain the commutative diagram
1 x p.' 1 ,
E/Tn n-) E/U 1 X U n - 1
P.n
BUn.
That y: is a monomorphism follows now from the fact that (Pn)* and
{I XP -l}* are both monomorphic. Moreover, im f1 ! 1 consists of elements
symmetric in t 2 , ..., tn. Hence the elements in im y: have that property.
Similarly, by considering the imbedding U n - 1 X U; c: Un, one shows that
the elements in image y: are symmetric in t 1 , ..., t n - 1 also. Hence
imy: c:f/P[t 1 ,...,t n ].
To complete the proof of the theorem we must show that
9' P [t 1 , ... , tn] c: im y: .
So consider first the projection
q; : P [t 1, ..., t n] P [t 1, ..., t n - 1 ]
which takes t k to t k for k n - 1 and t n to o. If aI, a2, ..., an are the prim-
itive symmetric functions in P [t 1 , ... , t n ], and G 1, G 2, ... , G n -1 are the prim-
itive symmetric functions in P [t 1 , ..., tn-I], then q; induces a map from
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 73
P [aI' ..., an] c: P [t 1 , ..., tn] to P [G 1 , ..., G n - 1 ] c: P [t 1 , ..., tn-I] which
takes ak to G k for k n - 1 and an to O. Now consider the commutative
diagram
i'*
P [t 1 , ..., tn-I] = H* (BTn-l) +- n-l H* (BUn-I)
i t
P [t l' ... , tn] = H (BTn) (
7*
n
H* (BUn).
Observe that
i) : is an isomorphism in dimension 2n - 1;
ii) deg G i = 2i, i = 1, ..., n - 1;
iii) q; is an isomorphism in dimensions 2n - 1.
These facts imply that aI, ...,a n -l eimy:. To finish the proof, we must
show that an e im y: . We know that the kernel of : in dimension 2n is iso-
morphic to Z. Let x e H2n (BUn) be the generator. Note that :(x) = 0
and x =t= O. Since ')I: is injective, y:(x) =t= O. Hence, if we write
y:(x) = c · an + f(al, ..., an-I),
c and f(al, ..., an-I) cannot be both zero. Notice thatf(al' ..., an-I) = 0;
otherwise q;y:(x) = f(G 1 , ..., G n -l) =t= 0, contradicting the fact that
q;y: (x) = 0 which follows, since :(x) = O. Therefore
y:(x) = can, c =t= O.
Finally we must show that y:(x) = an.
LEMMA (1.4) H2n (BTn)/y: H2n (BUn) is torsion-free.
Proof The spectral sequence of BU 1 XBU -1 BUn with fiber CP n - 1 is
trivial. Now
Ei'* = H* (BUn) @ H* (CPn-l)
is torsion-free and, hence,
Eo (H* (BU 1 XBU _I)) = Eoo
is also torsion-free. Since Hk(BU n ) = E 'o c: H*(BUl'xBU -I)' we can
conclude that
H* (BU 1 XBU _I)/H* (BUn)
74
THE TOPOLOGY OF CLASSICAL GROUPS
is torsion-free. Finally, an inductive argument shows that H* (B (U 1 x U:- 1 ))
is a direct summand of H* (BTn). Hence H* (BUn) is a direct summand of
H* (BTn) also.
Problems
The cohomology algebra H* (BOn; Z2)
Suppose that (Xn : E -+ BOn is a universal bundle for the orthogonal group On, and let,
(Ol)n = 0 1 X ... X 0 1 be the subgroup of On consisting of those elements a such that
a = + , i = 1,..., n,
where e-;', ..., e;, are the basic vectors in R n . Then 0 1 X ... x 0 1 acts freely on E, and
Pn: E -+ EI(Ol)n
is a universal bundle for (Ol)n. Observe that we also get the commutative diagram
E EI(Ol)n
17n
BOn.
1 Show that H* (EI(Ol)n; Z2) = P [t l , ..., t n ], with deg t i = 1, and that
y : H* (BOn; Z2) -+ H* (E[( 01)n; Z2)
is an isomorphism of H* (B0 2 ; Z2) onto the algebra of symmetric functions in the poly-
nomial algebra P [t 1, ..., t n ].
The cohomology algebra H* (BS0 2n + l ; Zp), p 9= 2
By definition, let
Tn = S02 X ... X S02 n times, n 2:: 1,
and suppose that Tn is imbedded as the subgroup of all elements a e S02n+l such that
j-l = COS 2n0 J "i;,j-l - sin 2nO je-;'}
a j = sin 2nO/i 2 }-1 + sin 2nOj j
for j = 1, ..., n, and
..
aC2n+l = e2n+l.
H -..... . f 2 +1 - - 1
ere el, ..., e2n+l are the basIc vectors 0 R n . Let Ul, ..., un e H (Tn; Zp) be the
generators defined as in he case of unitary group. Now suppose that
tXn : E BS0 2n + 1
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 75
is a universal bundle for S02n+l. Consider the following commutative diagram
E /l2n+ 1 . EIT n = BTlI
"'2" 1 '1'2,,+1
BS0 2n + l .
Again, as in the unitary case, define t j E H 2 (BT n ; Zp) by the equations
t j = -7:(Uj), j = 1, ..., n,
where 7: is the transgression in E -+ EfT n = BT n . Let N Tn be the normalizer of Tn in S02n+l .
2 Show that NTnfT n is the group generated by the permutations of the set {O l , ..., On} and
the changes of sign OJ -+ - OJ.
It follows that the induced action of NTnlT n on H* (BT n ; Zp) = P [t 1, ..., t n ] is to per-
mute the t/s and to change the sign t j -+ - t j for any number of t/s; and, hence, the sub-
algebra of P [t l , ..., t n ] left invariant by NTnlTnis P [0"1' ..., O"n] where O"j is thejth primitive
c, t " f 2 2
1unc Ion 0 t l' ..., tn.
3 Prove that
(Y2n+l)* : H* (BS0 2n + l ; Zp) -+ H* (BTn; Zp)
is an isomorphism of H* (BS0 2n + l ; Zp) onto P [0"1, ..., O"n].
[The proof proceeds by induction along the lines of the unitary case. The proof of the
induction step is done in two stages. First, one proves that (Y2n+l)* is a monomorphism
and, since the effect of NTn/T n on BS0 2n + l is trivial, that im (Y2n+l)* c: P [0"1, ..., O"n].
Second, one proves that (Y2n+l)* is onto P [0"1, ..., O"n]. The proof of the first stage depends
on the consideration of the fibration
f-l : Ef S0 2n-l X S02 -+ BS0 2n + l
induced by the imbedding S02n-l X S02 S02n+l. The fiber is S02n+l/ S02n-l X S02 .
Now one shows that H* (S02n+l/S02n-l X S02; Zp) is a truncated polynomial algebra
generated by a 2-dimensional element by considering
S02n+l/ S0 2n-l -+ S02n+l/S02n-l XS02
and observing that H* (S02n+l/S02n-l; Zp) H* (S4n-l; Zp), where p 9= 2. This implies
that the spectral sequence of f-l is trivial, which is enough to imply that (Y2n+l)* is a mono-
morphism. The second stage of the proof depends on considering the fiber bundle
BS0 2n - l -+ BS0 2n + l
with fiber S02n+l/S02n-l and is entirely similar to the unitary case, since S02n+l/S02n-l
is a homology (4n - I)-sphere modp, with p 9= 2.]
The cohomology algebra H* (BS0 2n + 2 ; Zp), P 9= 2
By definition, let
Tn+l = S02 X ... X S02 (n + I)-times,
76
THE TOPOLOGY OF CLASSICAL GROUPS
and suppose that Tn+l is imbedded in S02n+2 as usual. Again let Uj E HI (T"+l; Zp),
j = 1, ..., n + 1, be the basis defined as above. Suppose that NTn + 1 is the normalizer of
T ,,+l. SO
In 2n+2.
4 Show that NTn+ IfT n +l is the group of transformations generated by the permutations of
the set {Ul, ..., Un+l} and the map
Uj -+ -Uj
for an even number of uis.
Next consider the commutative diagram
E /l2n+2 EfTn+l = BTII+l
"'2" 1' 12n + 2
BS0 2n + 2 .
defined as above, where E -+ BS0 2n + 2 is a universal bundle for S02n+2. Now note that if
we put H* (BT n + l ; Zp) = P [t l , ..., t n + 1 ], where tj = 7:(Uj) as above, then (Y2n+2)* takes
H* (BS0 2n + 2 ; Zp) into the subalgebra P [0'1, ..., an; x] c: P [t l , ..., t"+l]' where aj is the
jth primitive function of tI, ..., t;+ 1 and
x = t 1 ... t"+l .
Observe that P [0'1' ..., an; x] is the subalgebra left invariant by N T 2n+2/T 2n + 2 .
5 Prove that
(Y2n+2)* : H* (BS0 2n + 2 ; Zp) -+ H* (BT" + 1 ; Zp)
is an isomorphism of H* (BS0 2n + 2 ; Zp) onto P [0'1, ..., an; x].
[Again the proof proceeds by induction on n, and the induction step is done in two
stages. One proves first that (Y2n+2)* is a monomorphism and, second, that it is onto
P [0'1, ..., an; x]. For the first stage, consider the bundle
f.l2n+2 : BS0 2n X BS0 2 -+ BS0 2n + 2
with fiber S02n+2/ S02n X S02 . Show that the spectral sequence of f.l2n+2 is trivial by show-
ing that
i* : H* (BS0 2n X BS0 2 ; Zp) -+ H* (S02n+2/S02n XS02; Zp)
is an epimorphism. (One needs to know that H* (S02n+2/ S02n X S02; Zp) is generated by
two elements y and z such that deg y = 2, deg z = 2n, and z E im i*.)
The second stage is proved by considering the fiber bundle
2n+2 : BS0 2n + l -+ BS0 2n + 2
with fiber S02n+2/ S02n+l = s2n+ 1. Show that d 2n + 2 (x2n+l) = X E H 2 n+2 (BS0 2n + 2 ; Zp) is
not zero, where d 2n + 2 is the differential E 2n + 2 -+ E 2n + 2 of the spectral sequence of 2n+2
and X2n+l E H2n+l (S2n+l ; Zp) is the generator. Deduce that
Eoo = E 2n + 3 H* (BS0 2n + 2; Zp)f f(x) H* (BS0 2n + 1 ; Zp),
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 77
where (x) is the polynomial algebra generated by x. These considerations imply that n+2
is surjective. The completion of stage 2 now follows lines such as those of the unitary
case. ]
The cohomology algebra H* (BSPn; Z)
Consider the usual imbedding
SPl X ... X SPl -+ SPn
and the resulting commutative diagram
E fJn ) EISP1X...XSP1=BSPIX ".XBSPl
17n
BSPn
where <Xn: E -+ BSPn is a universal bundle of SPn. Then H* (BSPl x ... X BSPl ; Z)
= P [t 1 , ..., t n], where tiE H 4 (BSPl X ... X BSPl ; Z).
6 Prove that
(Yn)* : H* (BSPn; Z) -+ H* (BSPl X ... X BSPl ; Z)
is an isomorphism onto the subalgebra P [0'1' ..., an], where aj is the jth prilnitive function of
tl,...,t n .
The integral cohomology algebra H* (BOn; Z)
7 Suppose that H* (X; Z) is finitely generated in every homogeneous degree, and let
A
H* (X; Z) be H* (X; Z)/torsion. Assume
{Jp : H* (X; Zp) -+ H* (X; Zp), P prime,
is the Bockstein operation. Show that
A
ker {Jp/im {Jp H* (X; Z) @ Zp
if, and only if, the elements of the p-primary subgroup of H* (X; Z) are all of order p.
[Consider the exact triangle
H* (X; Z) H* (X; Z)
a\
H* (X; Zp)
induced by the exact sequence 0 -+ Z ip Z --:,. Zp -+ 0 where i p is multiplied by p.
Recall that {Jp = jpo. Now, by looking at the successive derived exact triangles or, equi-
valently, the spectral sequence Er that is naturally associated with the exact couple above,
A
we see that the spectral sequence converges and that its limit is isomorphic to H* (X; Z)
@ Zl'. One easily shows that £2 E 00 if, and only if, the statement to be proved is true.]
8 Show that H* (BOn; Z) and H* (BSO n ; Z) have two-torsion only and that every torsion
element has order 2.
78
THE TOPOLOGY OF CLASSICAL GROUPS
[According to the computations above, H* (BSO n ; Z) has two-torsion only, and, since
OnlSOn is of order 2, H* (BOn; Z) also has two-torsion only. To prove the second assertion,
write
H* (BOn; Z2) = P [Wi, ..., W n ] c: P [t 1, ..., t n ],
where Wi is the ith primitive function in t 1, ..., tn. Next show that
H*(BSO n ; Z2) = P [W2, ..., w n ].
Using the fact that the Bockstein P2 is a derivation of algebras, show that
P2Wi = (i + 1) Wi+l
in H* (BSO n ; Z2)
and
P2Wi = Wi Wi + (i + 1) Wi+l in H* (BOn; Z2).
Now the rest follows by applying 7. For example, in the case of H* (BSO n ; Z2), ker P21
im P2 is generated by wt, i = 1, 2, . ... The case of H* (BOn; Z2) is done similarly after
changing the generators Wi, ..., w" according to the rule
U l = W i U 2 " = W 2 " i < [ !!..- ]
, " ", - 2 '
U21+1 = W2i+1 + W1 . W2i, i < [ ]. ]
9 Suppose that 'V n = (E, p, BSO n , SOn, SOn) is a universal SOn-bundle, and let T c: SOn
be the standard maximal torus. Then the map
I' : EIT -+ BSO n
induces a homomorphism
1'* : H* (BSO n ; Z) H* (EIT; Z)
whose image is the subalgebra of H* (EIT; Z) left invariant by the Weyl group of BOn, i.e.,
NTITwhere NT is the normalizer ofT, and the kernel of 1'* is exactly the subalgebra oftor-
sion elements.
There is an exactly similar proposition for On. [The proof depends on the computations
of H* (BSO n ; Zl').]
2 Characteristic classes
Suppose that
= (X,p, B, en, GL(n; C))
is a complex vector bundle over the finite cell-complex B. After introducing
a Hermitian metric in , we can assume that the group of is reduced to Un.
According to the classification theorem of bundles, is determined up to
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 79
equivalence by the homotopy class of a suitable map
f: B BUn,
where BUn is the basespace of a universal Un-bundle E BUn. The char-
acteristic classes of are, in a manner of speaking, computable invariants
attached to f. The ideal characteristic classes, of course, should determine
the homotopy class off uniquely, and hence they vanish if, and only if, the
class [f] is trivial. The characteristic classes based on the usual cohomology
theory fall short of that: their vanishing is a necessary condition for [f] to
be trivial, but not sufficient in general.
In this section we shall introduce the characteristic classes based on
H* (BUn; Z) (the so-called Chern classes). The characteristic classes of real
and quaternionic bundles will be taken up in the exercises. Suppose that
= (X, p, B, Un, Un)
is the principal bundle of , and let
'YJ = (E, lX, BUn, Un, Un)
be a universal Un-bundle. By definition, put
y*Ci = ith primitive function 11'''.' t n
where t 1 , ..., t n E H2 (EIT"; Z) are the generators defined as in the previous
section and y is the projection Eln BUn. Now let
f:B BUn
be the classifying map of . By definition, let
Ci( ) = f*(Ci), i = 1,..., n.
cl ) is called the ith Chern class of , and
n
C( ) = L Ci( )'
i=O
where co( ) = 1 E HO (B; Z), is the total Chern class of .
Observe that Tn c: Un acts freely on X, the total space of , and that
: XI Tn B
is a bundle with fiber UniTn. Moreover, by comparing with y : EIT" BUn
of the universal bundle 'YJ, one sees that
* : H* (B; Z) H* (XITn; Z)
80
THE TOPOLOGY OF CLASSICAL GROUPS
is a monomorphism whose image is a full symmetric sub-algebra of
H* (X/Tn; Z). (Un/Tn acts on X/Tn in a natural fashion.) Thus
n
*c ( ) = n (1 + Xi)
j= 1
where Xi E H* (X/Tn; Z). The classes Xi are called the roots of . For simplic-
ity of notation we shall write
n
c( ) = n (1 + Xj)
i= 1
and regard every symmetric function in the roots Xi as an element of H*(B; Z).
THEOREM (2.1) The correspondence c( ) is uniquely determined by the
following properties:
i) naturality, i.e.,
g*c ( ) = c (g* )
where g* is the bundle induced by a map g E B' B;
ii) Whitney duality, i.e.,
c ( <8> ') = c( ) c( ')
where ' is a Una-bundle on Band
iii) c(v) = 1 + t, where 'YJ is the Hopf-bundle over cpn, n > 1, and
t E H2 (cpn; Z) is the generator whose restriction to CP1 = 8 2 is minus the
suspension of the generator of H1 (8 1 ; Z) as in 1.
Proof We shall prove only that the Chern classes satisfy (ii) above, (i) and
(iii) being obvious.
To prove (ii), let
'YJn+m = (E, lX n + m , BU n + m , U n + m , U n + m )
be a universal Un+m-bundle, m being the dimension of the fibers of '. Now
we let Un X U m act on E as a subgroup of U n + m and consider the commutative
diagram,
E ' n.," ) EjU n x Una = BUn xBU m
l On,m
all+m
BUn +171
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 81
Observe nowthaten,mcorrespondstothe natural imbedding Un X Um U n + m -
Denote Cen,m)* ('Y}n+m) by 'Y}n IT] 'Y}m, where 'Y}n and 'Y}m are universal U n - and
Um-bundles respectively. Next consider the subgroup of diagonal matrices
p+m of U n + m as Tn X 1"", and consider the commutative diagram
E ) EIT"xTm = BTnxBTn&
17 .m =)'.X.7 m
EjUnxU m = BUnxBU m
1 p..m
BU n + m
induced by the inclusions Tn X 1"" c: Un X U m c: U n + m .
Observe now that
e:'m (c('Y}n+m)) = c('Y}n) c('Y}m).
Now to finish the proof observe that e:,m is a monomorphism and
dri(fxf')* (1J" [£) 1Jm) = ',
where f: B -+ BUn and f' : B -+ BUm are the classifying maps for and '
respectively, and d is the diagonal map B B X B.
Q.E.D.
Proof of uniqueness can be constructed along similar lines.
Consider now the problem of computing the Chern classes of complex
vector bundles constructed from others by the operations which generalize
the corresponding operations on vector spaces.
THEOREM (2.2) Suppose that and ' are complex vector bundles.
n
cC ) = n (1 + Xi)
i= 1
and
m
c( ') = n (1 + Yj);
j=l
then
n
i) c( *) = n (1 - Xi);
i= 1
ii) c ( <8> ') = n (1 + Xi + Yj); and
i ,j
iii) c ( (f) ') = n (1 + Xi) (1 + Yj).
i,j
6 Husseini (0216)
82
THE TOPOLOGY OF CLASSICAL GROUPS
The proof of Theorem (2.2) is similar to the proof of the Whitney duality
theorem and will therefore be omitted.
There is another characteristic class. Suppose that is a complex vector
bundle, and assume that
m
c( ) = IT (1 + Xi).
i=l
m
Then L e Xi is a symmetric rational class in the elements Xl' ...., X m and,
i=l
hence, defines a cohomology class in H* (B; Q), B being the basespace of
and Q, the rational numbers. Put
m
ch ( ) = L eXt
i= 1
(ch ( ) is called the Chern character of ).
The proof of the following theorem is similar to that of Theorem (2.1) and
therefore will not be given.
THEOREM (2.3).
ch ( ') = ch ( ) + ch ( ')
and
ch ( @ ') = ch ( ) U ch ( '),
where and ' are complex vector bundles.
Problems
Stiefel- Whitney classes
Suppose that
V n = (E,p, BOn, Om, On)
is a universal On-bundle. With the notation of Problem 1 of the last section, we knO\V
that H* (BOn; Z2) can be described as the full symmetric sub-algebra of the polynomial
P [t 1 , ..., t n ]. By definition, let Wi(V n ) = ith primitive symmetric function in t l , ..., tn.
Wl(V n ) is called the mod 2 universal ith Stiefel-Whitney class, and
n
W (v m ) = L Wl(V n ),
;=0
where wo(v n ) = 1, the mod 2 universal total Stiefel-Whitney class.
Now let
E = (X,p, B, R n , On)
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 83
be a real vector bundle over the finite complex B, and assume that
g = Cx, p, B, On, On)
is its associated principal bundle. Then g is determined, up to equivalence, by the homotopy
class of a suitable map
f: B -+ BOn.
By definition, the mod2 ith Stiefel-Whitney class of g, Wi (g), isf* Wi (V n ), and the mod 2
total Stiefel-Whitney class of g, W(g), isf* W(v n ). Note that W 2i + h i 2:: 1, the reduction
mod 2 of integral classes (see Problem 9 of g 1).
1 Show that the correspondence g -+ W(g) is uniquely determined by the following condi-
tions:
a) naturally, i. e.,
W(g*g) = g* (W(g»
where g: B' -+ B is a map and g*g is the bundle inducedfrom g by g;
b) Whitney duality, i. e.,
W(g g') = W(g) W(g');
and
c) W( 1Jm) = 1 + t, where 1Jm is the Hopf-bundle sm -+ RPm, m 2:: 1.
Syn'lpletic Pontryagin classes
Using an exactly similar procedure, one can define characteristic classes for symplectic
bundles. Suppose that V n = (E,p, BSPn, SPn, SPn) is a universal bundle for SPn. We know
that H* (BSPn; Z) is the full symmetric sub-algebra of the polynomial algebra P [t 1 , . . . , t n ] ,
the notation being that of 1. Define the universal ith symplectic Pontryagin class et(v n )
to be the ith primitive function in t 1, ..., t n (deg tJ = 4), and the universal total symplectic
n
Pontryagin class to be L ei(v n ), where eo(v n ) = 1. Now we assign to a given symplectic
i=O
bundle g the class f*e (v n ), where f is the classifying map for g.
2 Show that the correspondence g -+ e(g) is uniquely determined by the three conditions:
a) naturality, g*e (g) = e (g*g);
b) Whitney duality, e (g E') = e(g) e(g'); and
c) e(1Jm) = 1 + t, where 1Jm is the Hopf-bundle s4n+3 -+ HPn.
An alternative approach to characteristic classes
Now let
E = (X,p, B, en, Un)
be a complex line bundle over the finite cell complex B. The associated projective bundle is
defined by taking the space
P(X) = {(b, 1)1 b e B, I a line through 0 ep-l(b)}
as the total space and the natural map
q : P(X) -+ B
84
THE TOPOLOGY OF CLASSICAL GROUPS
which sends (b,1) to b as the projection. Observe that there is a natural line bundle 1: (i.e.,
a vector bundle of dimension 1) over P(X): the total space consists of all pairs (x, u) such
that x = (b, I) € P(X) and u is a vector in the space I. Suppose that y € H 2 (P(X); Z) is
negative of the first Chern class of 1:.
3 Show that
q* : H* (B; Z) -+ H* (P(X); Z)
is a monomorphism and that H* (P(X); Z) is a free H* (B; Z)-module with 1, y, y2, . . . , )"'-1
as a basis.
[Hint Note that, in the universal bundle, P(X)is E{U n - l X U l and q is EIU n - l x U l -+ EIU n
= BUn.]
4 Suppose at(E) € H 2i (B; Z) to be the elements uniquely defined by the equation
n
yn + L at(E) yn-i = o.
i=1
Show that
ai(g) = Ci(g), i = 1, ..., n.
[Hint Prove it for the Universal Un-bundle 'V n . Consider the polynomial
n
L Ci(V n ) x' = (x + tl) ... (x + t n ),
i=O
where t 1, ..., t n are the roots of 'V n . Observe that EfT n -+ BUn is the bundle of flags; i.e.,
- -
EIT n can be considered as the set of pairs (b,f), where b e BUn and f = (11' ..., In), an
ordered set of n lines through the origin in the fiber at b of the associated vector bundle.
Therefore EIT" can be obtained from this associated vector bundle by repeating the process
of forming P(X) n times. This means that the lifting of'V n to EfT n splits 1:1 E9 ... E9 1: n as
the sum of n live bundles, and the lifting of 1: from P(X) to EITn is one of these 1:t's. This
implies II (y + tt) = 0, which is enough to prove the assertion.]
i
5 Formulate and prove 4 for the mod 2 Stiefel- Whitney classes, and for the symplectic
Pontryagin classes.
6 Show that 4 and 5 are alternative methods for defining characteristic classes.
Characteristic classes as obstructions
Suppose that
g = (X, p, B, Cn, Un)
is a complex vector bundle, and let
i = (i,p, B, Un, Un)
be the associated principal bundle.
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 85
7 Show that cn(g) is the obstruction to find a section on the 2n-skeletonfor the associated
sphere bundle
A
Pn: XIU n - l -+ B.
8 Show that
p:-lCi (Pn *g) = Ci(g), i < n.
[To prove 7 and 8, assume that g is the universal bundle and use the computation of
H* (BUn; Z) obtained in the previous section.]
9 Formulate and prove Problems 7 and 8 for the sympletic Pontryagin classes.
The Euler-Poincare class
Suppose that
g = (X, p, B, R 2n , S02n)
is a real vector bundle, and let
go = (Xo,p, B, 8 2n -I, S02n)
and
gl = (Xl, p, B, D 2 n, S02n)
be the associated sphere and disk bundles. The Euler-Poincare class W 2m (g) of g is, by
definition, - 'z'([S2n-l ]), where [s2n-l] is the generator of H2n-l (s2n-l ; Z) while 7: is the
transgression in go. (The orientation of s2n-l is that induced by Sl on the (n - I)-fold
joins Sl* ... *Sl = S2n-l of S l with itself.)
10 Consider the Thom space Thorn (g) = X1IX o , and let
(/): Hi (B; Z) -+ Hi+2n (Thorn ( ); Z)
be the Tizom isomorphism. Show that
W 2m (g) = (/)-1 (4)(1) U (/)(1».
Deduce that
W 2n + 2m (g g') = W 2n (g) W 2m (g'),
where g' is the bundle
(X',p', B,R 2m , S02m).
11 Prove that the universal W 2m (i. e., W 2m of the universal bundle) is uniquely characterized
by the following two properties:
a) W 2m = W2m mod 2 and
b) Y* W 2m = tl, ..., t m ,
where y is the projection EIT = B T BS0 2n and t 1, ..., t n are the generators of H* (EIT;Z)
determined by T.
[The proof of (a) is a direct consequence of the computation of H* (BOn; Z2). To prove
(b), note that y*v n (v n being the universal bundle) is a direct sum of n S02-bundles. But the
Euler-Poincare class of the Hopf-bundle is just the generator in H 2 (BS0 2 ; Z).]
86
THE TOPOLOGY OF CLASSICAL GROUPS
12 Suppose that
g = (X,p, B, en, Un)
is a complex vector bundle, and let
g' = (X', p', B, R 2 n, S02,,)
be the bundle obtained by enlarging the group to 80 2n . Show that
cn( ) = W2n( ').
Pontryagin classes
Suppose that
g = (X,p, B, Rn, 0,,)
is a real vector bundle, and let g @ C be the bundle obtained by enlarging On to l]". By
definition, the ith Pontryagin class is
Pi( ) = (_I)i C2i (g @ C), i = 1, ..., n,
and the total Pontryagin class is
p(g) = L Pi(g), Po(g) = 1.
i O
13 Suppose that T is the standard maximal torus of On defined in S 1 of this chapter. Let
'V n = (E,p, BOn, On, On)
be the universal bundle and consider
y: EIT -+ BOn.
Then the Pontragin classes are uniquely determined by the two conditions:
a) y*(p) = II (1 + t'f) and
b) PI = w mod 2,
where the t/s are as in S 1 and W2i is the Inod2 Stiefel-Whitney 2ith class.
[The first part can be proved by considering the imbedding On Un, which is the stand-
ard one composed with the conjugation by a direct sum of 2 X 2 matrices
1 ( 1
J2 1
-:) ,
and of (1) when n is odd. Then the standard maximal torus is taken to be the diagonal
matrices n even
- .
e21U81 0
o e- 21Ci8 t
-
e 21Cj8 m 0
o e- 21Ci8 m
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 87
Now (a) follows immediately from considering the commutative diagram
EIT E' {Tn
! !
BOn BUR,
where E' is the total space of a universal bundle for Un and Tn is the standard maximal torus
for Un.
Part (b) is done by considering the commutative diagram
EI(Ol)n -:;.EI(Ol)n
1 E+'
BOn ) BUn,
where (Ol)n C On C Un is the subgroup of diagonal matrices with + 1 in the diagonal. One
also needs to know how the map RP OO -+ CPOO induced by 0 1 C U l behaves at the cohomo-
logical level mod2: it takes the generator of H* (CP OO ; Z2) = P[t l ] onto the square of the
generator of H* (RPoo; Z2) = P[t ], i.e., t l -+ t 2.]
14 By definition, let
Pt +t( ) = C2i+l ( @ C)
where is a real vector bundle. Show that
Pi+t( ) = W +l mod 2.
3 The homology and cohomology of the loopspaces of the classical groups
The loopspaces of Lie groups play an important role in the study of their
homotopy theory. Bott's original approach was to look for combinatorial
equivalents for the loopspaces by means of Morse theory. In fact he proves
that if G is a compact Lie group with a trivial center, then there is a circle 1:s
in G such that the map (J (J1: s (J-l1:; 1 induces a map
G/G1: -:;. QG,
G1: being the centralizer of 1:, which has the property that the image of the
induced homomorphism
H* (G/G1:; Z) -:;. H* (QG; Z)
is a set of generators of the Pontryagin algebra H* (QG; Z). Bott calls G/Gf:
a generating variety for QG. It is different from the notion of a generating
complex which was introduced in 7 of Chapter 2 inasmuch as it is not
88
THE TOPOLOGY OF CLASSICAL GROUPS
required that H* (G/G1:; Z) H* (QG; Z) be an isomorphism onto a set of
generators.
The purpose of this section is to show that the complex projective spaces
are generating varieties for the loopspaces of the unitary groups. This can be
done by elementary means.
Consider the matrix
e 2XiS 0
1
'is =
o 1
in U" + 1, where 0 < < 1. The commutator
U,,+l QSU,,+I
which sends a to a'i sa- I 'i -; 1, Q SU" + 1 being the loopspace of SU" + 1 , induces
amap
f,,: CP" = U n + I /U 1 X U QSU,,+l.
We want to show that CP" is a generating complex for QSU,,+ 1.
THEOREM (3.1) The map f" induces an isomorphism
f,,: P [H* (CP II )] H* (QSU,,+I)
of Hopf-algebras, where P [H* (CP,,)] is the polynomial algebra generated by
H* (CP,,). Here homology is with integral coefficients.
Proof The proof is quite similar to that of Theorem (1.1) of 4 of Chap-
ter 2. Consider the fibration
QSU" QSU II + I QS2"+I,
where p" is the loopmap induced by the usual projection SU,,+I 8 2 "+1
which sends a matrix to its last col umn. We shall prove the theorem by induc-
tion. If n = 1, then SU 2 = S3, and H* (QS3) is a polynomial algebra gener-
ated by the class of t e H 2 (QS3) which corresponds to S2 C QS3. Hence the
theorem is true when n = 1. Now suppose that the theorem is true for
n = k - 1. We would like to show that it is also true for n = k. First ob-
I
serve that, because
Pk : {JSU k + 1 -+ QS2k+1
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 89
is a homomorphism of associative H-spaces, the chain group of C* (QSU k + 1)
is actually a filtered DG-algebra, and hence each term Er, r 2, of the spec-
tral sequence of Pk is a bigraded DG-algebra. Moreover, the diagonal map
QSU k + 1 -+ QSU k + 1 X QSU k + 1 induces a comultiplication in each Er and
one can easily show that in this fashion each Er becomes a bigraded
DG-Hopf-algebra. Now consider
E 2 = H ( QS2k+ 1 ) I){\ H ( QS )
*.* * \CJ * k ,
where the equality is one of Hopf-algebras. Observe that H* (QS2k+l) and
H* (QSU k ) are generated by elements of even degree. Hence the spectral
sequence of Pk collapses, and
E2 = ... = E(» = EO (H* (QSU k + 1 )).
Proceeding as in the computation of H*( Un) in Chapter 2, one shows that
the sequence of graded algebras
1 -+ H* (!JSU k ) -+ H* (!JSU k + 1 ) -+ H* (!JS 2k + 1 ) -+ 1
is exact, and that the sequence of graded modules and homomorphisms
o -+ q (H* (QSU k )) -+ q (H* (QSU k + 1 )) -+ q (H* (QS2k+l)) -+ 0
is exact. We also know that the sequence of co algebras
o H* (CP k - 1 ) -+ H* (CP k ) H* (S2k) 0
is exact. Now, by considering the commutative ladder
o -+ H* (CP k - 1 ) -+ R* (CP k ) ) H*(S2k) ) 0
1 <fk-I). 1 <fk). 1
o -+ qH* (QSU k ) qH* (QSU k + 1 ) -+ qH* (QS2k+l) -+ 0
induced by the commutative diagram
CP k - 1 -+ CPk ) S2k
lf k - 1 lf k 1
QSU k --+!JSU k + 1 -+ QS2k+l,
we conclude that
(!t)*: R* (CP k ) qH* (!JSU k + 1 )
is an isomorphism.
90
THE TOPOLOGY OF CLASSICAL GROUPS
But H* (QSU k + 1 ) is commutative as an algebra since ,QSU n + 1 is a homo-
topy-commutative H-space. Hence (!t)* can be extended to an epimorphism
It: P [H* (CP k )] -+ H* (QSU k + 1 )
of Hopf-algebras. To finish the proof, one shows thatlt is an isomorphism
by showing that P [H* (CP k )] and H* (QSU k + 1 ) have the same rank in a
manner similar to the computation of H*( Un) in Chapter 2.
Problem
The loopspace of the symplectic group SPn
1 Show that H * ({JSPn; Z) is a polynomial algebra on generators X2, X6, . . ., X4n-2, where
degxt = i.
[Proceed by induction, using the fibration SPn-l -+ SPn -+ s4n-l.]
4 The Dott periodicity theorem for U
Suppose that cn is a complex n-space with the usual inner-product, and let
en be another copy. Then c n + C n is naturally isomorphic, as an inner-prod-
uct space, to C 2n . Let U (C n + cn) denote the group of unitary transforma-
tions of C n + C n . By definition, let
IX : U (C" + C") -'j> QU (C" + C")
n,n U(Cn) X U(Cn)
be the map induced by the map
8 1 X U(C n + cn) U(C n + cn)
which takes (e 21tiS , a) to a'isa- Lr; 1, 'is being the unitary transformation of
c n + C n which takes (x, y) to (e 21tiS x, y) .
THEOREM (4.1) The induced homomorphism
U(C n + cn)
( ex ) · 'Jt 'Jt k QU ( C n + cn )
n,n *. k U(Cn) X U(C n )
is an isomorphism for k < 2n.
Proof To prove the theorem, consider first the commutative diagram
U(C n + Cn)/U(C n ) x U(C n ) an,n ) QU(C n + cn)
l l m
U(C n + Cm)/u(c n ) x U(C m ) an,m ) ,Q (U(C n + cm)
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 91
with m > n, where the vertical maps are induced by imbedding C," into C m
as the subspace generated by the first n basic vectors. By considering the
commutative diagram
U(C n + Cn)jU(c n ) ) U(C n + Cm)ju(c m )
1 1
U(C n + Cn)ju(c n ) X U(C n ) 1n,m --:,. U(C n + Cm)ju(c n ) X U(C m )
where the vertical maps are the fiber bundle maps of the Stiefel manifolds
over the corresponding Grassmannian, one proves that the induced homo-
morphism
(Yn,m)*: nkU(Cn + Cn)ju(c n ) X U(C n ) -+ nkU(Cn + Cm)ju(c n ) X U(C m )
is an isomorphism for k < 2n.
Thus it is enough to prove the theorem for the maps cXn,m, with m large.
So let 81' ..., 8n and 8 1 , ..., 8m be the usual basic vectors c n and cm, and
imbed the nth-fold direct sum
(C 1 + CD) + ... + (C1 + CD), p large,
of C 1 + CD with itself into c n + C m by sending thejth summand to the subspace
generated by the basic vectors
A A
8 j , 8(j _ 1) D , ..., e j D ·
The preceding imbedding induces the following commutative diagram
U(C 1 + CD)
x ... X
U(Cl) x U(t,,) 1
U(CI + CD)
U(Cl) x U(t,,)
U(c n + em)/u(c n ) x U(e m ) «n,m) QU(C n + em)
with m > (p + 1) n, wherecX ,m is induced by the same map that inducescXn,m-
Observe that U (C 1 + C D )jU(C 1 ) x U(C D ) is the complex projective p-space
CpD, and that U (C n + Cm)ju(c n ) X U(C m ) is the Grassmann manifold Gn+m,n
of n-places in (n + m)-space. To simplify the notation let us write CpD for
the former space and Gn+m,n for the latter. Now note that the map cXn,m can
be factored through a suitable cartesian product of the spaces Q Up + 1, where
92
THE TOPOLOGY OF CLASSICAL GROUPS
we write U p + 1 for U(C 1 + CP), and hence yields the commutative diagram
CPP x ... x CPP rl.l.PX."Xrl.l.P QUp+l x ... x QU p + 1
Ip - Ip m
Gn+m,n = Un+mlU n x U m rl.n,m --:,.QU n + m
where U m stands for U(C m ), and P .m is induced in the same fashion as Pn,m.
Now note that Gn+m,n, for large m, represents a large skeleton of the classify-
ing space BUn, while CPP x ... x CPP, for large p, represents a large skeleton
of BTn, Tn being the standard maximal torus of Un. Moreover, Pn,m represents
the map BTn BUn induced by the imbedding Tn Un. Hence, according
to Theorem (1.2) of 1 of this chapter,
(Pn,m)* : H* (Gn+m,n; Z) H* (CpP x ... x CPP; Z)
is an isomorphism of H* (Gn+m,n; Z) onto a subalgebra of H* (CpP x...
x CpP; Z), which is, in the stable range,the full symmetric subalgebra
p [<11' ..., <1n] of H* (CP x ... xCP ; Z), of P [t 1 , ..., t n ]. Recall now
that Whitehead's comparison theorems imply that in order to prove the
theorem it is enough to show that (Xn,m induces an isomorphism in the stable
range on the integral cohomology groups. In view of what we know about
Pn,m, it suffices to show that a similar thing holds for P ,m. To do this we con-
sider the diagram
CpPx... xCpP QUp+IX". xQU p + 1
1 Y p Ip .m
QU p + 1
c
--:,. QU n + m
where YP takes (Xl' ..., X n ) to (Xl,p(X 1 ), ..., (Xl,p(X n ), the product being that of
loops on H-spaces. We wish to show that this diagram is homotopy-commu-
tative. To show that, it is enough to prove the following proposition.
PROPOSITION (4.2) Consider the diagram
U r X U r U r
! Il' !
U 2r U 2r
where f1 is the multiplication in U r , and f1' is the map which takes the first
factor onto the subgroup »,hich leaves the last r basic vectors fixed and the second
onto the subgroup that leaves the first r basic vectors fixed. Then the diagram
commutes up to homotopy.
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 93
Proof Let us identify U 2r with U(C r + C r ), and the second factor of
U r x U r with U (C r ). By definition, let Xt E U (C r + C r ) be the map such that
A 1'& A . 1'&
Xt8J = cos - t8j + SIn - t8j
2 2
. 1'& A 1'&
X t8. = -sin - 18. + COS - 18..
J 2 J 2 J
Then the map Ur(C r ) x Ur(C r ) U (C r + C r ) which sends (<1, 0') to <1X-; laXt
induces a homotopy as desired.
Now let
U = lim U(C n ),
n
and assume that U has the weak topology. Note that U and lim U (C n + cn)
n
are naturally isomorphic, where the isomorphism is induced by sending the
basic vectors 8) and 8j to 82j-l and 8 2 j respectively. Hence Theorem (4.1)
implies by passing to the limit that
(lX)* : 1'&kBU 1'&kQU
is an isomorphism for all k, where BU = lim G 2n ,n. Observe next that there
is a Serre fibration n
lim U 2n lU n lim U 2n lU n X Un = BU
n n
with fiber U and contractible total space. Hence we can think of Bu as a
classifying space for U, and hence there is a homotopy-equivalence q;: U
QB u such that
(Qq; OlX)*:1'&kBu 1'&kQ2Bu, k > 1
is an isomorphism. Since Bu and Q2 BU are of the homotopy type of CW-
complexes, (Qq; 0 lX) is actually a homotopy-equivalence.
Now recall that 1'&lBu = {O}, 1'&2BU = Z. Hence we have the following
theorem.
THEOREM (4.2) (Bott periodicity theorem) The map
lX : Bu QU
94
THE TOPOLOGY OF CLASSICAL GROUPS
is a homotopy-equivalence, and hence
{ Z,
'JtiBU =
0,
for i even
for i odd.
Problems
1 Suppose that
' : Ux U -+ U
is the limit of'the composite
U(C n ) X U(e n ) -+ U(C n + cn) -+ U(C 2n ).
Prove that ' is a multiplication with a two-sided homotopy-identity, and which is also homo-
topy-associative and homotopy-commutative. Show also that ' is homotopy-equivalent to
the usual multiplication.
2 Show that the isomorphism
(C n + en) + (C m + em) -+ (c m + cn) + (en + em) = c m + n + e m + n
induces a map
u(c n + en)
'V n m : A x
, U (C n ) X U ( C n )
U(c m + em)
-+
U(C m ) X u(e m )
u(c m + n + e m + n )
u(c m + n ) X u(e m + n )
which in the limit induces a multiplication
'V: BUxBU -+ BU
with a two-sided homotopy-unit, which is also homotopy-commutative and homotopy-asso-
ciative. Show also that the Bott map
<x:BU-+QU
is a homotopy-homomorphism.
3 Suppose that
<Xl : Cp oo -+ BU
.is the map induced by the imbedding of U l in U as the unitary group which leaves all but the
first basic vector fixed. Prove that the induced additive homomorphism
«(Xl) * : H * (CP OO ; Z) -+ H * (BU; Z)
can be extended uniquely to an isomorphism
(X: P [H * (CPOO; Z)] -+ H * (BU; Z)
of HopI-algebras. Here P [H * (CP OO ; Z) 1 is the polynomial algebra generated by ii * (CPOO ;Z)
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 95
5 The Grothendieck group K(X)
Suppose that d is the category of finite cell complexes and continuous maps.
The object of this section is to define a contravariant function K on d with
values in the category J 0 of abelian groups and homomorphisms. K is the
beginning of cohomology theory which satisfies all but the dimension axiom
(see Chapter IV). We shall see that the Bott periodicity theorem can be inter-
preted as a Kiinneth formula.
Let F(X) be the free abeIian group generated by the equivalence classes [ ]
of complex vector bundles over X, where X is in d By definition, the
Gothendieck group K(X) of X is the quotient group of F(X) by the subgroup
generated by the relations
[ ] + [1J] - [ + 1J].
Ifjis a map X Yin d, then we define
j*: K(Y) K(X)
to be the map induced by the correspondence which assigns to a given com-
plex vector bundle .,over Y its pull-back (the induced bundle) f* .
There is a closely related group K(X). We shall say that two complex vec-
tor bundles and 1J over X are stably equivalent if, and only if,
Ef) ml I'"oJ 1J Ef) nl
where ml and nl are the trivial bundles of dimension m and n respectively.
Denote the set of equivalence classes by K(X). Observe that the operation of
Whitney sum induces an operation 'on K(X). This operation can be easily
seen to be, commutative and associative with a two-sided identity 0 = [1].
Hence K(X) is an abelian semi-group.
PROPOSITION (5.1) K(X) is an abelian group.
Proof To prove the proposition we need show that every element [ ] in
K(X) has an inverse. Because of the classification theorem for bundles it is
enough to define an inverse for the universal bundle. So consider the bundle n
Pn,m: E Gn+m,n, m large,
where Gn+m,n is the Grassmann manifold of n planes in (n + m)-space c n + m ,
and where
- ....
E = {(x, u)1 x E Gn+m,n, u vector in x}.
96
THE TOPOLOGY OF CLASSICAL GROUPS
Define * to be the vector bundle whose total space is
- -
E = {(x, v) f x E Gn,m+n, v orthogonal to x}.
Then clearly
+ I'"oJ (n + m) 1.
Define - [ ] to be the stable class of [ .l]. To show that - [ ] is well-defined,
let , ' be two bundles such that
+ s 1 I'"oJ ' + t 1 .
Choose E and ,l. as above. Then
+ + s I I'"oJ l. + ' + t 1 ,
(m + n) 1 + sl I'"oJ
(m + n + s) 1 I'"oJ
Hence + ' is stably trivial. But
l. + ' + ,.l I'"oJ .l + (m' + n') 1.
This implies that is stably equivalent to ' , which proves that -l ] is
well-defined and, hence, K(X) is a group.
Q.E.D.
What is the relationship of K(X) to k(X)? Consider now the homomorphism
cp : K(X) [((X)
which is induced by the maps which take to its equivalence class. Clearly q;
is onto. Define "P to be the map
k(x) K(X)
which sends [ ] to - ml, m being dim . One easily checks hat 'ljJ is well-
defined and
ffJ1P = identity.
Hence'ljJ imbeds [((X) as a direct summand in K(X).
Observe that ml, where m = dim , can be regarded as the restriction of
to a point in X. This suggests that we can regard 1<. as the functor K reduced
in the sense of cohomology theory. Then we have the following proposition.
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 97
PROPOSITION (5.2) Suppose that XEd and let Xo be a O-cell in X. Then the
sequence
,.., 1p e
o K(X) ----:,. K(X) ---:,. K({xo}) 0
is exact and splits. Here E is the map induced by the injection {X o } X.
Proceeding with the analogy with cohomology theory, let by definition
K (X, {xo}) = X(X)
and
K(X,A) = [«(X/A) = K(X/A,A.)
where A is a subcomplex of X, and X/A is the complex obtained from X by
collapsing A to a point A .
We shall need the following in what follows. Suppose that X and Yare
finite cell complexes, and let Xo and Yo be O-cells in X and Y respectively. We
shall regard Xo and Yo as basepoints. As usual, let
X v Y = Xx {Yo} U {xo} x Y
and
X A Y= XxY/Xv Y.
PROPOSITION (5.3) The sequence
j* i*
o K(Xx Y, X v Y) K(Xx Y) ---:,.K* (X A Y) 0
is exact and splits. Here i and j are the natural maps X v Y X x Yand
Xx Y Xx Y/X v Y.
The proof is completely analogous to that in cohomology and will there-
fore be omitted.
Next we would like to introduce products in K(X) and K(X). SO let and 'YJ
be bundles over the finite cell complexes X and Y. Denote by [K] 'YJ the ex-
ternal tensor product. (The basespace is Xx Y; the fiber at (x, y) is the ten-
sor product of the fiber of at x and the fiber of'YJ at y. If X = Y, then
(8) 'YJ is the restriction of 0 'YJ to the diagonal X c X x X.) It is quite easy
to check that the operation [25] induces a homomorphism
f3: K(X) (8) K(Y) K(Xx Y).'
If X = Y, then the composite of f3 with the restriction map to the diagonal
K (X x X) K (X) defines in K(X) a multiplication which makes K(X) a
ring. Again in a fashion similar to chomology one proves that f3 is a map of
7 Husseini (0216)
98
THE TOPOLOGY OF CLASSICAL GROUPS
rings (or algebras over Z) where the product in the tensor product is the usual
coordinate-wise multiplication.
Observe next that 'YJ takes K(X, {xo}) (8) K(Y, {Yo}) to K(Xx Y, X v Y)
and hence induces a map
p: k(X) (8) K(Y) -+ k(x A Y).
The following proposition is essentially a restatement of the classification
theorem for bundles. Recall that BU = lim G 2n ,n. Hence a mapf: X -+ BU
n
can be thought of as a map in G 2n ,n for some n, and hence determines a
bundle f : over X, where n is the universal bundle over G 2n ,n. Recall also
that BU admits a multiplication. Hence the set of homotopy classes [X, BU]
of basepoint preserving maps is a group.
PROPOSITION (5.4) The correspondence
r/J: [X, BU] -+ k(X)
which assigns to a map f: X -+ BU the equivalence class of f* n is an iso-
morphism.
The proof is quite straightforward and will not therefore be given.
Now let us return to the homotopy equivalence
Q(q;) 0 lX : BU -+ Q2 BU
of the preceding section. Clearly Q(q;) 0 lX induces an isomorphism
(Qq; 0 lX)# : [X, BU] [X, Q2 BU],
where X is a finite cell complex. But
[X,Q 2 BU] = [8 2 A X, BU] = K(8 2 A X).
Thus (Qq; 0 lX)# induces an isomorphism
K(X) K (8 2 A X).
THEOREM (5.6) Consider the map
p : K(S2) (8) K(X) -+ f<. (8 2 A X).
Then
p ([1J] (8) [ ]) = (Qq; 0 lX)# [ ],
where'YJ is the Hopf Sl-bundle over 8 2 . Hence p is an isomorphism if, and only
if, p is a homotopy-equivalence.
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 99
Proof First we need to deform the map lX: BU !JU into one which is
more suitable. Recall that lX is, by definition, the limit of the maps
lXn,n: U 2n lU n X Un QU 2n ,
of 4, where U 2n , Un and Un are the unitary groups of c n + cn, C n and C n
respectively. Again as in 4, let
XS : en + C n C n + C n
be the map which takes (x, y) to (e 2XiS x, y).
THEOREM (5.7) lXn,n is homotopic to the map
n,n: U 2n lU n X On QU 2n
which is induced by that which takes (s, (1) of 8 1 X U 2n to <1Xs<1-1.
To prove the assertion, just observe that
lXn,n = n,n · C
where c is the constant map which takes everything to the loop s X; 1. But
QU 2n = QSU 2n is connected. Hence lXn,n is homotopic to n,n.
Let {81, ..., 8n} and {8 1 , ..., 8n} be the basic vectors of c n and C n respec-
tively. Every <1 E U 2n determines an n-plane
[<1] = [<181' ..., <18n]
in c n + en generated by <18 1 , ..., (f8 n , and the mapping <1 [<1] induces a
homeomorphism
U 2n lU n X On G 2n ,n
where G 2n ,n is the set of n-planes in C n + cn.
Let us consider next the unitary space C 2 = C' + C", and let 8' and 8" be
its standard basic vectors. Then (C n + Cn) (8) C2 is a unitary space of 4n di-
mensions. We shall identify U 4n with the group of unitary transformations of
(cn + cn) (8) C2 and the Grassmannian G 4n ,2n with the set of 2n-planes in
(cn + cn) (8) C 2 by sending <1 E U 4n to the 2n-plane
[<1] = [<1 (8 1 (8) 8'),- . .. , <1 (8n (8) 8'), <1 (8 1 (8) 8'), ..., <1 (8n (8) 8')] .
Hence G 4n ,2n = U4n1U n X U;n, where U n is the subgroup of U 4n leaving
" "A" A " .. . 1 " . b
81 (8) 8 , ... , 8n (8) 8 , 8 1 (8) 8 , ..., 8n (8) e InvarIant, whi e U 2n IS the su group
A A
I . 10\ , 10\' 10\' 10\ ". ·
eaving 8 1 8 , ..., 8n 8 , 81 B , ..., 8n 8 InvarIant.
100
THE TOPOLOGY OF CLASSICAL GROUPS
Next we need an expression for U n €P2n ) QG 4n ,2n. To do that let
(!t : C 2 C 2
be the rotation defined by the equations
() 8' = cos !!..- t8' + sin t8"
t 2 2
{) t 8" = - sin j(; t8' + cos j(; t8"
2 2'
where 0 < t < 1. Then
1 (8) (!t: (C n + en) (8) C2 (C n + en) (8) C 2
is a unitary transformation and, by sending (j E U 2n to the 2n-plane
[(1 (8) (!t)-1(j (1 (8) (!t)]
= [(1 (8) (!t) -1 (j (1 (8) (It) 81 (8) 8', ..., (1 (8) (It) -1(j (1 (8) (It) En (8) 8'],
we get a map
q;2n: U n QG 4n ,2n,
which in the limit induces a homotopy equivalence U QBU.
Observe that
(1 (8) (It)-1 (j (1 (8) (It) (8j (8) 8')
= 1 @ e;1 (ue j @ cos te' + ej @ sin te")'
This expression suggests the following description of the 2-sphere 8 2 . Sup-
pose] is the unit interval and let (x, t) E ]2. Then the mapping
[ j(; 2. j(; ]
(s, t) --+ cos"2 te' + e ,,'s sin "2 te" ,
where the expression on the righthand side is the line in C 2 generated by the
indicated vector, defines a map of ]2 onto the Gra5smannian G 2 ,1 of lines
through the origin in C 2 . Now if we identify the edge s = 0 with the edge
s = 1 in]2 and then identify the boundary circles t = 0 and t = 1 with two
distinct points we obtain a homeomorphism
2 "'"
8 G 2 ,1.
HOMOLOGY AND COHOMOLOGY OF CLASSIFYING SPACES 101
Now consider the composite
lJ n . n n D(J)2n n2
G 2n ,n ---:,. U2n ---:,. G 4n ,2n
where U 2n is identified with U n C U 4n as above. Let Yn,n denote the compo-
site Qq; n 0 n,n' and consider the adjoint
Y;:n: 8 2 A G 2n ,n -+ G 4n ,2n.
A direct calculation shows that
Y n (s, t, [<1]) = [<1] 0 P + [<1].1. 0 P',
where P is the line in C 2 corresponding to (s, t). According to the preceding
convention, [<1].1. is the n-plane in c n + C n orthogonal to [<1] and P' is the
line in C 2 generated by the vector cos!!.- tB' + sin n tB". Now we regard
2 2
the Orassmannian G Sn ,4n as the set of 4n-planes through the origin in
(C n + cn) 0 C2 + (cn + cn) 0 C 2 ,
and we consider G 4n ,2n imbedded in G Sn ,4n, according to the imbedding of
(C n + Cn) 0 C 2 , as the first factor (C n + cn) 0 C2 + (cn + cn) 0 C 2 .
Then Y n is homotopic to the map
8 2 A G 2n ,n -+ G Sn ,4n
(5.8)
which takes (s, t, [<1]) to the 4n-plane
[<1] 0 p + [<1].1. 0 P' + [1] 0 C 2 ,
where [1] is the n-plane [Bl, ..., En] of C n + Cn, and C2 the 2-plane [B', B"]
of C2.
Next we note that
C 2 = P + p.1.,
and hence the map of (5.8) is homotopic to the map that takes (s, t, [<1]) to
[<1] 0 P + [<1].1. 0 P' + [1] 0 P + [1] 0 p.1.. (5.9)
This latter map is homotopic to that which takes (s, t, [<1]) to
[<1] 0 P + [<1].1. 0 P' + [1].1. 0 p' + [1] 0 p.1.,
(5.10)
102
THE TOPOLOGY OF CLASSICAL GROUPS
where [1].L is the It-plane [81' ..., en] of en + en. But we regard the corre-
spondence
p --+ p'
which sends the line [cos ; te' + e 21tis sin ; te"J to [cos ; te' + sin ; le"J
as a map G 2 , 1 = 8 2 --+ G 2 , 1 = 8 2 , which is obviously homotopic to the con-
stant map [P] --+ [8']. To finish the proof of the theorem, just compare the
resulting expression with that for the reduced tensor product of bundles.
Bibliography
[1) M.F.Atiyah, K-Theory, W.A.Benjamin, 1967
[2] A. Borel, "Sur Ie cohomologie des espaces fibres princip2.ux et des espaces homogenes
de groupes de Lie compacts", Ann. of Math. 57 (1953), 115-207
[3] A.Borel and F.Hirzebruch, "Characteristic classes and homogeneous spaces", Amer.
J. Math. 80 (1958), 458-538, and 81 (1959), 315-82
[4] R. Bott, "Quelques remarques sur les theoremes periodicite", Bull. Soc. Math. France,
87 (1959), 293-310
[5] Seminaire H. Cartan 1959-60, Ecole Normale Superieure
[6] H. Toda, "A topological proof of theorenls of Bott and Borel-Hirzebruch for homo-
topy groups of unitary groups", Mem. Co!/. Sc. Univ. Kyoto, 32 (1959), 103-19
CHAPTER 4
K-theory
THE GROTHENDIECK GROUPS introduced in 5 of the preceding chapter can
be fitted into an extra-ordinary cohomology theory, the so-called K-theory,
which satisfies all but the dimension axiom. The aim of this chapter is to show
that this can be done. In 1 and 2 extra-ordinary cohomologies are defined
following [3]. These considerations are specialized for the K-theory.
In the last section we show how the Chern character represents the K-
theory in terms of cohomology with rational cohomology. Some other topics,
such as the spectral sequence induced by the :filtration by skeletons, are out-
lined in the exercises. The reader is referred to [1] and [2], where these topics
are developed further and many applications are given.
1 Preliminaries
Let d be the category of all finite cell-complexes and continuous functions.
Let (d, d) denote the category of pairs whose objects are.pairs (X, A) of a
complex Xed and a subcomplex A c X, and whose maps are continuous
functions of pairs. Finally, let J be the category of graded groups (negative
grading is allowed) and degree-preserving homomorphisms.
An extra-ordinary cohomology theory on (d, d) is defined by a contra-
variant functor H* of (d, d) into J and a natural transformation of
degree 1,
* : H* 0 7: --+ H*,
where 7: is the functor of(d, d) to itself which takes (X, A) to (A, 4» (4) being
the empty set) such that
(1) if/and g are homotopic maps in (d, d), then
H*(/) = H*(g);
(2) if (X; A, B) is a triad such that (X, B) E (d, d), (A, A n B) E (d, d),
and X = A U B, then the natural maps
k : (A, A n B) --+ (X, B)
103
104
THE TOPOLOGY OF CLASSICAL GROUPS
induce an isomorphism
H*(k): H* (X, B) -+ H* (A, A n B);
(3) if (X, A) E (d, d), then the triangle
H* (A, q,)
:;/ ,,\*(j)
H* (X, A) H*(j») H* (X, 4»
is exact. Here i andj are the inclusion maps.
As usual we shall writef* for H*(f). Condition (1) is the so-called Homo-
topy Axiom; condition (2) is the Excision Axiom; and condition (3) is the
Exactness Axiom. Observe that an extra-ordinary cohomology theory is
required to satisfy the Eilenberg-Steenrod axioms for ordinary cohomology,
but not necessarily the dimension axiom.
How do these theories arise? Roughly speaking, cohomotopy theory is an
extra-ordinary cohomology theory. But to be able to exhibit more examples,
we need the concept of a spectrum. So let us define it.
A spectrum E is a sequence { E n, 8n}, indexed by the set Z of all integers,
where E n is a space with basepoint of the homotopy-type of a countable
CW-complex, and 8n a map
En -+ QE n + 1 ,
- -
of an into the loopspace Q + 1. A map .[: E -+ E of the spectrum
E = { E n' en} to the spectrum E ' = { E , 8 } is a sequence {};,} where};, is a
map E n -+ E such that the diagram
E f n ) E'
:II ._11
1 £n 1 £
n E Dfn+l n E '
=n+ 1 ) =:3I+ 1
commutes. Here Qfn+l is the map induced by fn+l.
One need only assume that the spaces E n are of the homotopy type of a
CW-complex, but the countability condition makes some of the details easier
to carry out. Also note that a spectrum could very well be defined by specify-
ing that the maps 8n take the reduced suspension S A E n of E n to E n + 1 ,
because of the duality between maps En -+ QEn +l and maps S A E n -+ En+l.
Let us give some common examples of spectra.
K-THEOR Y
105
I) Suppose that E n = sn, the n-sphere in Euclidean (n + I)-space, and let
en : = sn -+ Q + 1 = Qsn+ 1 be the adjoint of the identity map sn = sn.
The resulting spectrum S = {£Z, en} is called the sphere spectrum.
2) Suppose that U is the infinite unitary group. By definition, put
{ U, n odd
u=
_n QU, n even
and let the connecting map en: Un -+ Q U n + 1 be, for odd n, the Bott map,
defined in 4 of Chapter 3, and the identity Un = QU -+ Q U n + 1 = QUwhen
n is even. The resulting spectrum U = {Q,., en} is called the unitary spectrum.
3) Suppose that n is a countable abelian group, and let K (n, n) denote an
EiIenberg-MacLane space; i.e., njK(n, n) = {O} ifj =F n, andnjK(Q, n) =n.
Now we define K(n )n to be K(n, n), and en: K( n)n -+ Q K (n)n+1 to be a
homotopy equivalence K (n, n) -+ QK (n, n + 1) which induces the identity
homomorphism onnnK(n, n) = n -+nnQK(n, n + I) = n. K( n) = {K(n)n,e n }
is called the Eilenberg-MacLane spectrum.
4) Suppose that E = { , en} is a spectrum, and let Xbe a finite cell com-
plex. We can obtain a new spectrum X A E by putting
(X A IDn = X A ,
and by defining the connecting map
n : (X A IDn -+ Q (X A + 1)
as follows. Let en: s A -+ +1 be the adjoint of en. Then the adjoint n
of n is by definition the composite
a 1 A ';n
8 A (X A E) n X A (8 A ) ) X A E n + 1
where the first map is the switching map which takes s A X A e to x A S A e.
The resulting spectrum X A E = {(X A E) n, n} is called the reduced join
of X and E . If X is the circle 8 1 then S A E is called the suspension of E .
Before describing the next example, let us recall a few well-known facts.
Let F (X, Y) denote the space of basepoint-preserving maps of X to Y, with
the compact open topology. It is well known that F(X, Y) is of the homo-
topy type of a countable CW -complex when X is finite cell complex and Y is a
space of the homotopy-type of a countable CW-complex.
Let
f[J: F(X A Y, Z) -+ F(X, F(Y, Z))
106
THE TOPOLOGY OF CLASSICAL GROUPS
be the map which sendsfto the map defined by
rp(f) (x)(y) =f(x A y).
The following result is well known.
(1.1) Suppose that X and Yare compact. Then rp is a homeomorphism.
The fifth example is as follows. Suppose that E = { , en} is a spectrum,
and let X be a finite cell complex. Then we can obtain a new spectrum
F (X; ID by putting
F (X; IDn = F(X, )
and defining the connecting maps 'YJn: F (X; E n) -+ QF(X; E + 1 ) as follows.
Observe first of all that en induces a map
(e n )#: F(X, ) -+ F(X, QEn+1).
But
<p -1
F(X,Q E n + 1 ) = F(X, F(S, +1) --)- F(X A S, E n + 1 )
a# <P
---:,. F(S A X, +1) ---:,. F(S, F(X, E n + 1 ) = QF(X; +1)
where a# is the map induced by the switching map S A X -+ X AS. Let
'YJn: F(X, ) -+ QF(X; +1)
be the composite of e n # and the sequence of equivalences given above. In this
fashion we obtain a spectrum F (X, E) = {F (X; E n), 'YJn}. If X = S we shall
write Q E for F(X; ID.
2 Extra-ordinary cohomology defined by a spectrum.
Suppose that E = { , en} is a spectrum. Now for every integer k, the map
ek: Ek -+ Q +l
induces a homomorphism of homotopy groups
e : nn+k( ) -+ nn+k (Q E k + 1) f"ttJ nn+k+ 1 ( + 1)
whenever k > -no Moreover, the groupsnn+k(gk) and the homomorphisms
E define a direct system
no E -n -+ n1 E -n+ 1 -+ ... -+ nk E -n+k -+ nk+ 1 E -n+k+ 1 -+ ....
K-THEOR Y
107
Now, by definition, let
nn E = limnkE-n+k = limnn+q E q .
k q
nn E is called the nth homotopy group of E .
For example, for the unitary spectrum U ,
{ Z, n odd
nn( U) =
- 0, n even,
the homotopy groups nn S of the sphere spectrum S are just the stable homo-
topy groups of the spheres. Also note that
n_n F (XjA, K( n)) = lim n-n+q F(XjA, K(n, q)) = Hn (XjA; n)
q oo
= Hn(X,A;n), n> 0,
where K(n ) is the Eilenberg-MacLane spectrum and H* (XjA; n) is the ordi-
nary cohomology of XjA with coefficients in n. (The preceding equality is of
course just Hopf's theorem which identifies ordinary cohomology with
homotopy classes of maps.) Finally,
n-n F (XjA; S ) = lim n-n+q F(XjA; sq);
but
q oo
n-n+q F(XjA; sq) = noF(s-n+q, F(XjA, sq)) = no F(s-n+q" X/A, sq)
= noF (XjA, Q-n+qSIl)
= noF(XjA, sn), for large q.
Hence, if dim XjA < 2n - 2,
n-n F (XjA; = nn(XjA),
where nn (XjA) is the nth cohomotopy group of (XjA).
These examples which identify ordinary cohomology, and cohomotopy,
with the homotopy groups of appropriate spectra are the key to the general-
ization .of ordinaOry cohomology. So suppose that E = { E n, en} is a spectrum,
and let (X, A) E (d, d). By definition, let
Hn (X, A; ID = n_nF(XjA; E) .
If A = 4>, then we write Hn (X; E ) for Hn (X; 4>; E ). Note that
HII (X; ID = n-nF(X+; ID,
108
THE TOPOLOGY OF CLASSICAL GROUPS
where X+ is the disjoint sum of X and a space consisting of one point only
which we consider as the basepoint of X+.
Now suppose next that
f: (X, A) -+ (X', A')
is a map in (d, d), and let
/: (XjA, A) -+ (X' jA', A')
be the induced map on the spaces obtained by identifying A and A' with the
basepoints of X and X' respectively. Thenjinduces a map
1#: F(X'jA'; +n) -+ F(XjA; E +q)
and, hence, a homomorphism
(f*): = H;(f): nqF(X'jA'; E n + q ) -+ nqF(XjA; E n + q ).
It is quite easy to check that the diagram
nqF(X'jA'; E n + q )
! (e n + q).
nqF(X'jA'; Q +q+l)
!=
(I*)n
q --:,. nqF (XjA; ¥3.+ q )
1 (e n + q)*
nqF(XjA; Q +q+l)
!=
(1*): + 1
nqF(X jA', +q+l) nq+1F(XIA; E n + q + 1 )
is commutative. Hence {(f*):} is a map of the direct system {nqF(X'IA' ;E n + q )}
to the direct system {nqF(XjA; E n + q )} and therefore induces a homo-
morphism
(f*)n = Hn (f, ID: Hn (X', A'; ID -+ Hn (X, A; E) .
(2.1) The correspondence
H* ( ; E) : (d, d) -+ J
which assigns H* (X, A; E) to (X, A) andf*" tofis a contravariant functor.
This assertion is quite easy to check, and therefore will be omitted.
Out next object is to define a natural transformation
. * ( , ) = *: H* ( ; ID 0 7: -+ H* ( ; ID
K-THEOR Y
109
of degree I. So l t (X, A) E (d, d) be a given pair. Observe now that there
is a homotopy-equivalence
Of : Xj A -+ X UTA
where X UTA is the space obtained from X by attaching the reduced cone
TA along A c X. Let
A" : X UTA -+ S A A +
be the composite of the map which collapses X to a point and the natural
injection S A A -+ S A A +. Consider now the composite
( )# i
F(A+, En) --> F(A+, QEn+l) ---:,. F(S A A+, +1) F(XjA, E n + 1 )
"'"
where a is induced by a" Of. Denote the composite map of F (A + , E n)
-+ F(XjA, E n + 1 ) by . It is quite easy to check that the diagram
F(A+, E n )
l(eS n )#
F(XjA, +1)
(e n )#
) QF (A+, +1)
! .Q(eSn + 1)#
(en + 1)#
) QF(XjA, +2)
is commutative. This implies that {( n)#} induces a map of the direct system
{nn+qF (A +, E n)} to the direct system {nn+q+lF (XjA; +l)} and, hence, a
homomorphism
* (X, A): H* (A, cfJ; E) -+ H* (X, A; E)
of degree I. The fact that * is a natural transformation
H* ( ; E) 0 7: -+ H* ( ; ID
is quite easy to check.
THEOREM (2.2) The contravariant functor H* ( ; E) and natural trans-
formation
* : H* ( ; E) 0 7: -+ H* ( ; ID
define a cohomology theory on (d, d).
Proof Axioms 1 and 2 are quite easy to check. To check the exactness
axiom for H* ( ; E ), let (X, A) E (d, d) and consider the sequence
i j
A X XjA.
110
THE TOPOLOGY OF CLASSICAL GROUPS
This sequence induces the Hurewicz fibration
j# i#
F ( X j A. E ) -----:,. F ( X. E ) --> F ( A. E ) fP
, _n , _n , =n n
Moreover, the connecting map Bn : E n --+ QE n + 1 of the spectrum E takes the
fibration fP n to the fibration fPn+ 1 and, hence, yields the commutative diagram
j# i#
F(XjA; E n) -----:,. F(X; En ) ---> F(A; E n)
! (e n )# 1 (e n )# ! (e n ">#
j# i#
F(XjA,QE n + 1 ) ---:,. F(X,Q E n + 1 ) --> F(A, QE n + 1 )
! .Q(e n + 1)# ! .Q(e n + 1)# ! .Q(e n + 1)#
112 E j# 112 i# ( 112
F (XjA, & _n+2) ---)0 F(X, & E n + 2 ) -----:,. F A, & E n + 2 )
! ! !
where the rows are Hurewicz fibrations. Now, passing to homotopy, we
obtain the commutative diagram
· .. --+ nn - k F (Xj A, En )
j*
---:,. nn-kF (X, En)
i*
-----:,.
nn-kF(A, )
!(e n )* !(e n )* !(e n )*
j* i*
... --+ nn-kF(XjA,Q +l) -----:,. nn-k F (X,Q E n + 1 ) -----:,. nn-k F (A, QE n + 1 )
! .Q(e n + 1)* ! .Q(e n + 1)* ! .Q(e n + 1)*
j* i*
... --+ nn-kF (XjA, Q2 En +2) ->nn_kF(X,Q2 En +2) -----:,.nn-kF(A, Q2 E n+2)
! ! !
Note that the rows in the diagram are exact, since each row is just the homo-
topy exact sequence of a Hurewicz fibration. Also,
nn-kF (X, Q qE n + q ) nn-kQqF (X, E n + q ) nn-k+qF (X; E,, +q).
With this substitution, each column in the preceding diagram of homotopy
groups gives the direct sequence used in the definition of H* ( , E ). Passing
to direct limits, we get the exact sequence
j* i* 0
u. --+ n_kF(XjA; ID ----+n-kF(X;ID ---)o7(;_k (A, ID ----+nk-lF(XjA; --+ ...
II j* II i* II II
... --+ Hk (X, A; E ) ----+ Hk (X; E ) ---)0 Hk (A; ----)0 Hk+ 1 (X, A; --+ ...
which is what is to be proved.
K-THEORY
111
Suppose now that (X, A, B) is a triple in (d, d), and let
: H* (A, B; -+ H* (X, A;
be the composite
cS*
H* (A, B; E) -+ H* (A; E) H* (X, A; ID.
Then as in ordinary cohomology, one proves the following.
PROPOSITION (2.3) The triangle
j*
H* ( X A. E ) J H* ( X B- E\
, '- , ,
H* (A, B,;
is exact. Here j'* and i* are the homomorphisms induced by the natural injec-
j i
tions (X, B) c (X, A) and (A, B) c (X, B).
Sometimes it is convenient to define the cohomology theory on the cate-
gory do whose objects are the pairs (X, xo) of a finite CW-complex X and a
O-cell Xo E X which is regarded as a basepoint. The maps of d 0 are base-
point-preserving maps X -+ X'. A cohomology theory on do is defined by a
contravariant functor jj* of do to J and a natural transformation of
degree -1,
(/* : H* 0 8 -+ H*,
where S is the functor do -+ do which sends X to its reduced suspension
S A X, such that the following conditions are satisfied:
1) iff, g E do are homotopic, then fIlI(f) = H"(g);
2) if XEd 0, then
a*(X) : H* (8 A X) H*(X)
is an isomorphism; and
3) if (X, A) is a pair in do, then
H* (X/A) jf*(p) · H*(X) finco) Hn(A)
is exact, where i is the imbedding A -+ X and p is the projection X -+ XI A .
(As usual, write f* for H*(f).) There is a 1-1 correspondence H* -+ H*
between the cohomology theories on (d, d) and those on do which goes
like this. Suppose that {H*, *} is a cohomology theory on (d, d). The
112
THE TOPOLOGY OF CLASSICAL GROUPS
corresponding cohomology theory {1I*, *} on do assigns H* (X, {xo}) to
(X, {xo}), and the homomorphism f*: H* (X, {Yo}) H* (X, {xo}) to
f: (X, xo) (Y, Yo). The natural transformation
...,
<5* : jj* 0 S Fi*
is, by definition, the negative of the composite
* lJ* - 1
H* (S A X, {xo}) H* (TX, X) H* (X, {xo})
where <5* is the boundary operator of the triple (TX, X, {xo}). Conversely,
if {iI*, *} is a cohomology theory on do, we obtain a cohomology theory
on (d, d) by putting
Hn (X, A) = Fin (X/A)
and defining <5 n (X, A) to be the negative of the composite
* - 1 "'" k* "'"
(1 ) H* (S A A) ---:,. H* (X/A)
II
H* (X, A)
H*(A)
II
H* (A, 4»
where k is the composite
X/A XU TA XU TA/X= S 1\ A,
T A being the reduced cone on X UTA.
Problems
Loop-spectra
It is q1;1ite convenient to work with a spectrum E = { , en} .such that the connecting maps
Tt E o -+ Tt l E i -+ ... -+ Tt k E k -+ ...
n_ n+_ n+_
are all isomorphisms. This happens if the connecting maps
en: -+ .Q +l' all n,
are homotopy equivalences. A spectrum which satisfies this property is called a loop-
spectrum. Can an arbitrary spectrum be replaced by a loop-spectrum?
1 Suppose that E = { , En} is a spectrum. Then there is a loop-spectrum E ' = { , e }
and a map f: E -+ E ' of spectra which induces an isomorphism
f* : n TtnE'
for all n.
K-THEORY
113
[First define a spectrum E = { En , en} by putting
,..,
E n = D o, for n < 0,
and
-
En = E n'
for 11 2:: 0,
and defining en to be the identity map for n < 0 and en for n 2:: O. Then there is a natural
,..,
map E -+ E which induces an isomorphism on the homotopy groups. Next by applying the
,.., ,..,
mapping cylinder trick to the adjoints ofek, S 1\ E k -+ E k+l adjust things so that the con-
necting maps
ek: E k DE k+l
,..,
are imbeddings. Finally, define E to be lim D nE n+k with the weak topology, and e :
n
IDe -+ DIDe+l to be the natural map. Then the resulting spectrum E ' = { E ,e } has the de-
sired properties.]
Convergent spectra
A spectrum E is said to be convergent if, and only if, there is an integer no, such that
Ttk no+l) = {OJ for all k i. For example the sphere spectrum Sand the Eilenberg-Mac-
Lane spectrum K( Tt) are both convergent. Can an arbitrary spectrum E be replaced by a
homotopically equivalent convergent spectrum? The answer is no in general, but we can do
the following.
2 Suppose that E = { , en} is a spectrum, and let mo be an integer. Then there is a spec-
trum E ' = { , e } and a map f: E ' -+ E such that
a) E = E i andfiistheidentitymap,for i mo,
b) Ttk +i) = {O}, k i, and
c) Ii * : Ttk E :n o + 1 -+ Ttkg,.O+l is an isomorphism for k 2:: i + 1.
[Let us recall first how the Moore-Postnikov decomposition of a countable CW-com-
plex X can be obtained. Define Xl to be the universal covering space of X and let PI : Xl
-+ Xbe the projection. Then TtlXl = {OJ and PI *: TtlXl -+ Xis an isomorphism for i 2:: 2.
Assume we have defined a sequence
pn X PI
Xn--+X n - l 000 l---+X
(M - P}n
such that (a) TtkXJ = {OJ k j, and (b) (pJPJ-l 000 Pl) *: nk:K.i TtJX is an isomorphism
for k 2:: j + 1. Now Hn+l (X n ; Ttn+1X n ) = Hom (H n + 1 (X n ; Z); nn+1Xn). Let 0n+l e Hn+l
(X n ; nn+1Xn) be the class which corresponds to the inverse of the Hurewicz isomorphism.
Define Xn+l to be the fibration over X n induced by On+l : X n K (Tt"+1Xn, n + 1) from the
path-space fibration DK(Ttn+lXn, n + 1) -+ K(nn+1Xn' -+ n + 1) K(Ttn+1X n , n + 1). Let
Pn+l: Xn+l X n be the natural projection. The (n + I)-stage decomposition is
X Pn+I X Pn X X X
n+l --+ n ---:,. n-l -+ 0 0 0 -+ 1 -+ .
(M - P)n+l
8 Husseini (0216)
114
THE TOPOLOGY OF CLASSICAL GROUPS
To prove 2, show that we can assume each E i is connected, and define
E :"o + i = mo+i)i'
where (Emo+l)l is tlte ith stage of the decomposition of E mo + i , described in the preceding
paragraph.]
The suspension homomorphism
We know that every spectrum E = { , en} gives rise to a new spectrum S 1\ E = {S 1\ En, n }
which plays a role similar to that of ordinary suspension in the category of spaces. How
are the homotopy groups Tt .E and Tt*S 1\ E related? To answer this, let
(] * : Ttn+k k) -+ Ttn+k+ 1 (S A E k )
be the ordinary suspension homomorphism. Then the homomorphism
Ttn+k<Ek) -+ Ttn+k+l <Ek+l)
which is equal to the composite
(-l)k ('£k) * m_ )
Ttn+k<Ek) ) Tt n +k+l (S 1\ E k ) ) Ttn+k+l +1
induces a homomorphism
S *: Ttk E -+ Ttk+1S 1\ E .
(ek is the adjoint of ek; the negative sign is due to the switching map.)
3 Prove that
8 *: Tt* E -+ Tt*S 1\ E
is an isomorphism of degree 1.
[Apply 2 above.]
The homology of spectra
Suppose that E = {En, en} is a spectrum. The homology group HkOO is, by definition,
lim Hk+n ). (Note that the composite homomorphism
n
H k + n <&.) tT. --:,. Hk+n+l (S 1\ En) (en). ) Hk+n+l (E,,+1)
is the connecting map of the direct system {Hk+n<En)}. Here (] * is the usual suspension
homomorphism and en is the adjoint of en.) One defines the suspension homomorphism
8. : H .@ -+ H * (8 1\ ID
in a manner analogous to homotopy.
4 Prove that S * is an isomorphism of degree 1.
Extra-ordinary homology theory
An extra-ordinary homology theory on do is defined by a covariant functor
H*: do - co,
K-THEORY
115
and a natural transformation
,.., ,..,
an : Hn -+ Hn+l 0 S
of degree 1 such that:
,.., ,..,
1) if fis homotopic to g, then H *(f) = H *(g);
2) if X e do, then
,.., ...,
a .(X) : H *(X) H * (S A X)
is an isomorphism of degree 1; and
3) if (X, A) is a pair in do, then the sequence
iI *(A) H *(0 ) ii *(X) H .(p) ) if * (XI A)
is exact. Here i is the imbedding A -+ X and p the map X -+ XI A which identifies A with
the basepoint.
Suppose that E = { , en} is a spectrum and X is in do. By definition let
,..,
Hn (X; ID = Tt n (E 1\ X).
If f: (X, xo) -+ (Y, Yo) is in do, let
..., ,.., ,..,
H*(f) = f*: H* (X; E ) -+ H* (Y; ID
be the induced homomorphism Tt n (E 1\ X) Tt n 1\ Y).
If Xis in do, let
,.., ,..,
a* (X; E ): H* (X; ID -+ H* (S 1\ X; E)
be the composite
s
Tt* (E 1\ X) --:,.n* (S 1\ E 1\ X) n. 1\ S 1\ X)
where T is induced by the switching map s 1\ e 1\ x e 1\ s 1\ x.
,..,
5 Show that H * ( ; E) and a. ( ; E) define an extra-ordinary homology theory.
[The verification of (1) and (2) is immediate. To verify (3) one needs the Blakers- Massey
theorem which allows one to identify Ttl (B, Y)with Ttl (Bly) for highly connected spaces.
First one verifies (3) when E is convergent, and then for the general case by applying
Problem 2 above.]
Ordinary homology theory
,..,
We know that ordinary cohomology is H* (X; K( n». How about ordinary homology?
,..,
6 Prove that H * ( ; K(n» is naturally isomorphic to the ordinary homology theory with
coefficients in Tt.
[Hint lI n (SO 1\ K (n» = Tt nK(n ) and hence if * ( ; K (n» satisfies the dimension axiom.]
7 Suppose that X, Y andf: X -+ Yare all in do. Prove that
..., ,..,
f*: H* (X; ID -+ H* (Y; ID
116
THE TOPOLOGY OF CLASSICAL GROUPS
is an isomorphism for all spectra E if, and only if,
,...
f*: H* (X; K (Z» = H* (X; Z) -+ H* (Y, K( Z» = H* (Y; Z)
is an isomorphism.
[Hint By applying the mapping cylinder trick, one reduces the problem to proving that
,... ,." ,...
if H * (X; Z) = {O then H * (X; E ) = {OJ. But H * (X; Z) = {OJ implies that S 1\ X is
contractibl .]
3 K-theory
Suppose (X, A) is a pair, and let K (X, A) be the Orothendieck group of
(X, A). Then, according to 5 of Chapter 3,
K(X, A) = [XI A, Bu] = 7l o F(X/A; ¥).
where U is the unitary spectrum. By definition, let.
Kn (X, A) = 1t:_ n F (X/A; U), all 11"
= Hn (X, A; U ).
Note that
KO (X, A) = K (X, A),
and
K(X) = K (X, 4» = KO (X, 4» = KO(X).
We shall call K* (X, A) = EnKn (X, A) the K-groups of the pair (X, A).
What are the coefficient groups of this theory? Recall that the coefficient
groups are
Kn(p) = n-nF (P + ; u. ) = n-n U
where P is the complex consisting of a single point, and 4> is the basepoint
of P/4>. Hence the coefficient groups of the K-theory are
{ Z,
Kn(p) =
0,
11, even,
11, odd.
Since Kn(p) = j(n(so), where So is the O-sphere, and since sm = sm 1\ So, it
follows that
,... ,... Sm ,...
K*(sm) = K* (sm 1\ SO) -=- K*(SO),
K-THEORY
117
where sm is m-fold cohomology suspension. Hence, for m even,
and, for m odd,
{ Z,
Kn(sm) =
0,
n even,
n odd,
Thus
{ Z,
Kn( sm) =
0,
n odd,
n even.
K*(S2m) K*(SO)
and
K*(S2m+l) K*(Sl).
Those last two equalities are an expression of the Bott periodicity theorems.
They are special cases of the following general considerations. Consider the
spectrum Q2 U . We would like to define a suitable map of spectra Y -+ Q2 U .
Recall therefore that
Q 2 U = {Q2 U n , Q2s n}
where Un = BU, when n is even, and Un = U, when n is odd. Moreover
en : BU -+ QU is the Bott map tX, and q;: U -+ QBU is the homotopy equival-
ence q; (see 4 of Chapter 3). Now define
'P: U -+ Q2 U
to be the map
1pn: Un -+ Q2l!n
which is equal to the composite
BU QU [)fP ->Q2BU
when n is even, and
U QBU >Q2U
when n is odd. One can easily check that 'P is a map of spectra, i.e., that
Q 2 e n Vln = Q1pn+ 1 en.
Hence 'P induces a homomorphism
'P*: Hn (X, A; U ) -+ Hn (X, A; Q2 U ).
But each 'Pn is a homotopy-equivalence. Hence 'P* is an isomorphism. Now
note that
Hn (X, A; U ) = Kn (X, A)
118
THE TOPOLOGY OF CLASSICAL GROUPS
and
Hn (X, A; Q2 U ) = n-nf(X/A; Q2 U )
= n_nfeX/A;Q2 U o )
= n_n F (S2 1\ (X/A), U o )
= n F ( X j A. Qn+2U )
o , _ 0
= 7t- n -2f(X/A; U o )
= Hn+2 (X, A; U ) = Kn+2 (X, A).
Thus P induces an isomorphism
P*: Kn (X, A) Kn+2 (X, A), all n.
4 Products in K-theory
We wish to introduce products in the K-groups which correspond to the cup-
product in ordinary cohomology. Recall that the procedure in ordinary
cohomology is broken into two stages. First one defines an external product,
H* (X; G) <8> H* (X; G) H* (XxX; G <8> G'),
which is then composed with the homomorphism induced by the diagonal
map X X x X and the homomorphism induced by a pairing G <8> G' G"
to yield the cup-product
H* (X; G) <8> H* (X, G') H* (X; G").
Since spectra in some sense play the role of a coefficient group, a procedure
for defining products in extra-ordinary cohomology along the preceding line
would call for an appropriate notion of pairing of spectra.
The appropriate notion is the following. A pairing of the spectra
A = { A m' lX m } and B = { , Pm} to the spectrum C = { , Ym} is given by
a sequence of maps
fm,n: 4m 1\ 4.n Q",+n
for all m, n with the following property. Consider first the following diagram
4m+l A Bn -. 4".+1 A B n
;mAl / .n
S A B lAfm;n S C 1m+n
A _m A ) . A _m+1I ) C m + JI + 1
A S 'f B A B +1
m A A. II ""') m+1 A :11+1
- - 1 A /In -
K-THEOR Y
119
where the two triangles are commutative; the maps m' Pn and Ym+n are the
adjoints of cX m , Pn and Ym+n respectively; and T: S 1\ A m 1\ A m 1\ S 1\
is the map which interchanges the first two factors. The diagram provides us
with t ee maps Im+l,n 0 ( m 1\ 1) = ', Ym+n 0 (I 1\ Im,n) = and (/m,n+l)
o (I 1\ {3,,) 0 T = " whose adjoints induce three
#' J#, #: [X, Am ] X [Y, ] [X 1\ Y, Q +n+l]
where [, ] is the set of base-preserving homotopy classes of maps. The
condition that {1m,,,} are required to satisfy in order to define a pairing is
# = J#,
( 4.1 )
,.., ....
# = ( - l)m #.
We shall denote the pairing by
I:( A , B) C .
The reader can easily check that I induces a homomorphism
1* : nm A (8) nn B -+ nm+n C
for all m, n. One can also check that I induces a pairing of spectra
1#: (F(X, A ), F(X', ID) F(X 1\ X', 9
where X and X' are finite cell complexes. These two preceding facts imply the
following theorem.
THEOREM (4.2) Suppose that f: ( A , B ) C is a pairing 01 spectra, and let
(X, Xl) and (Y, Y I ) be pairs in d. Then I induces a pairing
1* : H* (X, Xl; A ) (8) H* (Y, Y I ; B ) H* ((X, Xl) X (Y, Y I ); g
where (X, Xl) x(Y, Y I ) is the pair (Xx Y, Xx Y I U Xl X Y).
Suppose next that K(n ) = {K(n, n), en}, K( n') = {K(n', n), e } and
K( n") = {K (n", n), e:} are the Eilenberg-MacLane spectra corresponding
to the countable abelian groups n, n' and n", and let
A : n (8) n' n"
be a pairing. Note that
Hn+m (K (n, n) A K (n', m); n") = Horn (Hn+m (K (n, n) 1\ K (n', m)), n"),
120
THE TOPOLOGY OF CLASSICAL GROUPS
and that the Hurewicz homomorphism
nn+m (K (n, n) 1\ K (n', m)) -> Hn+m (K (n, n) 1\ K (n', m))
is an isomorphism. But
nn+m (K (n, n) 1\ K (n', m)) = n (8) n'.
Hence we can think of the pairing of groups A : n' (8) n' n as an element
A ,m E Hn+m (K(n, n) 1\ K(n, m); n"). By definition, the map
An+m: K(n, n) 1\ K(n', m) K(n", n + m)
is one such that A;+m(i n + m ) = A +m, where i n + m is the fundamental class of
K(n", n + m). One can easily check that the maps {An,m} define a pairing
( K( n), K( n')) K( n")
which induces the usual external product in ordinary cohomology.
Next we show that the unitary spectrum U = { Un , en} admits a suitable
paIrIng.
THEOREM (4.3) There is a natural pairing
A : ( U, U ) U
of spectra.
Proof We would like first to define maps
f) : BU 1\ BU -). BU,
1p: BU 1\ U -). U,
C: U 1\ BU U,
: U 1\ U U,
which will lead to a pairing A : ( U, U ) U such that the pairing induced on
the K-groups
A ,O: /(O(X) (8) j(O(y) /(0 (X 1\ Y)
is that given by the reduced tensor product (see 5 of Chapter 3). Since BU
is of the homotopy-type of a countable CW -complex, we can find an ascend-
ing sequence of finite cell-complexes
· .. C X n C X n + 1 c...
K-THEORY
121
such that X = lim X n = U X n is of the homotopy type of BU. Suppose that
n n O
i: X BU is a homotopy-equivalence, and let
in : X n -+ BU
be the restriction of i to Xn. Hence in represents an element [in] E K(X n ), the
group of stable complex vector bundles over Xn. By definition, let
O:n.n: X m 1\ X,. -+ BU
be a map such that
where
[O l.n] = f3 ([i m ], [in])
f3 : K(X m ) (8) K(X n ) K (X m 1\ X n )
is the reduced tensor-product defined in 5 of Chapter 3. Now notice that
O:n.n I Xl' A Xq, with p < m, q < n, is homotopic to O .q, since f3 is natural.
Thus, by first reordering the double indices (m, n) of the set {X m 1\ X,.} so as
to obtain an ascending sequence whose union is X 1\ X, and then by deform-
ing the maps O:n. n, stage by stage, we obtain a map
0' : X 1\ X BU.
Observe that
o = 0' (i- 1 1\ i- 1 ): BU 1\ BU BU
has the property that the induced map
:r&oF (Y, B U) (8) 7loF ( Y', B U) :r&oF (Y 1\ Y', B U)
is just the reduced tensor-product
f3: K(Y) (8) K(Y') -+ K(Y 1\ Y')
after identifying the corresponding groups.
To define the map
VJ: BU A U -+ U,
proceed as follows. Let
q;: U -+ QBU
be the homotopy-equivalence considered in 5 of Chapter 3. Then the ad- ,
joint of the composite
1 ,, 9
S 1\ BU 1\ U BU 1\ S 1\ U ) BU 1\ BU BU ,
122
THE TOPOLOGY OF CLASSICAL GROUPS
where the first map interchanges 8 and BU and is the adjoint of q;, defines
amap
1p' : BU A U -+ QBU.
The required map is
1p = q;-lVJ': BU A U -+ U.
The map
C: U A BU -+ U
is, by definition, the adjoint of the composite
q;Al (J
8 A U A BU ) BU A BU BU
followed by the mapq;-l : QBU -+ U. Here is the adjoint ofq;: U -+ QBU.
The last map,
: U A U -+ BU,
is defined to be the inverse in the group noF(S2 A U A U, BU) of the com-
posite
A (J
8 A 8 A U A U -+ 8 A U A 8 A U ---:,. BU A BU BU
here the first map interchanges the second and third factors.
Now we define the map
Am,n: U m A Un -+ U m + n
by the rule
Am,n = (j, when m und n are even
= 'ljJ, when m is even, and n odd
= C, when m is odd, and n even
= , when m and n are odd.
To prove that (Am,n) define a pairing A : (U , U ) -+ U proceed as follows.
Suppose X and Yare finite cell complexes. Recall that
K-m(x) = K (8 m A X)
and
K-n(y) = K(sn A Y)
where m and n are > O. Then the composite
K(sm A X) @ K (sn A Y) -+ K (sm A X A 8 n A Y) = ) K(sm+n A X A Y),
K-THEOR Y
123
defines a pairing
{3m,n: K-m(x) @ K-n(y) -+ K-(m+n) (X A Y).
Now one checks easily that the maps {3m,n with m, n = 0, -1, are equal to
those induced by 0, 'ljJ, C and . This is enough to show that the sequence of
maps {Am,n} defines a pairing
A : ( U, U ) -+ U
with the desired properties.
5 Relation of K-theory to another based on the rationals
......
By definition, let
Qn = X K(Q, 2i),
i O
n even,
= X K(Q, 2i + 1), n odd,
i O
where Q is the field of rational numbers; and let
Kn : Qn -+ QQn
be the product of the homotopy-equivalences K(Q;j) QK(Q,j + 1)
which induce the identity on the homotopy groups. Then Q = {Qn, Kn} is
- -
a spectrum. Let H* (X; Q) be the cohomology groups of X with coefficients
in Q. There is a natural pairing
It : (Q, Q) -+ Q
which is induced by the pairing
( K( Q), K( Q)) -+ K( Q)
of the Eilenberg-MacLane spectrum of Q. Certainly fJ induces a multiplica-
tion on H* (X; Q) which turns it into an algebra. Observe that
H O (X; Q) = L H 2i (X; Q)
- . o
and
HI (X; Q) = L H2i+l (X; Q),
- i O
124
THE TOPOLOGY OF CLASSICAL GROUPS
where H'* (X; Q) are the ordinary cohomology groups with rational coeffi-
cients. Note also that the Chern character ch defines a homomorphism
KO(X) = K(X) -+ L H 2i (X; Q) = HO (X; Q)
i O -
Kl(X) -+ L H2i+l (X; Q) = HI (X; Q).
i O -
This Chern character provides us with a way to represent KO + Kl in terms
of ordinary cohomology.
THEOREM The map that takes a bundle to its chern character ch ( ) induces
a homomorphism
K*(X) -+ H* (X; Q)
of algebras.
Problems
A spectrum E with a pairing
(E,ID E
is called a ring spectrum.
1 Prove that the diagram
H* (X, A; ID @ H* (X', A'; ID H* «X, A) x (X', A'), E )
1 1 a*
H* (X', A'; ID @ H* (X, A; ID H* «X', A') X (X, A); E )
is commutative. Here E is a ring spectrum; the horizontal maps are the external products
induced by the pairing in E ; 1;' is the map which takes a @ b to (-l)pq b @ a, with deg a = p
and deg b = q,. and (] is the map which sends (x, x') to (x', x).
2 Suppose that E is a ring spectrum such that
H k (P; ID = to),
when k is odd, where P is the space consisting of a single point. Show that H* (X, A; E ) is
an algebra over H* (P; ID.
3 With the same assumptions as for (2), show that in the exact triangle
"*
11* (A; E) H* (X; E )
H* (X, A; ID
the homomorphisms are all H* (P; ID-homomorphisms.
[Hint. To show that is an H* (P; ID-homomorphism, note that it is definable in terms
of the map XIA S A A which corresponds to XU TA XU TAIX = S A A, TA
being the cone on A.]
K-THEOR Y
125
The spectral sequence
Suppose that E is a spectrum, and let X be a finite cell-complex. By definition, let
AP,q = Hp+q (XCp); ID, A = L AP,q
P.q
Also let
Ef,q = Hp+q (XCP), X CP - 1 ); E ), El = L Ef,q.
P.q
i: A A
be the map induced by the restriction
AP,q = Hp+q (XCP); ID -+ AP-l,q+l = Hp+q (XCP-l); E ),
and let
Ef,q = Hp+q (XCP), XCp-l); E ) Hp+q (X CP ); ID = AP,q.
As an immediate consequence of the exactness axiom, we obtain that
A A
\
E 1
is an exact couple. Here is the co boundary homomorphism
AP-l,q = HP+q-l (X(P-l); ID -+ Ef .q = Hp+q (X(P), X CP - 1 ); E ).
Observe that
bidegj = (0,0), bideg i = (-1, 1), and bideg = (1,0).
Let {E" d,} with r 2:: 1, be the spectral sequence defined by the preceding exact couple.
4 Show that
Ef = L Ef.q = L CP (X; Hq({xo); E »
q q
as differential groups.
[Outline of proof By definition,
Hn (XCP), X CP - 1 ); ID = Tt_n F (XCP)IX CP - 1 ); E ) = TtoF(X(P)IX(P-l); ).
But XCP)IX(P-l) is a bouquet of p-spheres. Hence
E ,n-p = Tt o F(X CP )IX(P-l); En) = 2: nl'- . [e P ]
where the sum is taken over the cell XCp) - X Cp - 1 ). Thus an element in Ef.n- p can be
thought of as a cochain C p(X) -+ Tt p-n ) = Hn-p ({ Xo ) ; E ), where
C*(X) = L Cp(X)
P
is the cellular chain group of X. To prove that d l : Ef Ef-l coincides with the usual
coboundary of C* (X; H* ({xo); E » = L Hom (C p(X), H* ({xo); ID) consider the follow-
ing. Suppose that P
X = X' U f DP+l = X' U e P +l,
126
THE TOPOLOGY OF CLASSICAL GROUPS
where X' is a finite cell complex andf: ODP+l -+ X' is the attaching map for the cell e P + l .
Then the cofiber sequence
X' X XIX' = SP+l
leads to the map
0: XIX' -+ S A X'
such thatoh = h', where h is the homotopy equivalence XU TX' -+ XU TX'ITX' = XIX'
and h' is the natural projection h' : X U TX' -+ X U TX'I X = S A X'. Now show that the
co boundary
: Hn(x'; ID -+ Hn+l (X, X'; E)
is essentially the composite
a*
TtoF(X'; E n) -+ TtoF(S A X', E n + l ) ---:"1T, o F(XIX'; E n + 1 ).]
It follows from 4 that
E .q = HP(X; Hq({xo}; E ».
Now let
FP H* (X; ID = image (H* (XI X(p) ; ID H* (X; E »,
where X XI XCp) is the natural map. Clearly FP H* gives a descending filtration of
H* (X; E).
5 Prove that
EP,;: = Eg,q (H* (X; ID)
where Eo (H* (X; E ) is the bigraded group associated to the filtration given above.
6 Show that if E is a ring spectrum, then the pairing on E induces on each term E" r 2:: 1,
of the spectral sequence {Er, d r } a pairing
E, @ E, Er
which makes E, a bigraded differential commutative ring. Moreover, the equations
E '* = H* (X; H* ({xo); ID)
E * = Et.* (H* (X; E »
are those of bigyaded rings.
K*-theory
7 Show that the correspondence
K* Q : (d, d) -+ J
which assigns K* (X, A) @ Q to the pair (X, A) and the homomorphismf* @ 1 : K* (X', A')
-+ K* (X, A) to the map f: (X, A) -+ (X', A') is an extraordinary cohomology theory natur-
ally equivalent to H* ( ; Q) where Q is the spectrum defined in 6 above.
K-THEORY
127
[Hint Show that the Chern character ch defines a natural equivalence
K* @ Q -+ H* ( ; 0
by comparing the two spectral sequences defined by the two theories.]
8 Suppose that X is a cell complex such that the ordinary cohomology group H* (X; Z) is
torsion-free. Show that the spectral sequence {E" d,} defined by the K*-theory collapses, i.e.,
that d, : E, -+ E, is trivial for all r 2:: 2. Deduce that K*(X) is also torsion-free and that
ch : K*(X) -+ H* (X; Q)
is a monomorphism.
[Hint Compare E, with the spectral sequence' E of K* @ Q. Note that Z -+ Q induces an
injection
Ef'* = H* (X; K*({xo}) -+ 'Ef'* = H* (X; H* ({Xo); 0)
since H*(X) is free of torsion and Km({xo}) is either Z or 0.]
9 Suppose that X is such that H2l+1 (X; Z) = {O} for all i. Show that the spectral sequence
{Er, d r } of the K*-theory collapses and that
ch: K*(X) -+ H* (X; g)
is a monomorphism.
[Hint. E:.q = {O} if either p or q is odd.]
10 Show that in the spectral sequence {E" r 2:: 2} corresponding to the K*-theory, the
differentials
d 2i : E 2i -+ E 2i , i 2:: 1
are all zero.
K*-theory graded by Z2.
Observe first of all that
L Kft({xo})
n O
is a polynomial ring Z[y] where 1J e K- 2 ({xo}) = KO(S2) corresponds to the 1J - 1,1J being
the Hopf-bundle over S2. By definition, let
K * (X, A) = K* (X, A)/J = K O (X, A) + K l (X, A)
- -
where J is the ideal K* (X, A) Z[y] (Z[y] being the elements of degree different from zero).
According to Problem (3) above, the co boundary
* : K*(A) -+ K* (X, A)
is a Z[y]-homomorphism and hence induces a coboundary
* : K *(A) -+ K* (X, A).
11 Prove that the pair { K *, *} defines an extra-cohomology theory whose values are in the
category of abelian groups graded by Z2.
128
THE TOPOLOGY OF CLASSICAL GROUPS
12 Show that the filtration of X by its skeletons induces a spectral sequence {E" d,} of
graded differential algebras E, = L E such that
p
Er = H* (X; Z)
and
E = Eo ( K *(X).
Bibliography
[1] M.F.Atiyah, K-Theory, W.A.Benjamin, New York (1966)
[2] M.F.Atiyah and F.Hirzebruch, "Vector bundles and homogeneous spaces", Proc.
Symposia in Pure Math, III, Amer. Math. Soc. (1961), 7-37
[3] G. W. Whithead, "Generalized homology theories", Trans. Arner. Math. Soc. 102
(1962), 227-83
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