Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
Subseries:
1361
T. torn Dieck (Ed.)
Algebraic Topology and
Transformation Groups
Proceedings of a Conference held in
Gottingen, FRG, August 23-29, 1987
Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo

Editor
Tammo torn Dieck
Mathematisches Institut, Universitat Gottingen
Bunsenstr. 3-5, 3400 Gottingen, Federal Republic of Germany
Mathematics Subject Classification (1980): 57SXX, 55-XX
ISBN 3-540-50528-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-50528-8 Springer-Verlag New York Berlin Heidelberg
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TABLE OF CONTENTS
S. Bauer: The homotopy type of a 4-manlfold with finite
fundamental group. 1
C.-F. Bodigheimer and F.R. Cohen: Rational cohomology of
configuration spaces of surfaces. 7
1 1
G. Dylawerskl: An S -degree and S -maps between representation
spheres. 14
R. Lee and S.H. Weintraub: On certain Siegel modular varieties
of genus two and levels above two. 29
L.G. Lewis, Jr.: The RO(G)-graded equivariant ordinary
cohomology of complex projective spaces with linear z/p actions. 53
w. Luck: The equivariant degree. 123
w. Luck and A. Ranicki: Surgery transfer. 167
R.J. Milgram: Some remarks on the Kirby - Siebenmann class. 247
D. Notbohm: The fixed-point conjecture for p-toral groups. 253
1
V. Puppe: Simply connected manifolds without S -symmetry. 261
P. Vogel: 2x2- matrices and application to link theory. 269

List of Participants
ANDERSON, Douglas R.
Dept. of Mathematics
Syracuse University
Syracuse, N.Y. 13210
USA
BAK, Anthony
Fakultat fiir Mathematik
Universitat Bielefeld
Universitatsstr. 1
4800 Bielefeld 1
W-Germany
BAUER, Stefan
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3400 Gottingen
W-Germany
BODIGHEIMER, Carl-Friedrich
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3400 Gottingen
W-Germany
COHEN, Frederick
Dept. of Mathematics
Universitiy of Kentucky
Lexington, Ky 40506
USA
CONNOLLY, Frank
Dept. of Mathematics
University of Notre Dame
P.O. Box 3 98
Notre Dame, Ind. 46556
USA
DAVIS, James
Dept. of Mathematics
Indiana University
Bloomington, Ind. 47405
USA
torn DIECK, Tammo
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3400 Gottingen
W-Germany
DOVERMANN, Karl-Heinz
Dept. of Mathematics
University of Hawaii
Honolulu, HI 96822
USA
DYLAWERSKI, Grzegorz
Inst. Math.
Uniwersytet Gedanski
ul. Wita Stwosza 57
P-80-952 Gdansk
Polen
EWING, John
Dept. of Mathematics
Swain Hall East
Indiana University
Bloomington, Ind. 47 405
USA
FERRY, Steven
Dept. of Mathematics
University of Kentucky
Lexington, KY 40506
USA
FRANJOU, Vincent
Institut de Mathematique
Universite de Nantes
2, rue de la Houssiniere
F-44072 Nantes cedex
France
HUEBSCHMANN, Johannes
Mathematisches Institut
Im Neuenheimer Feld 288
6900 Heidelberg
W.-Germany
IGODT, Paul
Mathematik
Katholieke Universitet
Leuven
Fakulteit Wetenschappen
Campus Kortrijk
B- 8500 Kortrijk
Belgium
JACKOWSKI, Stefan
Wydz. Mat. i Mech.
Instytut Matematyki
Uniwersytet Warszawski
P-00-901 Warszawa
Polen
JODEL, Jerzy
Inst. Math.
Uniwersytet Gdanski
ul. Wita Stwosya 57
P-80-952 Gdansk
Polen

V
KOSCHORKE, U.
Lehrst. f. Mathematik V
Universitat Siegen
Holderlinstr. 3
5900 Siegen
W-Germany
LAITINEN, Erkki
Mathematik
University of Helsinki
Helsinki
Finland
LANNES, Jean
Ecole Polytechnique
- Mathematique -
F-91129 Palaiseau
France
LEE, Ronnie
Dept, of Mathematics
Yale University
Box 2155, Yale Station
New Haven, Conn 06520
USA
LEWIS, Gaunce
Dept. of Mathematics
Syracuse University
Syracuse, N.Y. 13210
USA
LOFFLER, Peter
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3 400 Gottingen
W-Germany
LOCK, Wolfgang
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3400 Gottingen
W-Germany
LUSTIG, Martin
Fakultat fur Mathematik
Universitatsstr. 150, Gebaude NA
4 6 30 Bochum 1
W.-Germany
McCLURE, James E.
Dept. of Mathematics
University of Kentucky
Lexington, KY 40506
USA
MAYER, K.H.
Institut f. Mathematik
Universitat Dortmund
Postfach 500 500
4 600 Dortmund 50
W-Germany
MILGRAM, R.J.
Dept. of Mathematics
Bldg. 380
Stanford University
Stanford, Cal. 94305
USA
MUNKHOLM, Hans J.
Matematisk Institut
Odense Universitet
Dk-5230 Odense M
Denmark
NOTBOHM, Dietrich
Mathematisches Institut
Bunsenstr. 3-5
3400 Gottingen
W.-Germany
ODA, Nobuyuki
Dept. of Appl. Mathematics
Jonan-ku
Fukuoka, 814-01
Japan
OLIVER, Robert
Matematisk Institut
Aarhus Universitet
Dk-8000 Aarhus C
Denmark
PEDERSEN, Erik
Matematisk Institut
Odense Universitet
Dk-5230 Odense M
Denmark
PESCHKE, Georg
Dept. of Mathematics
University of Alberta
Edmonton, Alberta
Canada,' T 6 G 261
PETRIE, Ted
Dept. of Mathematics
Rutgers University-
New Brunswick, N.J. 08903
USA
PUPPE, Volker
Fakultat fur Mathematik
Universitat Konstanz
Postfach 5560
77 50 Konstanz
W-Germany

VI
RANICKI, Andrew
Math. Dept.
The University
Mayfield Rd.
Edinburgh EH9 3JZ
Scotland
RAUSSEN, Martin
Inst. f. Elektr. Systemer
Aalborg Universitetscenter
Strandvejen 19
Dk-9000 Aalborg
Denmark
ROTHENBERG, Mel
Dept. of Mathematics
University of Chicago
5734 University Avenue
Chicago, 111. 60637
USA
SCHAFER, James A.
Dept. of Mathematics
College Park Campus
Mathematics Bldg. 084
College Park
Maryland 20742
USA
SCHNEIDER, Albert
Mathematisches Institut
Universitat Gottingen
Bunsenstr. 3-5
3 400 Gottingen
W-Germany
SCHWARTZ, Lionel
Dept. de Mathematique
Univ. de Paris/Sud, Bat. 425
F-91405 Orsay cedex
France
SMITH, Lawrence
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3400 Gottingen
W-Germany
SWITZER, Robert
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3400 Gottingen
W-Germany
TWISSELMANN, Ute
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3400 Gottingen
W-Germany
VALLEJ0, Ernesto
Mathematisches Institut
Im Neuenheimer Feld 288
6900 Mannheim
W-Germany
V0GEL, Pierre
Dept. de Mathematiques
Universite de Nantes
2, rue de La Houssiniere
F - 44072 Nantes
France
WEINTRAUB, Steven H.
Dept. of Mathematics
Louisiana State University
Baton Rouge LA 70803
USA
WEISS, Michael
Mathematisches Institut
SFB 170
Universitat Gottingen
Bunsenstr. 3-5
3 400 Gottingen
W-Germany
ZARATI, Said
Dept. de Mathematique
Universite de Tunis
1060 Tunis
Tunesia

The Homotopy Type of a 4-Manifold
with finite Fundamental Group
by Stefan Bauer*
ABSTRACT:... is determined by its quadratic 2-type, if the 2-Sylow subgroup has
4-periodic cohomology.
The homotopy type of simply connected 4-ma.nifolds is determined by the intersection
form. This is a well-known result of J.H.C. Whitehead and 3. Milnor. In the non-simply
connected case the homotopy groups -W\ and 7¾ and the first k-invariant k € H3(iri,iT2)
give other homotopy invariants. The quadratic 2-type of an oriented closed 4-manifold
is the isometry class of the quadruple [iri(M), TT2{M),k(M),y(M)], where j(M) denotes
the intersection form on ^(M) = /^(M). An isometry of two such quadruples is an
isomorphism of 7Tj and tt2 which induces an isometry on 7 and respects the k-invariant.
Recently [H—K] I. Hambleton and M. Kreck, studying the homeomorphism types of
4-manifolds, showed that for groups with periodic cohomology of period 4 the quadratic
2-type determines the homotopy type.
This result can be improved away from the prime 2.
Theorem: Suppose the 2-Sylow subgroup of G has 4-periodic cohomology. Then the
homotopy type of an oriented 4-dimensional Poincare complex with fundamental group G
is determined by its quadratic 2-type.
I am indebted to Richard Swan for showing me proposition 6. Furthermore I am
grateful to the department of mathematics at the University of Chicago for its hospitality
during the last year.
* Supported by the DFG

2
Let X be an oriented 4-dimensional Poincare complex with finite fundamental group,
/ : X —» B its 2-stage Postnikov approximation, determined by tti, tt2, and k, and let j(X)
denote the intersection form on H2{X), Then S4D(B, j(X)) denotes the set of homotopy
types of 4-dimensional Poincare complexes F, together with 3-equivalences g : Y —> B,
such that / and g induce an isometry of the quadratic 2-types.
The universal cover B is an Eilenberg-MacLane space and hence, by [MacL], H${B) =
T(iT2{B)), the Z7Ti(B)-module r(7r2(B)) being the module of symmetric 2-tensors, i.e. the
kernel of the map (1 - r): n2(B) ® tt2(B) -» tt2(B) ®tt2(B), (l-r)(a®6) = a®fe-6®a.
The intersection form on A' corresponds to f*[X] of the fundamental class [A] € Ht(X; Z).
Let H* denote Tate homology.
Proposition 1: If A' is a Poincare space with finite fundamental group G, then there
is a bijection //0(G;7r3(A)) *—* Sf D{B, j{X)).
The proof uses a lemma of [H-K]:
Lemma 2: Let (A,/) and (Y,g) be elements in S%D(B,j(X)). Then the only
obstruction for the existence of a homotopy equivalence h : X —* Y over B is the vanishing
oig.[Y]-f.[X]eH4(B).
Lemma 3: Given a diagram
Z A M
■nl
Z
such that the torsion in the cokernel of a is annihilated by n, then the torsion subgroup
in the pushout K is isomorphic to the torsion subgroup of cofcer(a).
Proof of 3: Since the torsion subgroup of M maps injectively into K as well as into
cofcer(a), we may assume it trivial. Then M is isomorphic to TV© < x > with a(l) = mx
for an integer m dividing n. The pushout then is isomorphic to (7V©Z©Z)/ < (0, m, n) >=
M®Z/m. A
Proof of proposition 1: Let (A',/) and (Y,g) be elements in S%D{B) such that /
and 5 induce an isometry of the quadratic 2-types. Let 7(A) = j{Y) = 7 denote the
intersection form on H2(X) and H2{Y). By [W] one has tt3(A) S r(7r2(X))/(7) S H4{B,X)

3
and 7t3(X) ®zg Z = H4{B,X). In the pushout diagramm:
0 —>
0 —>
0 —►
0
1
#4(1)®ZGZ
#4(X)
1
H4(X,X)
I
0
—
CM
0
1
H4(B) ®ZG Z
1
H4(B)
I
H4(B,B)
I
0
—
>
#<
0
1
,(B,A)®ZGZ — 0
1=
H4{B,X) —> 0
1
0
the torsion subgroup of H4{B, X) is isomorphic to the torsion subgroup of H4(E) by lemma
3: The module H4{B,X) is torsion free. Hence the torsion subgroup of H4{B,X) ®za Z
is annihilated by the order n of the group G, Note that 4> is just multiplication by n. In
particular one has
TorsioniH^B)) =■ ToTsion{H4{B,X)) = H0(G;tt3(X))
Since X and Y have the same quadratic 2-type, f»[X] = g»\Y\, hence we have
f*[X] — g»[Y] £ ToTsion(HiB). This gives an injection
S?D(Btl)^Ho(G;7c3(X)).
What about surjectivity? Let K C A* denote a subspace, where one single orbit is deleted.
Let a € ^(K) map via the surjection ttz{K) —> ir3{X) —> tt3(A') ®zg Z to a given element
a £ H0(G;ir3(X)). Let /3 be the image of 1 € ZG =■ H4(X,K) =- tt4(X, K) ^ tt3{K).
Now let k : S3 -» K represent a + P and define Xa := (K Uk (G x DA))/G. One has
to show that Xa is an orientable Poincare space. Orientability is clear, since H4{Xa) =
H4(Xa,K) = Z. Let / : Xa —> B extend J\k/g- The intersection form on Xa is
determined by
f*.[xa] = trf{fam[xa\) e h4(x).
But we have fa*[Xa] = f„[X] + a: In the following diagram 1 € Z = 7r4(A', B) is mapped
to h[X] e H4(B).
H4{X) s< H4(X,K/G) *-*- HA{X,K) S 7r4(A, X) —► jr3(/f)
/I I I I 1=
H4(B) <* H4{B,K/G) *-*- H4{B,K) £ jr4(B,/f) = t3(X)
If the upper row is replaced by the corresponding row for Xa and the vertical maps by
the ones induced by fa, then 1 € ZG is mapped (counterclockwise) to fa*[Xa] on the one
hand, on the other hand (clockwise) to /»[A'] + a.
Since the torsion element a lies in the kernel of the transfer, one immediately gets
fa.[Xa] = MX}- ' *

4
In the sequel all ZG-modules have underlying a free abelian group.
The short exact sequence
0 -» Z -1+ T(tt2X) — 7r3(X) -» 0
gives rise to an exact sequence in Tate homology:
#o(G;Z) — ff0(G;r(^A')) — ffo(G;ir3(A')) — £-i(G;Z) -^ ff_i(G;r(ir2X))
Here #0(G;Z) = 0 and //_i(G;Z) = Z/| G|.The sequence above gives the connection to
[H-K], theorem(l.l).
In order to analyze this sequence, I recall some facts from [H-K],§§2 and 3.
Facts:
1) T(ZG)= ©, Z[G/Hi\ © F, where the summation is over all subgroups Hi of
order 2 and F is a free ZG-module.
2) r(ZG)sr(f)®ZG^r(r)®ZG.Here/ denotes the augmentation ideal,
/* its dual.
3) The modules fi3Z and 53Z are (stably!) defined by exact sequences
0 -> fi3Z -> F2 -> Fi ^ F0 -> Z -> 0
and
0 -> Z -> Fi -> F2 -> -¾ -> 53Z -> 0
with free modules F,.
There is an exact sequence
0 -> fi3Z —► tt2(X) © rZG —► 53Z -> 0
Lemma 4:IfO—»^4—»B—»C—»0isa. short exact sequence of ZG-modules, which
are free over Z, then there are short exact sequences
0 -> T(A) —> T(B) —> D -> 0
and
0^A®zC —>D —> r(G) -> 0.
Proof: Given Z-bases {a;}, {cy} and {a,, cy } of .4, C and B, the map h : ai ® cy —»
Q-i ® cy + cy ® a; is well-defined and equivariant modulo r(.4). A
To prove the theorem, it suffices to show that Ho(G;tt3(X)) = 0. This in turn can be
done separately for each p-Sylow subgroup Gv of G.
Proposition 5: The map 7. : //_i(Gp;Z) —► H-i(Gp;T(tt2(X))) is injective, if
either p is odd or res§ 7r2(A') = A © B splits such that the rank of B over Z is odd. In
general the kernel is at most of order 2.

5
Proof: For the sake of brevity, let ir denote tt2(X) and also let F denote the module
r(ir). Now look at the following sequence of maps:
^:Z^L»r->7r®7rS Hom(irm,ir) £= Hom{ir, ir) ^ Z.
A generator of Z is mapped in Hom{ir*,ir) to the Poincare map a : ir* = H2(X) -—*
H2{X) = ir, and then to the element id € Hom(7r, ir). So we have ^(1) = ranfczC"')-
Fact 3) gives Tank?,{ir) — "~2 raorf | G |, hence the induced selfmap ipt of Z/|GP| =
H-i(Gp; Z) is multiplication by -2. This proves, that the kernel is at most of order 2. In
particular it is trivial, if p is odd.
In case p = 2 and res§ tt = A © B, such that the rank of the underlying group
of B is odd, one can replace the map Hom{ir,ir) > Z by the map Hom(ir,ir) -^-+
Hom(B, B) —► Z in the defining sequence for ip. A similar argument as above for p odd
gives the claim. A
Remark: The module resg 7r2(Ar) always splits, if #4(G;Z) ^ SxiZG(53Z,n3Z)
has no 2-torsion, in particular if G2 has 4-periodic cohomology.
Proposition 6: Let A denote either finZ or 5nZ and let r be the selfmap oi A® A
which permutes the factors. Then (—l)nr induces the identity on Hq(G; A ® A).
Proof: Let F. —> Z be a free resolution of Z and let F. be the truncated complex with
F{ = Fi for i < n— l,Fn = Un and F = 0 else. There is an obvious projection J : F. —* F.,
such that /n = 3n. The tensor product F.<2)F. = F. again is a free resolution of Z and F.
is a. truncatedfree resolution of Z with F22n = fiZ ® fiZ. The chain map / ® / induces an
isomorphism of Hm(F.2 ®zgZ) and H*(F. ®zgZ) in the dimensions * < 2n. The selfmap
i of F2., as usual defined by t(x ® j/) = (_l)des(*)<<es(?)x <g> ?/, is a chain automorphism,
inducing the identity on the augmentation, hence on all derived functors, in particular on
Hm(F. ®zg Z) = Hm(G;Z). In the same way an involution t can be defined on F. . and
/ ® / commutes with i. Obviously i2n = (~l)n,r- Hence ( —l)nr induces the identity on
^2n(F22n®zGZ) = //0(G;Z).
The proof for 5™Z is dual. ♦
Proof of the theorem: By proposition 1, it suffices to show that Hq{G;ttz(X))
vanishes. By proposition 4 and the remark following it, this group is isomorphic to
Ho(G;T(tt2(X))). In order to show that this group vanishes it suffices, by lemma 3, to show
that H0{G;A) vanishes for A £ {T{Q3 2)^(^2)^2 ® 53Z} But /70(G;fi3Z ® 53Z) °i
Ho(G;2i) = 0. Given a module B (with underlying free abelian group), there is a short
exact sequence
0 -> T(B) —> B ® B —* A2(B) -> 0.

6
The map r, which flips the both factors, induces, if applied to B £ {fl3Z,53Z} the
following diagram:
- Hi(G;A{B)) —* H0(G;T(B)) —* HQ(G;B ® B) -»
I (-id) i id 1 (-id)
- H!(G;A{B)) —* H0(G;T(B)) —* H0(G;B®B) -
The right vertical map is ( — id) by proposition 5. This diagram shows that any element
in Hq(G;T(B)) is annihilated by 4.In particular this group vanishes, if G is a p-group for
an odd prime p. That Hq(G2; T(B)) vanishes, if G2 has 4-periodic cohornology, follows at
once from the facts 1-3, since in this case fi3Z = I* (B nZG and 53Z — I © nZG A
Final Remark: An elementary but lengthy computation shows r(SQZ) S Z/20Z/2
and T(Q3Z) = 0 for G = Z/2®Z/2. In particular the group J/0(Z/2®Z/2;T(n3Z®53Z))
is nontrivial. Hence the argument above won't work in general.
REFERENCES
[B 1] K.S. Brown: Cohornology of groups. GTM 87, Springer-Verlag, N.Y. 1982
[B 2] R. Brown: Elements of Modern Topology. McGraw - Hill, London, 1968
[H-K] I. Hambleton and M. Kreck: On the Classification of Topological 4-Manifolds
with finite Fundamental Group. Preprint, 1986
[MacL] S. MacLane: Cohornology theory of abelian groups. Proc. Int. Math. Congress,
vol. 2 (1950), pp8- 14
[W] J.H.C. Whitehead: On simply connected 4-dimensional polyhedra. Comment.
Math. Helv., 22 (1949), pp 48 - 92.
Sonderforschungsbereich 170
Geometric und Analysis
Mathematisches Institut
Bunsenstr. 3-5
D-3400 GSttingen, FRG

Rational Cohomology of
Configuration Spaces of Surfaces
C.-F. Bodigheimer and F.R. Cohen
k
1. Introduction. The k-th configuration space C (M) of a manifold M is
the space of all unordered k-tuples of distinct points in M. In previous
k
work [BCT] we have determined the rank of H (C (M) ; IF ) for various
fields W . However, for even dimensional M the method worked for IF =IF_
* k
only. The following is a report on calculations of H (C (M);C) for M
a deleted, orientable surface. This case is of considerable interest
because of its applications to mapping class groups, see [ BCP].
Similar results for (m-1)-connected, deleted 2m-manifolds will appear
in [ BCM].
2. Statement of results. The symmetric group I, acts freely on the space
~ k
C (M) of all ordered k-tuples (z.,...,z ), z.eM, such that z. ^z; for
k
i4 j. The orbit space is C (M). As in [BCT] we will determine the
* k
rational vector space H (C (M);Q) as part of the cohomology of a much
larger space. Namely, if X is any space with basepoint x , we consider
the space
(1)
C(M;X) =( ^-cNM) fk x^/f
where (z1 , . . . f zki_;x1 ■ ■ ■ ■ >xk) « (z , . . . r zn_1 ;x1 , . . . -xk_1 ) if xk = XQ-
The space C is filtered by subspaces
/, k . . v
•k<~ (M;X) = I 4—L- C 3 (M) j X]
;2) F C(M;X) ={4—V C J(M) X XJ
h
and the quotients F.C/F C are denoted by D,(M;X).
Let M denote a closed, orientable surface of genus g, and M is M
g g g
minus a point. We study C(M ;S nj for n>1. H will always stand for

rational cohomology, and P[ ] resp. E[ ] for polynomial resp.
exterior algebras over Q.
Theorem A. There is an isomorphism of vector spaces
(3) H*C(M ;S2n) = P[v,u ,...,u2 ]siH#{E[w,z1 ,...,z2 ],d)
with |v|=2n, |u.l=4n+2, |wl=4n+1, |z.l=2n+1, and the differential
d if: given by d(wl =2(z.z, + ... +z. _1z2 ' "
Giving the generators weights, wght (v) = wght(z.) =1 and wght(u.)
=wght(w) =2, makes H C into a filtered vector space. We denote this
weight filtration by F H*C. The length filtration F C of C defines a
second filtration H F, C of H C.
k
Theorem B. As vector spaces
* 2n * 2n
(4) H F. C(M ;S ) = F. H C (M ; S ) .
kg kg
It follows that H D, (M ;S ) is isomorphic to the vector subspace
k g
of H (g,n) =P[v,u.]bH (E[w,z.],d) spanned by all monomials of weight
]^
exactly k. To obtain the cohomology of C (M ) itself, we consider the
vector bundle
k ~k,„ , k „k,
Lk
(5) <rf: 'CT(M )£ IR'V-C'V(M ) +
which has the following properties. First, the Thorn space of m times
^ is homomorphic to D, (M ;S ). Secondly, it has finite even order,
see [CCKN]. Hence
2nk 2nk'k k
(6) Dk(Mg;S^nK) =1 C (M )+
for 2n, = ord{\ ) . Thus we have

* k
Theorem C. As a vector space, H C (M ) is isomorphic to the vector
subspace generated by all monomials of weight k _in H (g,n,),
desuspended 2n, k times .
Regarding the homology of E = E[w,z,...,z. ] we have
Theorem D. The homology H (E,d) is as follows:
(7) rank H± (2n+1)= (2±g) - Q?2) for i=0,1,...g, and all (non-zero)
elements have weight i;
(8) rank H . ( 2n+1 ,+4n+f (2g) - (¾ for i -g,...,2g. and all
(non-zero) elements have weight i+2;
(9) rank H. =0 in all other degrees j.
Note the apparent duality rank H. = rank H . for N = 2g(2n+1)+4n+1 .
We will give the proof of Theorem A in the next section. The proof
of Theorem B is the same as for [BCT, Thm.B]. By what we said above
Theorem C folows from Theorem B. And Theorem D will be derived in the
last section.
3. Mapping spaces and fibrations. Let D denote an embedded disc in M .
There is a commutative diagram
(10) C(D?S2n) >Q2S2n+2
C(M ;S2n) >map (M ;S2n+2)
C(Mg,Dts2n) >(is2n+2)2g
where map stands for based maps. The right column is induced by
restricting to the 1-section, and is a fibration. The left column is
a quasifibration. Since S is connected, all three horizontal maps

10
are equivalences, see [M], [B] for details.
The E_-term of the Serre spectral sequence of these (quasi)fibrations
is as follows. From the base we have 2g-fold tensor product of
(11) H*fiS2n+2 = H*(S2n+1xfiS4n+3) = E[ zL] BPtUj] (i = 1,...2g),
where |z.I =2n+1 and |u.I =4n+2. From the fibre we have
(12) HVs2n+2 = H*(«S2n+1x «2S4n+3) =H*(«S2n+1xS4n+1)
= P[ v] a E[ w] ,
where |v| =2n and |w| =4n+1. The following determines all differentials
in this spectral sequence.
Lemma. The differentials are as follows:
(13) d2n+1(v)=0
(14) d4n+2(w) =2zlZ2 + 2z2z3+ ... +2z2g_lZ2g
Proof: Assertion (13) follows from the stable splitting of C(M ;S ),
on [B]. (14) results from symmetries of M and of the fibrations (10)
g
which leave d invariant. ■
The lemma implies E. ,, =E =H C (M ;S ). Furthermore, E. ,, is
4n+3 °o g 4n+3
a tensor product of the polynomial algebra P[v,u.., ... u? ] and the
homology module H (E,d) of the exterior algebra E = E[w,z., ..., z_ ]
with differential d. This proves Theorem A.
4. Homology of E.Let us write x . = z_ . - and y. = z_ . for i = 1 , . . .g.
■" l 2i-1 l 2i • i
The form d(w) = 2z.z_ + 2z2z + ... + 2z2 ,z is equivalent to the
standard symplectic form x y +x y + ... +x y . The vector space

11
E[g] = L[ g] a> wL[ g] with L[ g ] = E[ x y , ..., x . y ] . The differential is
zero on the first summand, and sends the second to the first. Hence
we regard d as an endomorphism of L[g], given by multiplication with
d(w) =xlYl + ... +xgYg.
Let L,[ g] denote the vector subspace spanned by all k-fold products
(15) z . z . ...z. with 1<i, <i_<...<i, < 2 .
1^2 in 12 kg
Since d(w) is homogeneous of weight 2, we have
(16) d = d[g] = © dk[g], dk[g]; Lktg] > Lk+2[g] '
k=0
The (co)kernelsof d,[g ] is determined by the (co)kerne 1 of d,[g-1]
2 2
and d^g-1] for l = k,k-1,k-2. The (co)kerne 1 of d^g-1] in turn is
determined by the (co)kernels of d [g-2]2 and d [g-2]3 for m = 1,1-1,1-2.
m m
Therefore we will study all powers d [g] and prove the following
(Lefschetz) lemma by simultaneous induction on g, k and r.
Lemma. For 0 < k < g the differential
dk[g]r :Lk[g] >Lk+2rCg] is
(17) a monomorphism for 0 < k < g-r,
(18) an isomorphism for k=g-r
(19) an epimorphism for g-r <k<2g
g r r r-1
Proof: For X =Z x. y, we have X = X , + x y and X = X , +rj, ,xy,
g i=1 iJi g g-1 gJg g g-1 g-1 g^g
in particular \g = g!oo where u =x.y.x„y0 ... x y is the volume
g g g 1 1 ^2 g^g
element. To facilitate the induction, we decompose L,[ g] further by
partitioning the canonical basis elements (15) into four types.
(20) ik<2g-2
(21) ik_1 <2g-2 and ik = 2g-1,

12
(22)
(23)
ik_i ^ 2g-2 and ik = 2g,
ik_1 = 2g-1 and iR = 2g .
Hence Lk[g]=L [g-1] a> LR-1[ g-1 ]x « Lk_i[g-1]yq * Lk-2[g~1 ]XgYg"
With respect to this decomposition d, [ g ] has the following matrix form
:24)
dk[g]J
d rg-i]J
k
0
0
0
k-1L
0
g-lf
o dk_
0
0 rdk[g-1]
0 0
^[q-1]* 0
0 dk-2[g~1]
r-1
A 0 0 A'
0 B 0 0
0 0 B 0
0 0 0 C
To start the induction consider the case g = 1 . The only non-zero
differential d [ 1] ; L [ 1] —*LJ 1] is an isomorphism. For g > 2 and k = 0,
r r
d [g] sends the generator of L [g] to A , and thus is monic. Assume
o J o g
the lemma holds for g-1. We distinguish three cases.
Case k < g-r: Then A, A', B as well as C in (24) are all monomorphisms
by hypothesis. Hence, from 0 = d, [ g ] (a,b..,b2,c)={A{a), B(b.), B(b2),
A'(a) +C(c)) we conclude a = b. =b2=0, an* so c = 0 as well. Thus
dj[g] is a monomorphism.
Case k = g-r: Here A is an epimorphism, A' and B are isomorphisms,
and C is a monomorphism. Assume 0 =d,[g] (a,b.,b,,c) =(A(a), B(b.),
A' (a) +C(c)). First, b1 = b2 = 0. We now have A (a) =dR[g-1 ]ra = 0
and dk_2[g-1] c = -rd.[g-1] a; writing this as d.[g-1] (-ra) =A(-ra)
r + 1
= 0. Thus, since d._[g-1] is an isomorphism, c=0. Therefore,
r ~1 r ~1
-rd [g-1] a = 0, and a = 0 since d _ [g-1] is an isomorphism. We
see that d,[g] is a monomorphism between vector spaces of equal
dimensions, hence an isomorphism.

13
Case k > g-r: This time A, A', B, C are epimorphisms. Given (a,b..,b2
£L, _ [g] we can first find a, b. , b_ satisfying A(a) =a, B(b.) = b.
and B(b2) =b2- Then we choose c such that C(c) =c-A'(a). Hence
d,[ g] is epimorphic. ■
The lemma completely determines H (E,d) as a vector space over
0- Theorem D now follows.
References
[B]
C.-F. Bodigheimer: Stable splittings of mapping spaces.
Proc. Seattle (1985), Springer LNM 1286, p. 174-187.
[BCM] C.-F. Bodigheimer, F.R. Cohen, R.J. Milgram: On deleted
symmetric products. In preparation.
[BCP] C.-F. Bodigheimer, F.R. Cohen, M. Peim: Mapping spaces and
the hyperelliptic mapping class group. In preparation.
[BCT] C.-F. Bodigheimer, F.R. Cohen, L. Taylor: Homology of
conf iguration spaces . To appear in Topology.
[ CCKN] F.R. Cohen, R. Cohen, N. Kuhn, J. Neisendorfer: Bundles
over configuration spaces. Pac. J. Math. 104 (1983) , p. 47-
[ M] D. McDuff: Configuration spaces of positive and negative
particles. Topology 14 (1975), p. 91-107.
C.-F. Bodigheimer
Mathematisches Institut
BunsenstraBe 3-5
D-3400 Gottingen
West Germany
F.R. Cohen
Department of Mathematics
University of Kentucky
Lexington, KY 40506
USA

An S -Degree and S -Maps Between Representation
Spheres
Grzegorz Dylawerski
Abstract. Let V be an orthogonal representation of G=S and let
S(V) , S(V©R) be the unit spheres in V , V®R respectively. In this
paper we classify S -equivariant maps S(V©R) * S(V) . More
precisely we construct an isomorphism [S(V©R),S{V}] G » A(V) where A(V) =
= [S(V®R) ,S(VJ ] ffi ((& Z) , HcS runs over all isotropy subgroups
of V different from S .
Introduction. Let V be an orthogonal finite-dimensional
representation of G=S ,<fte(V©R) an open bounded invariant subset and f:(<5l,3cft)
* (V.VS10>) an equivariant map. For the above f an S -degree ,
denoted Deg ( f,<ft) , was defined in work [3]. This is an element of
the group Z0 $(® Z) , where HCS runs over all the isotropy sub-
H 1
groups of ? different from S . It is natural to aak whether this
degree classifies the homotopy classes of equivariant maps (B,3B) —*■
-»(V,Vn{0}) , where B denotes the unit ball in V + R . This is not
true ingeneral. Anyway slightly modifying the first coordinate of
Deg (f,S) we get a new invariant which classifies these G-homotopy
clasaes . Since [(B,8B) , (V,V <• {o]\a*= [S (V S> R),S(V)]G , the new
S -degree classifies the Ghomotopy classes of G-maps between spheres.
The problem of classification of G-maps between representation
spheres was studied by G.B. Segal [5] , T. torn Dieck [2] ,
R.Rubinsztein[4]. We would like to mention that we came to the
method used here in a result of studying [2~\
In Section 0. we introduce notations , compile some basic facts
concerning group actions and the obstruction theory . In Section 1. we

15
recall the properties of Deg(f,5l) , sketch the definition of Deg(f,<2)
in the special case and define a new S -degree D(f,B) , (Deg(f,B)
and D(f,B) are distinct on the first coordinate only) .
In Section 2. we relate Deg(f,<52) to the obstruction theory . In
Section 3. we describe the group structure on CS(V 0 R) ,S0V)L and
define a homomorphism D : [S(V + R) ,S(V;]„ »A (V) . Section 4.
Is devoted to the proof of the main theorem of this paper . This
theorem says that homomorphism D Is an isomorphism .
0. Preliminaries. We begin by recalling some terminology and
facts concerning gropp action and obstruction theory .
Let G be a compact Lie group , and X a left G-space . We
shall denote by Gx the isotropy subgroup of x 6 X and by Gx the
orbit of x . For each subgroup H of G let X denote the fixed
point set of H i.e. X = [x^-X , H c Gx) . The set of points of X
for which G Is precisely H will be denoted by L. ,
Let V be an orthogonal representation of G and x«V . We
denote by Nx the normal space to Gx at x and Bjj ( x,r) = tyeWx , ly-xl<
<r) . If XCV Is an invariant subset such that G(XH) = X , then
the projection f : X —* X/G Is an N^H)/H - bundle and the homeo-
morphlsm
% ■ G xH U,. * (G/H) x U > G(UJ = I"1 (UJ
(where UA = Xn Bjjfx.r) , <*. = (.x,r) , r Is sufficiently small and
we Identify U4 with I(UA)) form a family of local trlvlallzations
of bundle T\ : X » X/G see £1] II 5.2,11 5.8 .
Suppose that the action of G on X Is free and Y Is a left G-
space . Consider the associated G-bundle p : Y XQ X * X/G . If
y . g*U » ff"*(U) Is a local trlvlallzatlon of the bundle
then (J7 : Y*U ■=■ (Y*GG) x U ♦ Yx (G*U) » YxGif1(U) **p~1 ( V)
^P (?,u) = [y.fCe.u)]

16
la a local trlvlallzatlon of p . Moreover , there la a one-to-one
correapondence between G-mapa f : X —» Y and croaa-aectlona sf
of p given by af(TT<x}) = [f (x) ,x] aee [1] II 2.4 , II 2.6
Now we aaaume that G Is a compact connected Lie group , G acta
freely on X and X/G la trlangulable $ X/G =|KI , K - trlangula-
tlon of X/G . Let Y be an n-almple G-apace and let L be a aub-
cemplex of K . Conalder a partial crosa-aectlon s s Knu L » Y xQ X.
Since G Is connected , the bundle of coefficients aaaociated
with the bundle p la trivial . So the obstruction z(a) to
extending a on Kn+1u L Ilea In Cn+1 ( K,L, %n (Y)) . Prom the above
facta It followa :
0.1 Lemma. Let f : ft"1 (Knul) » Y be a G-map and let
a„ : K^L >Y x„ X be the partial croaa-aectlon of p correapondlng
to f . Then z (af) «r) = [f *ye/ ] * Xn(Y) where la an n+1 - alm-
plex contained In U , y : G*u »Tf (.U) Is a local trlvlallzatlon
of ¥ and 4>e(x) = y(e,x) .
1. Let V be a real orthogonal repreaentatlon of G = S ,
0(VJ = [H fS1 i 3 x e 7 Gx = H} and Si c V © R an open bounded
Invariant aubaet . We denote by CASi.^8.) the apace of S -equlvarlant
mapa f : (M,ZSl) > (V.VHOl) with the atandard metric |f1-f2l =
= aup If., (x) -f2(x}| . We aay that fQ,f1 € CG(<51,3<&) are G-homotopic
If there exlsta a G-homotopy h:[<$i*[0,1] , 3&*ro,1]) -—► l?,?HO^)
(I.e. h(gx,t) = g-h(x.t) ) auch that hf,0) = f0 h^*,1) = f.,
For maps f e Cgdft.oA) we can define the S -degree Deg(f,<5!) fe
Z0 ©( ® , Z) . We denote by deg„(f,SL)eZ the H-coordlnate of
Deg(f,<5l) and by degg4(f,<») e Z~ Its first coordinate .
1.1. Theorem. Let Si, SlQ, Si., &2 c v ® R be open boimded
Invariant subsets and f (■ CG{£lf7><Sl) . Then the following properties
hold:

17
a) If degH(f,5l) ,1 0 , then f"1(0)n<# t <P .
b) If f-1 (0) c SLQc & , then DegCf ,<£) = Deg(f ,&e) .
c) If f"1 (0) c Si^u Sl^cSi and £., n 5¾ = <f> , then
Deg(f.Jl) = DegCf,^) + DegCf.&j) .
d) If h ! Ca*to,1] , 3£*[0,f|) —► (V,V*.{0^) Is a G-homo-
topy then Deg(h0,<ft) = Deg(h1(<£) .
e) Suppose W Is another representation of G=S and let
U be an open bounded Invariant subset of W such that
Oeu . Define P : U*£ » WOT by F(x,y) = (x,f(y)) .
Then Deg(F,Ux&) = Deg(f,<Si) .
f) let ItO(V)u(S1i and fH : (&H,3&H)—» (/,VHx [0\)
denotes the restriction of f . Then deg_(ffcft) =
= degH(fH,,aH) .
g) If <&jj = 0 , then degH(f,JZ) = 0 .
The properties a-e have been proved In ([3] Theorem 1.2 ) .
The properties f,g follow Immediately from ( [3] Definition 3.6 ,
3.7 ) .
let K e o£'ofTr)H . Assume now that SU. = c5i . We recall the
definition of deg„(f,<&) .
Suppose a e V © R and Ga t S . let T be a tangent vector
to M = Ga at a and Na = [x «7 © R , <x.v> = 0} the normal space
to Ga at a . Note that aeHa and V © R = Na ® spanlv} . let
A ; Na —» 7 be a linear Isomorphism . Define AA : V ffi R —» V ffi R
by N, ffi spanW a (x,*v) » (A (x), A)*V ffi R and sgn A -
■ sgn(det AA) . Recall the notation s BjjCa.r) = (xeHa ♦ Ix-al < r ^
B(r) a inVJ , Ixl < r }
1.2. Definition. let a eV ® R and Ga ^ S1 . We say that a
continuous map y : B(2) * V ® R Is a slice map at a If s
a) (¢(0) = a

18
b) there exlsta r>0 such that Bjj(a,r) Is a slice and
<*(B(2)) c BN(a,r)
c) there exists a linear Isomorphism A : N > 7 such
that <£(x) = a + A~ (x) and sgn A = +1
1.3 Definition. We say that an open Invariant subset £lBc SI
is elementary If there exlsta a finite family eft.,,...., SlT
of open Invariant subsets of SI such that :
a) SlQ c Si1 u ... u 5ir
b) ^ n &J = $ for 1 / j
c) for each 1 f U i4 r , there Is a slice map c^jBU) »
—> Jl such that $1^ c g. ^ ( B (1))
1.4.Definition. We say that f e c„(ca,a<a) Is an elementary map
if there exists an elementary subset <2 c <Jl such that
f"1 (0) c £q .
In ([3] Proposition 2.11 ) the following lemma has been
proved :
1.5 lemma. The set of all elementary maps Is an open and
dense subset of C„{<S,,qSI) .
Assume now that f «C-(ifl ,S<8) Is an elementary map and {£l{\
l<p;} satisfy the conditions of Definition 1.3 . let U^e^ (&]_)
-1
and Pj_ = f ° (jj_ : Uj^ * V . Clearly P., (0) Is a compact subset
of Uj^ , thus the Brouwer degree degCF^Uj) Is well defined .
Define
l.fi.Pgflnltlon. degK(f,<£) = ZZ degCFj^UjJ
let now f e C„ (<£ ,961) be any equlvarlant map and t= mln|f (x)l
By lemma 1.5 , -here exists an elementary map tee CQ (<5l,3<2) such
that lf-fel< £ • Define

19
1.7 Definition. degK(f,<&) = degK(fe,<5l)
Remark, We would like to point out that In Definitions 1.2 -
- 1.7 and In lemma 1.5 the set SI satisfied the condition <SL = <&
For the general case see ([3] Section 3.) .
let B=^x£V®R, Ixl < 1¾ . Now we define a new Invariant
for f £CG(B,aB) . It will be needed In Section 3,4 .
1.8.Definition. let 7 be a real representation of G = S ,
Define j
A 17)=.
{0\ © ( 9 Z )
HCO W
Z9( ® Z)
HeO(V7
z7e ( $ z)
c HeO(V)
If dim V° » 1 or 2
If dim Vs = 3
If dim t1 >, 4
For given fec„(B,9B) , by the same letter f we denote the
Induced map f/lff : S(V © R) * S (.V) , and the same for the
restriction map fGec(BG,9BG) . let [fG] denote the homotopy class
of the map fG : S(V ® R)G
S(VT
1.9 Definition. Let f£CG(B,0B) and H « 0C7) u [S1\ .
Define
degH(f,B) If HM1
dH(f,B) = -
0
TfGJ
If H = S1 and dim TG= 1 or 2
If H = S1 and dim VG>,3
and
D(f,B) = (dH(f,B)} 6 AtV)

20
2. Connection of deg„("f,&) with the obstruction theory .
Throughout this section we shall make the following
assumptions.
Let 7 be an orthogonal representation of S , dim 7 e n+1
2.1
SI c 7 ® R an open bounded Invariant connected subset auch
that cS is a smooth S -manifold with boundary , and <&H = SI
where (H) Is the main orbit type on 7 .
Under the above assumptions the orbit space <£/S Is a smooth
manifold , hence 62/S Is trlangulable f let K denote a trlengu-
latlon of &/S such that each simplex S«K Is contained In a chart
(U,y) of the S1-bundle &—»&/S1 (see Section 0.) . Note
that dim K = n+1 . Let S be any n+1 - simplex contained In a
chart (U,y) , U = I(Bjy(a,r)/i SI) and
cf i B(2) ^BjjCa.r) a slice map . We define yet U —» Bjj(a,r)
y9U) = yCe.x) and X = tp~1 (ye(«D) = <p "1 (1T"1C€r)) c B(1) .
Since X Is n+1-simplex there exists an orientation-preserving
homeomorphlsm hiB(1) * t . Let 6+ be the simplex oriented by
the homeomorphlsm y~ « t^s ° h .
Suppose feC„(S2f2&) Is an equlvarlant map such that
f"1(0)n TT"1(lKn/) =0 (Kn denotes n-skeleton) . Prom
definition 1.6. and Theorem 1.1c we have
degH(f,£) = JZ degH(f,TT\6)) = EIdeg(f=<f , r) .
«2Z deg (f <><p<>h,B) = iC deg(f«ye»H;g',o(<) <-h, B)
where 6 runs over all n+1 -slmplexes of K . Let rf,<fe/c«] 6
6 JCn(7st0j) denots the homotopy class of the map *°ye/as * &+ '
* 7^tO} . Identifying Xfi(7^(0¾) «• Z by
5Cn(V^t0i) =[3B,7^(0)] s= [(B,3B)iC7,V^C01)] ^2¾ %
we obtain

21
2.2 Lemma. Let Si c7 © R satisfy the assumptions 2.1 and
f ec0(<5l,3(a) be an equlvarlant map such that f"1 (05nff"1(Kn) s 0
Then
degH(f,j» = r[ffe/3e+] e xaiv«m
We shall denote : V0= V^tO^ , V0XG<R = E —>K - the bundle
associated with the bundle IT : 52 —* K and L = 3K = 2><S2/S
Any map f e 0^(52,352) Induces a partial cross-section sfs I —*B .
Since the fibre of the bundle p , 7Q Is n-1 - connected , there
exists an extension s£ : Kn u L * B of sf.
The cross-section s£ Induces a cocycle z(s£) Cn+1(K,L|J^ (V0))
2.3 Lemma. Let <& c 7 © R satisfy 2.1 and f e CGCtS2,3.51) .
Then degH(f,<ft) = ZIz(s£)(e+)
Proof. Let fA{7T~1 { Kn) » VQ be the equlvarlant map
corresponding to Bj and F : C&.aJl) » (V,VMO\) be an extension of
f* . Since f and 7 are equal on dSl , dee (f, £1) = degH(F,<&) .
Prom Theorem 1.1c we have degH(F,<R) = H deg^F.TTW)) . We
shall show that degH(r,tr~1(6+)) = z(a£)(e+) . Let (U,y) be a
chart of the bundle ¥ which contain 6 . Prom Lemma 2.2 and 0.1 it
follows that degH(rf7r~V<J+)) =l7°Ve/9J*l -^«^6/^1 = *(s?)(0 ,
Consider a homomorphlsm EI : Cn+1(K,L»I (V0)j —» &n(V0)
XH lz) = ZIzC^+) . Since IKI Is a compact connected and orlentable
manifold with boundary iLI , the homomorphlsm 1EL Induces an
Isomorphism C*: Hn+1(K,L»]i:nevon —» KnCV0) .
Let f e C„(StfdSL) and sf , s£ denote partial cross-sections
as above . We denote by cf the cohomology class of z(s£i j
cf e Hn+1(K,LjTlnlV0V) . Prom the obstruction theory It Is known
that cf Is Independent of the choice of extension sj . The
cohomology class cf Is called the first obstruction . It Is wellknown

22
that there exists an extension of sf on K = K if and only if
c_ = 0 . Prom the above considerations we deduce :
2.4 Corollary. 2^*cf = degH(f, SI)
2.5 Corollary. degH(f,<ft) = 0 if and only if cf = 0 .
2.6 Theorem. let SI c v ® R satisfy 2.1 and f e. aQ(SLf^Sl) .
If degH(f,£) = 0 then there exists f e CJSl,iSl) such that
fQ(x) = f(x) for xeaa and f0(a) c v^iOi .
3. The group structure on [S(W), S(VJ] r .
let W,V be orthogonal representations of a compact lie
group G ; S(W) , S(V) the unit spheres in W and 7 respectively j
xQ6 S(W)G , y0€S<V)G fixed points . let [S(Wi ,SlVf|G denote the
set of G-homotopy classes of G-maps f : S(W) > S(V) and let
[s(W), xQ j S(V), y 1G denote the set of G-homotopy classes (rel.
x ) of G-maps f .- S(W) —» S(V| with f(x) = y„ .
Suppose dim WG>/ 1 and dim V0^ 2 . let L=spanlx^ c w and
let W1 = J,-1 be the orthogonal complement of I in f . We may
identify S(W) with a non-reduced suspension ZlStW^ = [0,1] x SCW.,)/-
Under this identification x0=[0,x] , -xQ=l1 ,x] for xsSlW^ .
Let ^] . Cf2J e fscw) • s(v)l g * We can choose f1»f2 : sW
—* 3{T) in such a way that f1l-xQ) = yQ and f 2lx0) = yQ .
Define f3 : S(W) —» s(V)
3.1 f3[t,x]
f1t2t,x] for 0v<tH/2 , i'SII,)
_ f2[2t-1,x] for 1/2<t£1 , xeStS,)
Wow we define a group structure on ts(W),S(V)3 Q by
[f,] + [f2] = tf3l
The following lemma shows that the operation "+" is well defined

23
3.2 lemma. If f1.f2 s SlW) —» S(V) are G-homotoplc and
f1(xQ5 = f2(x0) , then they are G-homotopic ( rel. x0) .
The proof of this lemma ia given In [4] . The standard
computations show that the operation "+" yields a group structure
on CS(W) , SIV)] G . The G-homotopy class of the constant map
f = yo is the neutral element .
Now consider the case dim WC> 2 , dim Vs = 1 . Let
f : (SW,xQ) —» (SlV),y0) be a G-map . Observe that f(S(W)G) =
= y . Therefore we can define In the same way a group structure
on [s(W),x0 j S(V),yo]G .
The following lemma will be needed in the next section .
3.3 lemma. If dim WG> 2 and dim VG = 1, then there exists
a bisection
y : Z2x[SlW), xQ } SIV), y0]G > [SIW>, siv)]G
Proof. let L = span(y0^ , V-, = Ix and y;SlV) *■ S(V)
be given by yQ,y) = (-¾.y) , where U,y) elW^ V . Define
V(0"[f]) = [fl , V dx[f]) = Of] .
In the remainder of this section we assume that W = V © R
and G = S . let B denote the unit ball in V®R . Consider an
S1rmap f : S(V © R) »• StV) . Let f*j B * V denote an
extension of f . For fA the degree D(f\B)€A(V) haa been defined in
Definition 1.9 . It is easily seen that D(f\B) is independend of
the choice of the extension fA . Therefore we can define .
3.4 Definition. D : [s (V © R),SWlG ► A(V)
D[fJ = D(fA,B)
We shall denote t d„ff] = d (f^,B) .

24
3.5 Theorem, i) If dim Vu> 2 then
D : [SIV9R) ,SCV)] Q »AIV) ia a group homomorphism .
ii) If dim VG = 1 then
D : [s(VgR),x0 » SlVJ.y^g —* A UH is a group homomorphism .
Proof, i) Let [f.,] , [fg] £ [S(V+R) ,SiV)] G . We have to
prove that dH(ff1l + ff 2] ) = dR[f1] + dHTf2] for H £ OlV) u ls1i .
It ia evident for H = S1 (see Definition 1.9» 3.4) . Assume now
that H£0(V) . We identify B with [0,1] x Bl^l / ~ where
W1 = span{x \L f B(W.) - the unit ball in W1 . let f^f^f,
be as in 3.1 . We denote by f?.f2 : B —* v the S1-extensions
Define f*tB —* V by
for 0 i t ¢1/2 , x £ BIW.,)
for 1/2* t £ 1 , x 6BIW.,)
of f-j.f^ » respectively
f^ft.x] =,
f^f2t.il
fgfet-l.xl
Consider the sets :
ft-! = (tt.xl «B t 0 <t <1/2 , xtintBlf^)
Q,2 = (.[t,x] £B , 1/2<t <1 , xeintBtW.,)}
Prom 1.1c, 1.9 , 3.4 we have dR(tf1] +[t21) = ^^3! = d^f^.B)
= deg-tf^.a^) + deg (fi,S2) . let us define two maps
f} , f 2 : B —* V
f^f2t,x] for OUn/2 , iSB^)
y0 for 1/2 < t 4 1 , x eBlW.,)
fl ft.x]
f2tt,x] ..
y0 for 0itl1/2 ,i« B(W.,)
f 2 [2t-1 ,x] for 1/2€ti-\ , x e Biff.,)
Observe that f£ , f^ : (B,3B) ♦ (V,V^[0^) are S'-homotopic ,
the maps f!j , f? are equal on &., and f2 , f? are equal
,1
on
&2 . Therefore from 1.1 b,d it follows that deg^f^.S.,) +
+ degH(f^,^) = degH(f^,&1) + degH(f>jt«2) = degH(f^,B) + degH(f2,B)
= degH(f^,B) + degH(f2»B) - dR[f1] + dH[f2J . This proves i) .
The same proof works in the case iij .

25
4. A classification of S1-map3 S(VgR) » S(V) .
In thla section V denotes an orthogonal representation of the
1 c
group G = S . let *0,yo be fixed points of SlV®R)u f
5(.7) respectively . We now formulate our main result .
4.1 Theorem. 1) If dim VG >, 2 then
D s [S(V©R),S(V)] G —»A(V) is a group Isomorphism .
11) If dim VG = 1 then
D j [s(Vs>R),x0 ; SlV),y01G —» A(V) Is a group Isomorphism .
The proof of Theorem 4.1 Is baaed on the following two lemmaa
4.2 lemma. let f s (B,3B) —* (V.V^tOl) be an S1-map . If
D(f,B) = 0 then there exists an S1-map fQ : (B,8B) » (T,V^{0))
such that f(x) = fQ(x} for x e3B and fQ(B) c vMO) .
Proof. Choose an ordering H1,H2, ... fHk for the set
{s1l u o(V) such that If H,cH, then j 4 1 . let B,= U BHj .
1 u -1 Hi-
He will define S1-maps f1:(B,3B) » (V,V*{p}) f 1=1, ... ,k
such that f1(B1) c v^lO\ and fi(x) = f (x) for x e 3B .
Suppose first that 1=.1 . Since dgl(f,B) = [f/^G] = 0 ,
there exists an extension F:BG —> VG>lO} of f/aBC . Define
f.,: SBuB1 ► V*lOj by f 1 (x) = fix) for xOB and f1 lx) ■
= Fix) for x e. bg=B., . The map f.j can be extended on B by
Tletze-Gleason Theorem .
Assume the map fj^ Is defined . We will define fl+1 . let
H = H, 1 . Since fjCBj) c y~>(0\ f we can choose an open Invariant
connected subset & of (V®R)H such that f^ (0)Hc<R c Si c B^j
and £ Is a smooth S1-manifold with boundary . Denote by
g x (S,3&) —* IVH,VH-I0i) the restriction of fj_ . Theorem 1.1 b,f

26
dHCf]
Implies that degH(g,<ft) = degH(ff,BH) = degH(fltB) = djCfj.B) = 0 .
Prom theorem 2.6 It follows that there exists an S -map
gQt (SI,QSl) >(VH,VH^10}) such that gQ(x| = g(x) for x e 3&
and gQ(Sl) c VHnoi . We define f 1+1 t 3BuBi+1 —» V^ \0) by fi+1(x} =
= f^xS for x « 3B v (Bi+1v&) and fi+1 Ix) = gQ(x) for x«# .
The map fi+1 extends on B by Tietze-Gleason Theorem .
Observe that \ = B and fk(B)c y^{0} . We put tQ = fk
arid the proof is completed .
4.3.Lemma. 1) Let K (■ 0 (V) . Then there exists a G-map
f : SCVaR) > S(V) such that
1 f or H = K
0 for H t K
li) If dim V > 3 , then there exists a G-map
f i S(V®R) » SIV) such that
fl for H = S1
dH f = \ 1
H 10 for H ^ S1
Remark. If dim VG = 1 or 2 then dgl[fJ = 0 for any
G-map f (see Definition 1.9) .
Proof. 1) Choose xQ « int Bjj , r > 0 and y0*SlV)G such
that ixol+r<1 , lx0t+(3/2)-r > 1 and D(xQ,2r) c lVaR)R ia a
slice in the space lVs>R)K at the point xQ ( D(xo,2rl denotes a
disc in the space (NX)K ) . Let x1 = xQ+ (3/2)r^xQ/ tx0\) t
U, = G-D(x„,2r) . We shall define a G-map Y t V^R —» YK such
that Fix) = yQ for x e VK©R <• U^ , F"110) =Gx(>uGx1 and
„ f 1 for H = K
d„(?/RK ,BK) = \
H /B 1 0 for H * K
It is easy to construct a map g t D(xQ,2r) » V^ such that
g(3D(x0,2r)) = y0 , g"1(0) =ix0,x1) and deg(g,D(*0,r)) = 1

27
(Brouwer degree) . Define F(z-x) = yQ for z.x ^U2r and F(z«x) =
= z«g(x) for x€D(xQ,2r) , ztS . We now extend F on V®R by
the formula F(x,y) = F(x) +y where (x,y) € (V<SR)%(VK)X = V®R
and we define a G-map f s S(V©R) » SIV) by fix) = r(x)/|F(x)| .
Prom Theorem 1.1 and Definition 1.9 , 3.4 , it follows immediately
that
1 if H = K
dHM =
1 0 if H ^ K
ii) Choose a map fG : S(V®R)G »S(V)G such that ffG] =
= 1 e [SlVg>R)G,S(V)G] . let F : BG —» VG denote an extension of
fG . We extend f on B by formula F(x,y) = F(x)+y where
(x,y) e (VffiB)G © (V0)1 . The G-map f » S(VaiR) ► SIV) is given
by f(x) = F(x)/lfCx)| . This ends the proof .
Proof of Theorem 4.1 . (Mono) . Suppose that D[f] * 0 .
Prom Lemma 4.2 , it follows that there exists a G-extension
F:CB,3B) »(V,V^i01i) of f such that f(B) c V^O) . We define
a G-homotopy H : S(V©R)* [0,1] >SIV) by H(x,t) = F(tx)/IF(tx)|
It is easy to check that the homotopy H joins the map f and the
constant map HC«,0) = F(0)/IF(0)| . Therefore we have \f\ = 0 .
(Epi) . It follows immediately from lemma 4.3
Corollary 4.4
S V+R fS V Q
Z, 3> ( a> Z) Jf dim Vs = 1
d He 0(7}
HeO(V)Z if dim V° = 2
Z ¢( e Z) if dim ?G = 3
H6 0(V)
Zo ®( ® Z) if dim V° > 4
c HeOCV)
In this paper we have studied Deg(f,<ft) of an S -maps .
nevertheless we are able to define an analogous invariant of f -
equivariant maps , Tn - the torus . Moreover, the statement of

28
Theorem 4.1 extends on this case in following manner
[S(V©R),SIV)3 Tn = [s<Va>R)T ,SIV)T] ® (® Z)
H
where the last sum is taken over all isotropy groups H (on S(V))
with one dimensional orbits .
Institute of Mathematics
University of Gdansk
Wita Stwosza 57
80-952 Gdansk
References
fl] G.E. Bredon . Introduction to Compact Transformation Group ,
Academic Press , New York and London 1972 .
[2] T. torn Dieck , Transformation Groups and Representation Theory,
tect. Notes in Math. 766 , Springer , Heidelberg-New York,1979.
[3l C.Dylawerski.K.Geba,J.Jodel.W.Marzantowicz , An S -Bquivarlant
Degree An The Puller Index, Preprint No 64 , University of
Gdanak . 1987 .
[4^ R.I. Rubinsztein , On the equivariant homotopy of spheres ,
Disaertationes Mathematicae , No 134 , Warszawa 1976 .
[5] G.B. Segal , Equivariant stable homotopy theory , Actes f
Congres Inten. Math. Nice 1970 , Tome 2 , p. 59-63 .
[6] N.Steenrod , The topology of fibre bundles , Princeton
University Press , 1951 .

ON CERTAIN SIEGEL NODULAR VARIETIES
OF GENUS TWO AND LEVELS ABOVE TWO
Ronnie Lee and Steven H. Weintraub
*
In our previous work, we have studied spaces M, which are
moduli spaces of stable curves (i.e. Riemann surfaces) of genus 2
with level A structure, for two particular subgroups A of
PSp4(Z).
*
In general, M. is a three-dimensional complex projective
variety. It is usually non-singular, though for some choices of A
it has finite quotient singularities. (In Satake's language, it
is then a V-manif old). It is the Igusa compactif ication of the
variety M = S2/A, the quotient of S2, the Slegel space of
* 0
degree 2, under the action of A. Further, M => M => M., where
0 *
M. (a Zarlski open set in M ) is the moduli space of non-
singular curves of genus 2 with level A structure.
We follow our previous notation and write the complement
* 0
M. - M. = 3. u 0 , where 3. and 0. are unions of components
(each a complex surface) and are themselves moduli spaces for the
two kinds of singular but stable curves of genus 2 (see [LWi],
section 8.4).
In our papers [LWi], [LW2] we considered the case a = r(2),
the principal congruence subgroup of level 2. (In this case a
level A structure is more commonly known as a level 2 structure.)
* Partially supported by the National Science Foundation.
** Partially supported by the National Science Foundation and the
Sonderforschungsbereich fur Geometrie und Analysis (SFB 170).

30
In [LW3] we considered the case A = V, where T is a certain
subgroup of T(2). We define V precisely in (1.1) below. Here
we just observe that T(2) =• T => T(4), and [T(2): T] = 26,
[T(2): T(4)] =- 29. The quotient T(2)/r(4) is an elementary
abelian 2-group, and hence so is T(2)/r.
*
We wish to investigate the topology of these spaces M for
various A.
In [LW]J, [LW2] we proved the following theorem in case
A = T(2) (see also [G]).
*
Theorem 0.1. a) H (M ) = 0 for i odd.
*
b) The map H.(3, u 0 ) —► H (M.) is an epimorphism
for i < 6.
*
c) H,(M ) has a basis consisting of algebraic
cycles (so, in particular, in the Hodge
decomposition of H (MA), HP'q = 0 unless p = q).
*
d) The integral homology of M. is torsion-free.
In [LW3] we proved the same theorem in case A = T.
In addition to this qualitative ("soft") information, in the
above-mentioned papers we have the following quantitative ("hard")
information.
Theorem 0.2. a) In case A = T(2), rank H,(M ) = 16.
*
b) In case A = T, rank H4(MA) = 79.
(Of course, by Poincare duality, rank H^ =
rank Hj)•
Our main result in this paper is the determination of the
homology of M. (at least up to 2-torsion) for all T c a c T(2).
The line of argument is given to us by the following theorem:

31
Theorem 0.3. a) - d) The conclusions of theorem 0.1 a) - d) are
valid for any T c Ac T(2), except that homology must be taken
with coefficients in Z[y], rather than in Z.
e) For any such A there is an exact sequence
(with (co)homology having coefficients in 22[y])
0 - H^Mj) - H4(3A U 0A) — H4(M*) — 0.
Proof: Recall the following general fact: If a finite group G
acts on a space M, and if IF is any field of characteristic 0
or prime to the order of G, then (see [B, theorem III.2.4]):
H*(M/G: IF) = H*(M: TF)G
where ( ) denotes the elements fixed under the action of G.
* *
Here we have the 2-group A/r acting on M„ with quotient M. .
Let us for the moment take coefficients in Q. Then a) holds
immediately, as does b), since the quotient (3„ u 0„)/A is 3. u 0 .
Furthermore, c) holds as well, as the required basis of algebraic
* *
cycles for H,(M.) will be images in M. of those elements of the
*
basis in M„ whose fundamental classes are fixed under the action
of A/r. As the reader will see, if in the arguments In this paper
we replace $ by any field IF of characteristic not equal to two,
we obtain the same dimensions for all spaces, so in particular
dim H^(MA: IF) = dim H^(MA: Q)
*
and so the homology of M. has no odd torsion and d) holds.
Of course, it is the proof of part e) that requires work,
ider the exact sequence of the pair
coefficients in Q (or IF, char W * 2)
*
Consider the exact sequence of the pair (M , 3 u 0 ), again with
H5(3rU 0r) —► H5(Mr, 3r U 0r) —► H4(3r U 0r) —► H4(Mr)

32
The first of these groups is zero, as 3„ u 0 is a union of com-
1 *
plex surfaces, and the second is isomorphic to H (M - (3 u 0))
10 iii
= H (M„) by Alexander duality. Thus we have
r
H^mJ) -* H4(3r U 0r) -* H4(M*)
In fact, as we showed In [LW3], this last map is an epimorphism
(and Indeed, this is part of 0.1 b)). This follows from the
computation of dim H4(Mr) = 79, dim H (M°) = 27, and dim H,(3j,u 0p)
= 106 there. We shall see below how to make these last two
computations (cf. 1.16 and 3.1, and 1.12, 2.1 and 2.2).
Now let A/T act on this short exact sequence. We again
obtain a short exact sequence
0 —* H (Mr) —>■ H4(3f u 0r) —* H4(Mr)
which is nothing other than the sequence
0 - H^mJ) — H4(3A u GA) — H4(M*) — 0
as desired.
* 10
Corollary 0.4. dim H (MA) = dim H4(3A u 0 ) - dim H (MA), where
(co)homology is taken in an arbitrary field IF of characteristic
not equal to two (and these numbers are independent of the choice
*
of IF.) (Note that by Poincare duality this is also dim H„(MA)
*
as M. is an IF-homology manifold.)
*
This corollary tells us how to compute H,(M.) - compute the
two terms on the right-hand side and subtract. In principle, this
is the approach we follow.
In practice, as there are very many subgroups A, we choose a
somewhat different line of attack. Namely, what we actually deter-
*
mine is the action of G = T(2)/r on the space R. = H4(Mr).

33
Knowing R as a representation space of G of course tells us
dim H4(Mr) = dim H,(MA> for any T c Ac r(2). (It turns out
as well that the final result is much easier to state in terms of
R.) How do we compute R as a representation of G? It turns out
*
that we may do so by computing dim H,(M.) for a relatively small
number of subgroups A (of a kind we call "un-twisted").
Thus our paper is arranged as follows: In section 1 we
establish our notation, recall some basic results of [LW3], and
establish precisely what we what we need to compute. In section 2
and 3 we compute the two terms on the right-hand side of 0.4 for
certain A. In section 4 we assemble this information to obtain
our main result, theorem 4.2, which gives the action of G on R.
*
As a specific application we then give dim H,(M ) for the extreme
cases T of index 2 in A, and A of index 2 In I\2).
All (co)homology henceforth is to be understood as having
coefficients in Q. As we have remarked, this gives us information
complete except for 2-torsion. We discuss this question in 4.7.
*
In many cases we can show that the (co)homology of M. is torsion-
free .
As this manuscript is being photo-offset from typescript, the
reader will be able to appreciate the marvelous typing job done on
it by Nell Castleberry, to whom the author extends his deepest
thanks.

34
1. The situation at level T.
We begin by establishing notation and recalling some of the
results of [LW3].
Definition 1.1. Let T(n) = {M e PSp (Z) | M = I mod n}.
0 a b
Let r = {M e PSp,(Z) M5I+2 0 mod 4}.
^4 ' c -a
(Observe that T(4) c Tc r(2), and [T: T(4)] = 23, [T(2): T] « 26.)
Definition 1.2. Let S2 = {M e M (C) | M - CM and Im(M)
is positive definite }.
Let 0° c S2 = {(q °)} | Im(z) > 0, Im(w) > 0}
together with the union of its translates under the action of
PSp^(Z). We call 0 the Humbert surface in S2 and set S„ =
S2 - 0°.
Definition 1.3. For any subgroup A of PSp^(Z), set
MA = S2/A, mJJ = S°/A, 0° = 0°/A.
For any Ac F(2), A acts freely on S„. Also, M is the
moduli space of non-singular Riemann surfaces R of genus 2 with a
level A structure (i.e. a choice of symplectic basis for H^(R: Z)
modulo the action of A.)
*
Definition 1.4. Let M be the Igusa compactification of M..
_ A 0 * A
Set 3. = M. - M., and let 0. be the closure of 0A in H^.
The space M. is the moduli space of stable (in the sense of
Mumford [M]) Riemann surfaces of genus 2 with level A structure.
*
We call 0. the Humbert surface in M., and, by abuse of
language, 3. the boundary of M. (even though, as a projective

35
variety, this space has no boundary in the topological sense). Of
course, both the boundary and the Humbert surface are the union of
many irreducible components .
*
We now recall our results on the structure of M„. We begin
0
with m".
First we exhibit a 4-fold cover f: IP ■*■ W, branched over 3
points, each of whose inverse images has cardinality two.
Lemma 1.5. Let X = dm z > 0}/r (4) and XQ = {im z > 0}/T (2),
where r (n) is the principal congruence subgroup of level n of
PSL2(Z). Then X is a 4-fold cover of XQ with group T (2)/T (4)
= (Z/2) + (Z/2). Also, we may identify X with IP1 - {±i,0,<=°,±l}
and Xq with W" - {0,1,«}, and the covering projection f: X ■*■ Xfl
with the function
z = f(w) = ((w2 + l)/(w2-l))2.
Furthermore, f extends to a branched cover of X = IP1 to X„ =
IP1 with f(±i) = 0, f(0) = f(<=°) = 1, and f(±l) = «.
Proof. [LW3], 2.1 and 2.2.
The group of the cover (Z/2) + (Z/2) is generated by the
loops around 0 and 1 in the base, each of which has order 2. We
denote them by p and 2 respectively. Of course, the loop
around °° is then r = pq, the other non-trivial element of this
group.
Notation 1.6. Let V denote the Klein 4-group (isomorphic to
(Z/2) + (Z/2)) with elements {l,p,q,r} which appears in 1.5.
Theorem 1.7. Let Zq = {(x.,x ,x„) e Xn | x. not all distinct}
and Z = {(Xl,x2,x3) e X | (f(x1),f(x2),f(x3)) e ZQ}.
Then M°(2) = XQ * XQ * XQ - ZQ, and M° = X * X * X - Z,
where the map M ■*■ M is covering projection f x f x f.

36
Thus Mj, is a 26-fold cover of Mp/-?} wlth 8rouP G =
Vj x v2 x V_, with each V^ naturally isomorphic to V.
Proof. This is theorem 3.9 of [LW3].
We let V. have the non-trivial elements Pi»qi»ri = Pi^i'
where the isomorphism with V is the obvious one suggested by the
notation. We shall frequently identify V^ with V via the
isomorphism. Of course G is generated by {p ,q }, i = 1,2,3.
In order to do our computations, we need specific matrices
representing the generators (and hence elements) of G. These we
also obtained in the proof of this theorem in [LW3], and we quote
the result here. (In [LW3] we denoted p,, say, by 0 , as it is
given by the effect of a "Dehn twist" around a lift of a loop
representing p., but for simplicity of notation we drop the 0 here.)
Lemma 1.8. p, = I + 2
/0 1 1 01
0 0 0 0
0 0 0 0
\o 1 1 0,
qL - I + 2
p2 = T + 2 0 0 0 0 q2 = 1 + 2
0 0 11
P3=I + 2|oSioi ^3= I + 2
0 0 0 0
We now make a useful observation, which will reduce the number
of cases which we have to consider. Recall that the automorphism
group of V is £ , which acts by permuting the non-trivial
elements .
Lemma 1.9. Let E x E operate on G = V. x V x v, as follows:
An element of G is a word in p,,...,r-j. The first factor
operates on such a word by permuting the symbols p, q, and r (as
above). The second factor acts by permuting the subscripts 1,
2, and 3.

37
If A. and A are two subgroups of G equivalent under
* *
this action, then M and M. are equivalent as complex
algebraic varieties.
Proof. The action of £_ on V may be realized by the unique
automorphism of Xn = IF permuting 0, 1, and °° as specified,
giving an action of the first factor on X0 * XQ x Xn, and the
second factor acts on this product by permuting the factors. These
0 *
M , which extend to M„, as their
effect on 3„ u ° is to permute components. This automorphism of
* l l * *
M„ then descends to an equivalence between M, and M, .
r Al a2
We shall denote this £_ * £- subgroup of the automorphism
group of G by A(G).
* 0
Now we consider M, - M, = 3, u 0,. Each of 3. and 0. is
A A A A A A
A
a union of irreducible components which are complex surfaces. 3
is a union of complex surfaces D,(£), where the indexing set is
{±1 | I a non-zero primitive vector in Z }/action of A (1.10)
where A acts on I by ordinary matrix multiplication, (l)X = IX.
(In case A = T(n), the principal congruence subgroup of level n,
this set is just the set of non-zero primitive vectors in (Z/n) ,
taken up to sign.)
Similarly, 0 is a union of complex surfaces H.(A) where
the indexing set is
{A = {6,61} | A = ± ^aA 61 = ± S.[aV2, with 6 and 61
mutually orthogonal anisotropic subspaces (1.11)
of Z4, & + Sl = Z4}/action of A .
We refer to such a A as a anisotropic pair. Again A acts by
matrix multiplication, and in case A = T(n) this is the set of
pairs of such subspaces in (Z/n) .
We call the components D(£) of 3 boundary components and
the components H(A) Humbert surfaces. The justification for all
these remarks can be found in [LW2], for A = T(2), and in [LW3],

38
for A = T, but the set-up holds generally. The set in (1.5)
indexes one type of vertex in the Tits building for A. We do not
need to consider the Tits building here, (or the "Tits building
with scaffolding" of [LW3]) for it contains the further information
on how the various components of 3 (or 3. u 0 ) intersect, and
that is superfluous here.
Lema 1.12. ha(3a u Qa) is the free abelian group on the
generators {[D(£)]} of (1.5) and {[H(A)]} of (1.6), where [ ]
denotes the fundamental homology class.
Proof. As we are dealing with complex surfaces, which have a
canonical orientation, [ ] is well defined. Since the
intersection of two irreducible components is a complex subvariety (perhaps
singular or empty) it has real codimension at least two so the
lemma is immediate from the Mayer-Vietoris sequence.
Lema 1.13. As a representation space of G, H/.(3r u 9p) *s
isomorphic to +Z[£] + +Z[A], where Y e G acts on the latter
I A
by I —*■ ly, A —* Ay.
Proof. The action of y takes D(X.) to D( iy) and H(A) to
H(Ay), so it induces a map Y* on H,(3r u 0 ) by Y*([D(£)]) =
±[D(&y)], Y*([H(A)]) =±[H(Ay)]. Since the action of Y is a
complex automorphism, it preserves the canonical orientations, so
both signs are +, and the lemma follows.
We rephrase theorem 1.7 in a way that will be slightly more
useful.
Definition 1.14. Let Z = {(x.,x2,x ) e XQ x XQ x XQ|(x1,x?,x3) e Z(
or x. = 0, 1, or °° for some i}.
Let Z = {(x1,x2,x3) e X x x x xj (f (x^.f (x2>,f (x3>) e ZQ}.
Theorem 1.15. M°=XxXxx-Z.
Proof. This is immediate - compared with the description of M„
in 1.7, we are adding and then subtracting all points (xpX2»X3)
with (f(x1),f(x2),f(x1)) not in X0«

39
This latter description has the advantage that X x x x x is
a compact manifold (in fact Tr x p1 x p1) and Z is a union
(not disjoint) of irreducible components, each of which is compact.
Typical components are ((x,y,0)} or {(x,x,y)}, so each
component of Z is T!r x TSr, and different components intersect in a
W- or Jp, or not at all.
The group G = T(2)/r has an obvious action (by covering
translations on each factor X) extending its previously defined
action on M„. For any f(2) => A a V, we will let Z denote the
quotient of Z„ = Z under the action of the subgroup A/T of G.
Proposition 1.16. For any subgroup A of T(2), H (M ) is a
free abelian group of rank 3 less than the number of irreducible
components of Z. .
Proof. By Alexander duality, H (M ) = H,.(X x x x X,Z). We then
have the exact sequence of the pair:
H (X x X x X) —* H (X xxx X,Z) —* H (Z) —* H4(X x X x.X)
The first group above is obviously zero; the last map is obviously
onto a free abelian group of rank 3, and, by the same argument as
in 1.12, H,(Z) has rank equal to the number of components of Z,
so the proposition follows for A = r.
Now consider a group A with A/T non-trivial. Certainly
H is free abelian, so we need only compute its rank. Thus let us
take homology with coefficients in I}.
Then we have
0 -*• H5(X xxx X,Z)A/F -+ H4(Z)A/F —► H4(X x X x X)A/F -+ 0
Now, by the argument of 0.3, we may identify the first two terms
with h (M:1) and H,(Z.) respectively. Furthermore, A/T acts
trivially on the homology of X x X x x (as each generator
p,,...,qo of G does) so the last term has rank three and the
proposition follows.

40
We single out a special class of subgroups of G.
Definition 1.17. A subgroup of G is called untwisted if it has a
set of generators which are a subset of {p ,p ,p_,q ,q ,q_,r ,r ,r_}.
We call r c A c r(2) untwisted if A/T c G is.
Such subgroups are naturally distinguished. Also, we will see
that in order to determine the action of G on the homology of
*
M„, it suffices to consider relatively few subgroups, all of which
are untwisted.

41
2. The action oa lines and anisotropic planes.
By 1.13, the action of A on the fundamental classes of the
*
components of the boundary (resp. the Humbert surface) of M„ is
given by the action of A/T on lines I (resp. anisotropic pairs
A). In this section we determine these actions for certain
(enough) subgroups A.
These are long computations, so rather than working them out
in full we indicate how to do them in one or two illustrative
cases.
The boundary components (resp. Humbert components) at level A
are in 1-1 correspondence with the A-equivalence classes of lines
(resp. anistropic pairs), so it is this number we need to compute.
Of course, for A = T or T(2) we already know the answer; it is
for the intermediate levels that work must be done.
Proposition 2.1. For the following untwisted groups A, the
number of equivalence classes of lines at level A is as stated:
[A: T] Generators of A/T Number of equivalence classes
Pi
Pp
Pp
Pp
Pp
Pp
Pp
Pp
-
p2
^1
q2
P2>
P2»
P2>
P2»
P3
^1
^3
p3' qp q2> ^3
54
40
32
33
29
28
25
23
15
Proof. To determine the action of A/T on lines at level T we
of course must know the latter. They are given by [LW3], theorem
3.6 (and there are 54 of them). Recall they arise as follows:
There are 15 lines at level 2, given by (aj,a2,b1,b2) where
a. and b. are defined mod 2 and not all are zero. Each line
at level 2 is covered by 8 lines at level 4.
1
2
4
4
4
8
8
8
64

42
For example, (1,0,0,0) is covered by
{(1,0,0,0), (1,0,0,2), (1,0,2,0), (1,0,2,2),
(1,2,0,0), (1,2,0,2), (1,2,2,0), (1,2,2,2)} mod 4,
and (0,1,0,0) is covered by
{(0,1,0,0), (0,1,0,2), (0,1,2,0), (0,1,2,2),
(2,1,0,0), (2,1,0,2), (2,1,2,0), (2,1,2,2)} mod 4,
and (1,1,0,0) is covered by
{(1,1,0,0), (1,1,0,2), (1,-1,0,0,), (1,-1,0,2),
(1,1,2,0), (1,1,2,2,), (1,-1,2,0), (1,-1,2,2)} mod 4.
(Note we have resolved the ambiguity in sign by choosing some
entry to be +1 mod 4.)
Now on lines over (1,0,0,0), T/F(4) acts trivially, so there
are 8 equivalence classes of such lines at level V. On lines over
(0,1,0,0), r/T(4) acts by interchanging the first and second,
third and fourth, fifth and sixth, and seventh and eighth, so there
are 4 equivalence classes of such lines at level V. On lines over
(1,1,0,0), r/T(4) acts transitively on the first four, and
transitively on the last four, so there are 2 equivalence classes of
such lines at level T.
Now let us consider the action of a typical element p, of
G. On the equivalence classes of lines over (1,0,0,0) it acts as
follows: Interchanging (1,0,0,0) and (1,2,2,0), (1,0,0,2) and
(1,2,2,2,), (1,0,2,0) and (1,2,0,0), (1,0,2,2,) and (1,2,0,2).
Thus p, has 4 orbits on these. (On the other hand, as the reader
may check, there are also 8 lines at level V covering the line
(0,0,1,0) at level 2 and p, acts trivially on these, so there
are 8 orbits of p, there.)
On the four equivalence classes of lines over (0,1,0,0) p^
acts trivially , giving 4 orbits. Also, pj acts by interchanging
the two equivalence classes of lines over (1,1,0,0), giving 1
orbit here. Then adding the number of orbits over each of the 15
lines at level 2 gives 40 orbits for A with A/T generated by
p, , giving the second line of the table.
From this point on the computation is routine.

43
Proposition 2.2. For the following untwisted groups A, the
number of equivalence classes of anisotropic pairs at level A is
as stated:
[A: T]
Generators of A/T
Number of equivalence classes
1
2
4
4
4
8
8
8
64
Proof.
Pi
Pp
Pi'
Pp
Pi'
PP
Pp
Pp
This
-
P2
^1
q2
P2>
P2»
P2»
P2»
P3
^1
^3
P3» qp q2» q3
is entirely analogous
52
36
26
28
25
22
20
18
10
to th
proposition (only slightly more complicated as there are more
choices of representatives). By theorem 3.7 of [LW33 we know all
the anisotropic pairs of level T. (Each of the 10 anistropic
pairs at level 2 is covered by 16 at level 4, and by either 4 or 16
at level f. There are a total of 52 of these at level f.) For
example, the pair A = {6,6 } at level 2, with
6 = (1,1,0,0)a(0,0,1,0), is covered by the four equivalence
classes of pairs at level V whose representatives we may take
to be 6 and its orthogonal complement 6 , with
6 = (1,1,0,0)a(0,0,1,0), (1,1,0,0)a(0,0,1,2),
(1,1,0,0)a(0,2,1,0), (1,1,0,0)a(0,2,1,2) mod 4.
Then p, acting on the first one of these planes sends it to
(1,-1,2,0)a(0,0,1,0). But adding twice the second vector to the
first, we see that this is the same plane as (1,-1,0,0)a(0,0,1,0).
Also, in the proof of 2.1, we observed that the line (1,-1,0,0)
is equivalent to (1,1,0,0), so we conclude that this plane is
equivalent to (1,1,0,0)a(0,0,1,0), i.e. pl acts trivially on
(1,1,0,0)a(0,0,1,0).
Otherwise the computation is straightforward.

44
3. The action on excised components.
We see from 1.16 that we must count the number of components
of Z, , i.e. the number of orbits of components of Z = Z„ under
the action of A/T.
Proposition 3.1. For each untwisted subgroup A of T(2), the
number of irreducible components of A is as follows:
[A: T]
Generators of A/T
Number of components
64
^1
Pp
Pp
Pp
Pp
Pp
Pp
Pp
p2
^1
q2
P2» P3
P2» ^1
p2, q3
p2» P3» qp
q2, q3
30
24
20
21
19
18
17
16
12
Proof. Again there are many cases and we shall merely indicate a
few.
f_1(0),
Recall f: X—* Xn is a branched covering with group
(Z/2) + (Z/2) generated by p and q, with each of
f-1(l), and f-1(°°) having cardinality two. The following
schematic represents this cover:
P*<uMMl
t=(u\(a*)
00

45
X
Let X (resp. X ) denote the quotient of X by the action of p
(resp. of q). Then X (resp. X ) is a 2-fold branched cover of
'.q, branched over 1 and « (resp. 0 and °°) . We continue to let
f denote the covering projection. Then f (0), f (1), f~ (°°)
have cardinality 2, 1, 1 in X (resp. cardinality 1, 2, 1 in X2).
An irreducible component of A. projects onto one of the
following types of components in A : (*,x,y), (x,*,y), (x,y,*),
(x,x,y), (x,y,x), or (y,x,x), where * = 0, 1, or °° and x
and y are arbitrary. Thus we have six kinds of components, and
we will gather the number of each kind into a 6-tuple, whose sum is
the number of components of A . For example, when A = T(2),
A. = An has the 6-tuple (3,3,3,1,1,1,) and so has 12 components.
a) The case A = V. The 6-tuple is (6,6,6,4,4,4,), as
follows: Here the covering space is X x X x X. F (*,x,y) has 6
components as f (0) u f (1) u f (o°) has cardinality 6, giving
the first entry, and the second and third are identical. The
fourth entry is the number of components of X , the inverse of
the diagonal in X * X. But this inverse consists of {(x ,x„) j x
and Xj differ by a covering translation}, and so has 4 components
(as the group of covering translations has 4 elements), and the
fifth and sixth entries are identical.
b) The case A/T generated by pi . Now the 6-tuple is
(4,6,6,2,2,4). Here the covering space is X x X x X. F-1(*,x,y)
-1 -1 -1 ^
has 4 components as f (0) U f u f (°°) has cardinality 4 in
X giving the first entry. The second and third are as in a).
The fourth and fifth entries are the number of components of the
inverse image of the diagonal in X * X. But this inverse image
is the quotient of X (as in a)) by the group generated by
pxid:X*x—*"Xxx, and this quotient has two components. The
sixth entry is as in a).
c) The case A/T generated by Pj and p2« The 6-tuple is
(4,4,6,2,2,2) and the covering space is X x X x X. In
particular note that the fourth entry is 2 by the same argument as
in a).

46
d) The case A/T generated by p, and q,. Now the 6-tuple
is (3,6,6,1,1,4). Here the covering space is JL x x x x, and
the argument is similar to b).
e) The case A/T generated by pi and q2 • The 6-tuple is
(4,4,6,1,2,2) and the covering space is X xx x x. The only
(subtle) difference between this and case c) is the following: The
inverse image of the diagonal of Xnx x~ in X x x has 4
components . Under the action of the group generated by p, and
p2 they are Identified to two components in X x x (i.e. this
group acts on the 4 components with p, and p2 each acting non-
trivially but giving the same identification), while here, under
the action of the group generated by pi and q2 they are
identified to one component in X x x (i.e. this group acts on the 4
components with p, and p2 each acting non-trivially but giving
different identifications).
The remaining cases are similar.

47
4. The representation of G on the homology of M
In this section we obtain our main result. We use the
*
calculations of section 2 and 3, which give dim H,(M.) for some
*
A, to decompose H,(M„) into a sum of irreducible representations
of G = r(2)/r.
First we assemble some information.
Proposition 4.1. For the following untwisted subgroups
Tc Ac r(2), dim H4(MA: Q) is as stated:
[A: T] Generators of A/T
Dimension
64
Pi
Pi > P2
Pp qi
Pp q2
Pp P2» P3
pp p2» qi
Pp P2» q3
pp P2» P3» qp
q2» q3
79
55
41
43
38
35
31
28
16
Proof. Immediate from 0.4, 2.1, 2.2, 1.16 and 3.1.
Now G has 64 irreducible representations, all 1-dimensional,
which are obtained by letting each of the six generators Pp p2»
p3, qp qj, q3 act by multiplication by ±1. We will denote an
irreducible representation of G by e = (e1,...,e6) where each
e. is + or - according as the corresponding generator acts by
+1 or -1.
Let R = H4(M„) regarded as a representation space of G.
Let R(e) be the subspace on which G acts by the representation
e. Our problem is to determine dim R(e) = the multiplicity of e
In R. The answer is this:
Theorea 4.2. The multiplicities of the irreducible representations
-"""""^————— ^
of G in its action on H (M ) are given by the following table:

48
el»e2'e3
e4,e5,e6
+++
++-
+-+
-++
+—
-+-
—+
+++
16
3
3
3
3
3
3
1
++-
3
3
0
0
0
0
0
0
+-+
3
0
3
0
0
0
0
0
-++
3
0
0
3
0
0
0
0
+—
3
0
0
0
3
1
1
0
-+-
3
0
0
0
1
3
1
0
—+
3
0
0
0
1
1
3
0
1
0
0
0
0
0
0
1
(Thus for example, the multiplicity of the representation where
Pl»P2'p3 (resP- qi»q2>cl3) act by (+1,-1,-1) (resp. (-1,+1,-1))
is the intersection of the column labelled +-- and the row
labelled - + - and is 1. (Note in this representation (rp^,^)
act by (-1,-1,+1).)
Reaark 4.3. The reader will observe that the multiplicity of each
non-trivial representation is one less than a power of two. Why
this should be so, or what it means, is a complete mystery to us.
Gathering the irreducible representation of G into A(G)-
equivalence classes, we may rephrase the theorem as follows. (Note
that when we compare representations of different V^'s, we are
using their identification to V.)
Theorem 4.4. As a representation space of G = V. * V x V„,
*
R = H/(M„) decomposes as follows:
Type of irreducible No. of irreducibles Multiplicity Total
of this type of each in R Dimension
in R
Trivial 1 16 16
V± acts non-trivially 9 3 27
for one value of i

49
V^ acts non-trivially 9 3 27
for two values of i -
both act the same way
V± acts non-trivially 18 0 0
for two values of i -
they act differently
V. acts non-trivially 3 13
for all values of i -
all act the same way
V^ acts non-trivially 18 0 0
for all i - two act
same, one different
V. acts non-trivially 6 16
for all i - all act
differently
Proof. Since all representations of a given type are A(G)-
equivalent, they occur with the same multiplicity, so we must
determine this common value for each type. Let the multiplicities
of these types be niQ,...,m6 (i.e. niQ is the multiplicity of the
trivial representation, m, the multiplicity of each irreducible
representation in which V. acts non-trivially for one value of
i, etc).
It is easy to check that the number of each type of
irreducible appearing in R is as claimed. Thus by counting dimensions
we obtain the equation
mQ + 9m, + 9m2 + 18m., + 3m-j + 81m5 + 6m^ = dim R = 79.
Now consider the action of Pj on R. By 4.1, the dimension
of the subspace of R on which p, acts trivially is 55. This
subspace is a sum of copies of 32 of the 64 irreducible
representations of G, and it is easy to see that the number of these of
type 0 is 1, of type 1 is 7, of type 2 is 5, etc.

50
Proceeding in this fashion for all the subgroups A given in
4.1 yields the linear system
1 9
1 7
1 5
1 6
1 5
1 3
1 4
1 3
1 0
9
5
3
3
2
3
1
1
0
18
10
4
6
5
0
2
2
0
3
1
1
0
0
1
0
0
0
18
6
2
0
2
0
0
1
0
6
2
0
0
1
0
0
0
0
79
55
41
43
38
35
31
28
16
This (consistent) system has rank 7, and hence a unique
solution, (niQ.mj,.. .,1¾) = (16,3,3,0,1,0,1), yielding the theorem.
*
From this theorem we may of course determine dim H^(M.) for
any T c A c T(2). There are very many such A (even up to A(G)-
equivalence) so we content ourselves with listing the extreme cases.
Corollary 4»5» Let T c A c r(2) be any subgroup with [A: f]
Then A is a A(G)-equivalent to one of the following, and
*
dim H^(M ) is as stated:
= 2.
Generator of A/T
Dimension of H^(M.)
Pi
PlP2
PLq2
PlP2P3
PlP2(l3
55
51
45
59
43
41
Corollary 4.6. Let T c A c r(2) be any subgroup with
[T(2): A] = 2, so A is the kernel of a homomorphism
*.: r(2)/r —♦■ {±1}. Then A is a A(G)-equlvalent to one of the
following, and dim H,(M.) Is as stated:

51
Generators not in Ker($ ) Dimension of H^(M )
P! 19
Pp p2 19
Pp q2 16
pl' p2' p3 17
pl' p2' q3
Pp P2» ^1» ^3
16
17
*
(Note that here H,(M.) will be a sum of two types of
irreducible representations of G, the trivial one and one other. The
six cases of this corollary correspond, in order, to the six non-
trivial types of irreducibles in theorem 4.4.)
We close by considering the question of torsion in the ho-
*
mology of M. . As we have seen, the only possible torsion is
2-torsion.
*
Theorem 4.7. Suppose A is untwisted. Then H^(M.) is torsion
free .
Proof. If A is untwisted, then A/T may be written as a product
W1 x w_ x w_ with W c V . (The different V^ need not be
isomorphic.)
*
From theorem 1.15, we see that M„ is rational, and indeed,
0 *
this theorem shows that M„, a Zariski open set in M„, is
isomorphic to a Zariski open set in XxXxx= IP1 x jp1 x IP1.
But then M is isomorphic to a Zariski open set in
(X/W ) x (X/W ) x (X/W ), which is itself isomorphic to
p1 x up1 x p1, so M is rational.
Then by [AM, proposition 1], H,.(M ) is torsion-free.

52
References
[AM] Artln, M. and Mumford, D. Some elementary examples of unl-
ratlonal varieties which are not rational, Proc. Lond. Math.
Soc. 25 (1972), 75-95.
[B] Bredon, G. Introduction to compact transformation groups,
Academic Press, New York, 1972.
[G] van der Geer, G. On the geometry of a Slegel modular
threefold, Math. Ann. 260 (1982), 317-350.
[LV^] Lee, R. and Welntraub, S. H. Cohomology of a Slegel modular
variety of degree two, In Group Actions on Manifolds,
R. Schultz, ed., Amer. Math. Soc, Providence, RI, 1985,
433-488.
[LW2] Cohomology of Spi»(Z) and
related groups and spaces, Topology 24 (1985), 291-310.
[LW3] Moduli spaces of Rlemann
surfaces of genus two with level structures, to appear In
Trans. Amer. Math. Soc.
[M] Mumford, D. Stability of projective varieties,
L'Enselgnement Math. 23 (1977), 39-110.
Yale University
Louisiana State University and Universitat Gottingen

THE RO(G)-GRADED EQUIVARIANT ORDINARY COHOMOLOGY OF
COMPLEX PROJECTIVE SPACES WITH LINEAR Z/p ACTIONS
L. Gaunce Lewis, Jr.
INTRODUCTION. If X is a CW complex with cells only in even dimensions and R
is a ring, then, by an elementary result in cellular cohomology theory, the ordinary
cohomology H*(X; R) of X with R coefficients is a free, Z-graded R-module. Since
this result is quite useful in the study of well-behaved complex manifolds like
projective spaces or Grassmannians, it would be nice to be able to generalize it to
equivariant ordinary cohomology. The result does generalize in the following sense.
Let G be a finite group, X be a G-CW complex (in the sense of [MAT, LMSM]), and
R be a ring-valued contravariant coefficient system [ILL]. Then the G-equivariant
ordinary Bredon cohomology H*(X;R) of X with R coefficients may be regarded as a
coefficient system. If the cells of X are all even dimensional, then H*(X;R) is a free
module over R in the sense appropriate to coefficient systems. Unfortunately, this
theorem does not apply to complex projective spaces or complex Grassmannians with
any reasonable nontrivial G-action because these spaces do not have the right kind of
G-CW structure. In fact, if G is Z/p, for any prime p, and T] is a nontrivial
irreducible complex G-representation, then the theorem does not apply to S , the one-
point compactification of rj. Moreover, the Z-graded Bredon cohomology of S with
coefficients in the Burnside ring coefficient system is quite obviously not free over the
coefficient system.
The purpose of this paper is to provide an equivariant generalization of the
"freeness" theorem which does apply to an interesting class of G-spaces and to use
this result to describe the equivariant ordinary cohomology of complex projective
spaces with linear Z/p actions. These results are obtained by regarding equivariant
ordinary cohomology as a Mackey functor-valued theory graded on the real
representation ring RO(G) of G [LMM, LMSM]. To obtain such a theory, we take
the Burnside ring Mackey functor as our coefficient ring. Instead of using cells of the
form G/H x en, where H runs over the subgroups of G, we use the unit disks of real
G-representations as cells. Our main theorem, Theorem 2.6, then has roughly the
following form.
THEOREM. Let G be Z/p and let X be a G-CW complex constructed from the unit
disks of real G-representations. If these disks are all even dimensional and are
attached in the proper order, then the equivariant ordinary cohomology HqX of X is
a free RO(G)-graded module over the equivariant ordinary cohomology of a point.
To show that this theorem is not without applications, we prove in Theorem 3.1 that
if V is a complex G-representation and P(V) is the associated complex projective
space with the induced linear G-action, then P(V) has the required type of cell
structure. Theorems 4.3 and 4.9, which describe the ring structure of HqP(V), follow
from the freeness of HqP(V). As a sample of these results, assume that p = 2 and V

54
is a complex G-representation consisting of countably many copies of both the
(complex) one-dimensional sign representation A and the one dimensional trivial
representation 1. Then P(V) is the classifying space for G-equivariant complex line
bundles. As an RO(G)-graded ring, HqP(V) is generated by an element c in
dimension A and an element C in dimension 1 + A. The second generator is a
polynomial generator; the first satisfies the single relation
c2 = €2c + £C,
where € and £ are elements in the cohomology of a point. If, instead, V contains an
equal, but finite, number of copies of A and 1, then the only change in HqP(V) is
that the polynomial generator C is truncated in the appropriate dimension. If the
number of copies of 1 in V is different from the number of copies of A in V, or if p is
odd, then the ring structure of HqP(V) is more complex.
Equivariant ordinary Bredon cohomology with Burnside ring coefficients is
just the part of RO(G)-graded equivariant ordinary cohomology with Burnside ring
coefficients that is indexed on the trivial representations. All of the generators of
HqP(V) occur in dimensions corresponding to nontrivial representations of G. This
behavior of the generators offers a partial explanation of the difficulties encountered
in trying to compute Bredon cohomology. All that can been seen of HqP(V) with
2-graded Bredon cohomology is some junk connected to the RO(G)-graded
cohomology of a point whose presence in HqP(V) is forced by the unseen generators
in the nontrivial dimensions.
Using HqP(V), It is possible to give an alternative proof of the homotopy
rigidity of linear 2/p actions on complex projective spaces [LIU]. Moreover, the
"freeness" theorem should apply to complex Grassmannians with linear 2/p actions,
and it should be possible to compute the ring structure of the equivariant ordinary
cohomology of these spaces. Of course, it would be nice to extend the main theorem
to groups other than 2/p. Unfortunately, the obvious generalization of this theorem
fails for groups other than 2/p. The counterexamples have some interesting
connections with the equivariant Hurewicz theorem [LEI]. All of these topics are
being investigated.
All of the results in this paper depend on the observation that equivariant
cohomology theories are Mackey functor-valued. Therefore, the first section of this
paper contains a discussion of Mackey functors for the group 2/p. In the second
section, we discuss the RO(G)-graded cohomology of a point, precisely define what
we mean by a G-CW complex, and prove our "freeness" theorem. The G-cell
structure of complex projective spaces with linear 2/p actions is discussed in section
3. There the cohomology of these spaces is shown to be free over the cohomology of
a point. Section 4 is devoted to the multiplicative structure of the cohomology of a
point. The multiplicative structure of the cohomology of complex projective spaces is
discussed in section 5. The results stated in this section are proved in section 6. The
results on the cohomology of a point stated in sections 2 and 4 are proved in the
appendix.
A few comments on notational conventions are necessary. Hereafter, all
homology and cohomology is reduced. If X is a G-space and we wish to work with

55
the unreduced cohomolgy of X, then we take the reduced cohomology of X , the
disjoint union of X and a G-trivial basepoint. In particular, instead of speaking of
the cohomology of a point, hereafter we speak of the cohomology of S , which always
has trivial G action. If V is a G-representation, then SV and DV are the unit sphere
and unit disk of V with respect to some G-invariant norm. The one-point
compactification of V is denoted S and the point at infinity is taken as the
basepoint. If X is a based G-space, then £ X denotes the smash product of X and
S . Unless otherwise noted, all spaces, maps, homotopies, etc., are G-spaces,
G-maps, and G-homotopies, etc. We will shift back and forth between real and
complex G-representations; in general, real representations will be used for grading
our cohomology groups and complex representations will be used in discussions of the
structure of projective spaces. If the virtual representation a is represented by the
difference V-W of representations V and W, then |a| = dim V - dim W is the real
virtual dimension of a and a =V -W is the fixed virtual representation
associated to a. The trivial virtual representation of real dimension n is denoted by
n. Recall that the set of irreducible complex representations of G forms a group
under tensor product. If 77 is an irreducible complex representation, then vj~ denotes
the inverse of rj in this group. The tensor product of 77 and any representation V is
denoted 77 V. Many of our formulas contain terms of the form A/p, where A is some
integer-valued espression. The claim that A is divisible by p is implicitly included in
the use of such a term.
I would like to thank Tammo torn Dieck, Sonderforschungsbereich 170, and
the Mathematisches Institut at Gottingen for their hospitality during the initial
stages of this work. I would especially like to thank Tammo torn Dieck for
suggesting the problem which led to this paper and for invaluable comments,
especially on the main theorem, Theorem 2.6.
Equivariant cohomology theories graded on RO(G) are not universally
familiar objects, so a few remarks about what this paper assumes of its readers seem
appropriate. Equivariant ordinary cohomology with Burnside ring coefficients assigns
to each virtual representation a in RO(G) a contravariant functor Hq from the
homotopy category of based G-spaces to the category of Mackey functors. It also
assigns a suspension natural isomorphism
Hg+v(SvX) * H°(X)
to each pair (a,V) consisting of a virtual representation a and an actual
representation V. The isomorphisms associated to the three pairs (a,V), (a,W), and
(a,V + W) are required to satisfy a coherence condition. The functors Hq are
required to be exact in the sense that they convert cofibre sequences into long exact
sequences. The dimension axiom requires that HqS be the Burnside ring Mackey
functor and that HqS0 be zero if n € 1 and n^O. If a is a nontrivial virtual
representation, then HqS0 need not be zero, but it is uniquely determined by the
axioms. Note that because HqS0 is nonzero in dimensions other than zero, the
assertion that the cohomology of certain spaces is free over the cohomology of S is
very different from the assertion that the cohomology is free over the coefficient ring.
Our cohomology theory is ring valued; that is, any pair of elements drawn from HqX

56
and HqX have a cup product which is in Hq X. We will also work with
RO(G)-graded, Mackey functor-valued, reduced equivariant ordinary homology with
Burnside ring coefficients. This homology theory satisfies the obvious analogs of the
cohomology axioms. Also, it has a Hurewicz map, which we use to convert various
space level maps into homology classes. Finally, we assume that S and the free
orbit G+ satisfy equivariant Spanier-Whitehead duality [WIR, LMSM]; that is, for
any a in RO(G) there are isomorphisms
XLqj =Ji-ao and Hg10' — H-avj •
The proofs of all our results flow from these basic assumptions. In fact, most
of the proofs are simple long exact sequence arguments which would be left to the
reader in a paper dealing with a 2-graded, abelian group-valued, nonequivariant
cohomology. One of the points of this paper is that these simple techniques work
perfectly well in RO(G)-graded, Mackey functor-valued, equivariant cohomology
theories and yield useful results. The one serious demand made of the reader is a
willingness to work with Mackey functors. When the group is 2/p, these are really
very simple objects. Section one is intended as a tutorial on them.
1. MACKEY FUNCTORS FOR //p. Since the language of Mackey functors
pervades this paper, this section contains a brief introduction to Mackey functors for
the groups 2/p. For any finite group G, a G-Mackey functor M is a contravariant
additive functor from the Burnside category B(G) of G to the category Ab of abelian
groups [DRE, LE2, LIN]. However, since we are only concerned with G = 2/p,
rather than describing B(G) in detail, we simply note that a 2/p-Mackey functor M
is determined by two abelian groups, M(G/G) and M(G/e); two maps, a restriction
map
p : M(G/G) -► M(G/e)
and a transfer map
r : M(G/e) -► M(G/G);
and an action of G on M(G/e). The trace of this action and the composite p r are
required to be equal by the definition of the composition of maps in B(G); that is, if
x € M(G/e), then
pr(x) = £gx-
gcG
The abelian groups M(G/G) and M(G/e) are the values of the Mackey functor M at
the trivial orbit and the free orbit; or, if one prefers to think in terms of subgroups
instead of orbits, the values of M at the group and at the trivial subgroup. For
convenience, we abbreviate G/G to 1 and write M(e) for M(G/e). Frequently the
G-action on M(e) is trivial; in these cases the composite pr is just multiplication by
P-
A map f:M->N between Mackey functors consists of homomorphisms
f(l) : M(1)-»N(1) and f(e) : M(e)-. N(e)

57
which commute with p and r in the obvious sense. The map f(e) must also be
G-equivariant. The category 90¾ of Mackey functors is a complete and cocomplete
abelian category. The limit or colimit of a diagram in 90¾ is formed by taking the
limit or colimit of the corresponding two diagrams consisting of the abelian groups
associated to G/G and to G/e. The maps p and r and the group action on the limit
or colimit are the obvious induced maps and action.
We will describe Mackey functors diagramatically in the form
M(l)
-( ]•
M(e)
e
where M(l) and M(e) will be replaced by the appropriate abelian groups, p and r
may be replaced by explicit descriptions of the restriction and transfer maps, and 9
may be replaced by an explicit description of the group action. If p or r is replaced
by a number (usually 0, 1, or p), then the map is just multiplication by that number.
If 9 is omitted or replaced by 1, then the group action on M(e) is trivial. If p = 2
and 9 is replaced by -1, then the generator of G = 2/2 acts by multiplication by -1.
EXAMPLES 1.1 The following Mackey functors and maps appear repeatedly in our
cohomology computations.
(a) The Burnside ring Mackey functor A is given by
zez
(1,P) )(0,1)
I
where the notation (l,p) means that the restriction map p is the identity on the first
component and multiplication by p on the second. Similarly, (0,1) means that the
transfer map is the inclusion into the second factor. For any Mackey functor M,
there is a one-to-one correspondence between maps f: A->M and elements of M(l).
The correspondence relates the map f to the element f(l)((l,0)) of M(l). It follows
from this correspondence that A is a projective Mackey functor.
(b) The d-twisted Burnside ring Mackey functor A[d] is given by
zez
(d,p)l 1(0,1)
I

58
where d € 2. Note that A = A[l]. If d = ± d' mod p, then there is an
isomorphism f : Afd] = A[d'] of Mackey functors. The map f(e) is the identity and if
d' = ±d + np, then
f(l)(l,0) = (±l,n) € 2©2
f(l)(0,l) = (0,1).
If d = 0 mod p, then A[d] decomposes as the sum of two other Mackey functors; thus
Afd] is only of interest when d ^ 0 mod p. In this case, it is a projective Mackey
functor. An alternative 2-basis for A[d](l) will be used in some of our cohomology
calculations. To distinguish the two bases, we denote (1,0) and (0,1) in the present
basis by /J, and r respectively. Select integers a and b such that ad + bp = 1. The
alternative 2-basis consists of <r = a/j, + br and k = p/j, - dr. Note that p{cr) = 1,
p[k) = 0, and r(l) = r. In fact, k generates the kernel of p, and r generates the
image of the map r for which it is named. Of course, <r depends on the choice of a
and b; in our applications, these choices will always be specified.
(c) If C is any abelian group, then we use (C) to denote the Mackey functor
described by the diagram
C
0
(d) If dt and d2 are integers prime to p, then there is an isomorphism
g12:A[d1]©(Z)-»A[d2]©(Z).
Let /J,t and ri be the standard generators for A[dJ, and let zx and z2 be generators of
(2)(1) in the domain and range of g12. Select integers a,- and b{ such that
asds- + b,p = 1, for i = 1 or 2. The map g12(e): 2 -> 2 is the identity map, and the
maP 812(1)is siven by
812(1)(/½) = d!(a2^2 + b2r2) + (bi + b2 - bjb2p)z2
gi2(l)(7-i) = T2
and
Sl2(l)(zl) = P/^2 - d2r2 - ald2z2-
The inverse of g12 is just g21. The existence of this isomorphism will explain an
apparent inconsistency in our description of the equivariant cohomology of projective
spaces.
(e) Associated to an abelian group B with a G-action, we have the Mackey
functors L(B) and R(B) given by

59
L(B)
B/G
R(B)
BK
tr
tr
B
B
Here, i : B -> B is the inclusion of the fixed point subgroup and w : B -* B/G is the
projection onto the orbit group. The two maps tr are variants of the trace map. The
map tr : B -> B takes xgBto^gxGB. If x £ B and [x] is the associated
l/G -* B is given by
MM)
Egx € B.
gcG
These two constructions give functors from the category of 2[G]-modules to the
category of Mackey functors. These functors are the left and right adjoints to the
obvious forgetful functor from the category of Mackey functors to the category of
2[G]-modules. We will encounter these functors most often when B is 2 with the
trivial action or, if p = 2, with the sign action. Denote the resulting Mackey functors
by L, R, L_, and R_. These functors are described by the diagrams
/ L
R
1
1
L.
1/2
R.
If
If C is any abelian group, there is an obvious permutation action of G on C ,
the direct sum of p copies of C. Unless otherwise indicated, this action is assumed
when we refer to L(C ) or R(C ). These two functors are isomorphic and are
described by the diagram

60
c
A I 17
CP
\5
e
where A is the diagonal map. V is the folding map, and 6 is the permutation action.
(f) If M is a Mackey functor, then L(M(e)P) = R(M(e)P) is denoted MG.
There are two reasonable choices of a G action on M(e) , the permutation action or
the composite of the permutation action and the given action of G on each factor
M(e). These actions yield isomorphic 2[G]-modules, so the choice is not important.
The simple permutation action is always assumed here. The assignment of MG to M
is a special case of an important construction in induction theory [DRE, LE2] that
assigns a Mackey functor Mi to each object b of B(G) and each Mackey functor M.
The restriction map p: M(l) -* M(e) = MG(1) and the diagonal map
A:M(e)-»M(e) =MG(e) form a natural transformation p from M to MG.
Similarly, r : MG(1) = M(e) -» M(l) and the folding map V: MG(e) = M(e)P -» M(e)
form a natural transformation r: MG ->M. The Mackey functor AG = L(2 ) is
characterized by the fact that, for any Mackey functor M, there is a one-to-one
correspondence between maps f : AG -> M and elements of M(e). This correspondence
relates the map f to the element f(e)((l,0,0, ... ,0)) of M(e). It follows that AG is a
projective Mackey functor.
(g) If Y is a G-space, M is a Mackey functor, a € RO(G), and HG(Y;M)
and Ha(Y;M) denote the abelian group-valued equivariant ordinary cohomology and
homology of Y with coefficients M in dimension a, then the Mackey functor valued
cohomology HG(Y;M) and homology H»(Y;M) are described by the diagrams
HG(Y;M) Hg(Y;M)
**( )*■ .„d "i h
md
HG(GxY;M) H?(GxY;M)
where the maps 7r*and w+ are induced by the projection 7r : G x Y -> Y, and the
maps 7r, and ir- are the transfer maps arising from regarding the projection i as a
covering' space. The group HG(GxY;M) is isomorphic to the nonequivariant
cohomology group H a (Y;M(e)). If a is represented by the difference V - W of
representations V and W, then, under this isomorphism, the action of an element g of

61
G on Hq(GxY;M) may be described as the composite of multiplication by the
degrees of the maps g: SV -* SV and g: SW -* S and the actions of g on
H a (Y; M(e)) induced by the action of g on M(e) and the action of g^1 on Y. Similar
remarks apply in homology. If no coefficient Mackey functor M is indicated in
equivariant cohomology or homology, then Burnside ring coefficients are intended.
(h) For any Mackey functor M and any abelian group B, the Mackey functor
M ® B has value M(G/H) <g> B for the orbit G/H and the obvious restriction, transfer,
and action by G. If M* is an RO(G)-graded G-Mackey functor and B* is a 2-graded
abelian group, then M* <g> B* is the RO(G)-graded G-Mackey functor defined by
(M*® B*)a = E M^® Bn.
P + n = a
If a CW complex Y with cells only in even dimensions is regarded as a G-space by
assigning it the trivial G-action, then there is an isomorphism of RO(G)-graded
Mackey functors
H*GY = R'GS°®E*(Y;2)
which preserves cup products.
For any finite group G, there is a box product operation □ on the category
90¾ of G-Mackey functors which behaves like the tensor product on the category of
abelian groups. In particular, 90¾ is a symmetric monoidal closed category under the
box product. The Burnside ring Mackey functor A is the unit for □. If G = 2/p,
then the box product MDN of Mackey functors M and N is described by the diagram
[M(l) ® N(l) 0 M(e) ® N(ef|/«
M(e)®N(e)
0-
e
The equivalence relation ss is given by
x <g> ry « px <g>y for x € M(l) and y € N(e)
tv <gi w « v <gi pw for v G M(e) and w € N(l).
The action 0 of G on M(e) <8> N(e) is just the tensor product of the actions of G on
M(e) and N(e). The map r is derived from the inclusion of M(e) ® N(e) as a
summand of the direct sum used to define MDN(l). The map p is induced by p <g> p
on the first summand and the trace map of the action 9 on the second.
EXAMPLES 1.2(a) For any integers d1 and d2, there is an isomorphism
A[d,]DA[d2] = A[d!d2]

62
of Mackey functors.
(b) If B is a 2[G]-module and M is any Mackey functor, then there is an
isomorphism
L(B)DM = L(B®M(e)).
(c) For any Mackey functor M, the product RDM is described by the
diagram
M(l)/(p - rp)
M(e)
cr
e
where M(l)/(p - tp) is the cokernel of the difference between the multiplication by p
map and the composite rp. The maps p and r are induced by the restriction and
transfer maps for M, In particular, if M = R(B) for some 2[G]-module B, then
RDR(B) = R(B). Also, for any abelian group C, RD <C> = <C/pC>.
(d) If p = 2, then for any Mackey functor M, the product R.DM is
described by the diagram
M(e)/(image p)
M(e)
-9
Here 7r: M(e) -* M(e)/(image p) is the projection onto the cokernel of the restriction
map and v: M(e) -* M(e) describes the action of the nontrivial element of G on M(e).
The action -9 is the composite of the given action 9 of G on M(e) and the sign action
of G on M(e). In particular, R.DR. = L.
(e) For any abelian group C and any Mackey functor M,
<C>DM = <C<8> (M(l)/image r)>.
A Mackey functor ring (or Green functor [DRE, LE2]) is a Mackey functor S
together with a multiplication map \i : SDS -* S and a unit map rj : A -> S making
the appropriate diagrams commute. A module over S is just a Mackey functor M
together with an action map £ : S □ M -* M making the appropriate diagrams
commute. Since the Burnside ring Mackey functor A is the unit for D, it is a Mackey
functor ring whose multiplication is the isomorphism ADA-+A and whose unit is

63
the identity map A-»A. Every Mackey functor is a module over A with action map
the isomorphism ADM-»M. Note that if S is a Mackey functor ring and R is a
ring, then the Mackey functor S <g> R of Examples 1.1(h) is a Mackey functor ring.
Similar remarks apply in the graded case. The cohomology of any G-space Y with
coefficients a Mackey functor ring S is an RO(G)-graded Mackey functor ring whose
multiplication is given by maps
E£(Y;S) DJl£(Y;S) -» Hg+V;S),
for a and /? in RO(G).
The following result characterizes maps out of box products and allows us to
describe a Mackey functor ring S in terms of S(l) and S(e). This is the approach to
Mackey functor rings used in our discussion of the ring structure of the cohomology of
complex projective spaces.
PROPOSITION 1.3 For any Mackey functors M, N and P, there is a one-to-one
correspondence between maps h : MDN -* P and pairs H = (Hj, He) of maps
Hx : M(l) ® N(1)-»P(1)
He : M(e) ® N(e) -► P(e)
such that, for x € M(l), y € N(l), z € M(e), and w € N(e),
He(px®py) = />H1(xi»y)
H^rzgiy) = rHe(z<g>py)
Hj(x <g> rw) = rHe(px <g> w).
The second and third of these equations are called the Frobenius relations.
PROOF. The maps He and h are related by He = h(e). Given h, Hj is derived in
an obvious way from h(l) using the definition of MDN. Given Hx and He, h(l) is
constructed from the maps Hx and rHe on the two summands used to define
MDN(l).
It follows easily from the proposition that, if S is a Mackey functor ring, then
S(l) and S(e) are rings, p:S(l) -* S(e) is a ring homomorphism, and r:S(e) -* S(l) is
an S(l)-module map when S(e) is considered an S(l)-module via p. Moreover, if M is
a Mackey functor module over S, then M(l) is an S(l)-module and M(e) is an S(e)-
module. If we regard M(e) as an S(l)-module via p:S(l) -* S(e), then the maps
p: M(l) -> M(e) and r:M(e) -> M(e) are S(l)-module maps.
2. H*GS° AND SPACES WITH FREE COHOMOLOGY. Here, we recall Stong's
unpublished description of the additive structure of the RO(G)-graded equivariant
ordinary cohomology of S . We use this to show that if X is a generalized G-cell
complex constructed from suitable even-dimensional cells, then HqX and H, X are
free over HqS0. The additive structure of the cohomology HqG + of the free orbit is
also described. This is used to show that HqX and H, X are projective over RqS

64
when X is constructed from a slightly more general class of even-dimensional cells.
Since 2/2 has only one nontrivial irreducible representation, HqS° is very
easy to describe when G = 2/2.
THEOREM 2.1. If G = 2/2 and a € RO(G), then
if |a| = |aG| = 0,
R%S° =
if |a| = 0,
if |a| = 0,
if |a| = 0,
if |a| = 0,
la^l < 0, and
Gl
|aG| < 1, and
la G| > 0, and
|a<
> 1, and
if |a| ^ 0 and la'
0,
|a u| is even,
a G| is odd,
la G| is even,
|aG| is odd,
if |a| > 0, laG| < 0, and |aG| is even,
if |a| < 0, |aG| > 1, and |aG| is odd,
otherwise.
The most effective way to visualize R*GS° is to display RqS° for various a on
a coordinate plane in which the horizontal and vertical coordinates specify |a I and
|a|, respectively. In such a plot, given as Table 2.2 below, the zero values of HqS
are indicated by blanks. The only values in this plot with odd horizontal coordinate
are the R^ and L^ on the horizontal axis and the (2/2) in the fourth quadrant.
••• (2/2) (2/2)
••• (2/2) (2/2)
... (2/2) (2/2)
•■■ (2/2) (2/2)
••• R R. R R^
(2/2)
(2/2)
(2/2)
(2/2)
R R^
(2)
(2)
(2)
(2)
A R^
(2)
(2)
(2)
(2)
L L_
(2/2)
(2/2)
(2/2)
(2/2)
L L. L •••
(2/2)
(2/2)
(2/2)
(2/2)
TABLE 2.2. H>S° for p = 2.
Even though the representation ring of G is much more complicated when
p ^ 2, HqS0 is completely determined by the integers |a| and |a I except in the
special case where |a| — |a 1 = 0. In this special case, HqS° is A[d] for some integer

65
d which depends on a. Unfortunately, because of the isomorphism described in
Examples 1.1(b), d is only determined up to a multiple of p. The major source of
unpleasantness in the description of the multiplicative structure of the equivariant
cohomology of a point and of complex projective spaces is this lack of a canonical
choice for d. To explain the relation between a and d, we introduce several relatives
of the representation ring. Let R(G) be the complex representation ring of G and
RSO(G) be the ring of SO-isomorphism classes of SO-representations of G. Since any
real representation of G is also an SO-representation, the difference between RO(G)
and RSO(G) is that, in RSO(G), equivalences between representations are required to
preserve underlying nonequivariant orientations on the representation spaces. The
difference between R(G) and RSO(G) is that elements of RSO(G) may contain an
odd number of copies of the trivial one-dimensional real representation of G. Let
R0(G), RO0(G), and RSO0(G) denote the subrings of R(G), RO(G), and RSO(G)
containing those virtual representations a with |a| = [a 1 = 0. Note that
R0(G) = RSO0(G). Let R0(G) be the free abelian monoid generated by the formal
differences <f> - 77 of complex isomorphism classes of nontrivial irreducible complex
representations. Note that R0(G) is the quotient of R-o(G) obtained by allowing the
obvious cancellations and that RO0(G) is the quotient of R0(G) obtained by
identifying conjugate representations. Let A be the irreducible complex
representation which sends the standard generator of 2/p to e2lrl'v. The monoid
R0(G) is generated by elements of the form Am - An, where 1 < m, n < p-1.
Define a homomorphism from R0(G) to 2, regarded as a monoid under
multiplication, by sending the generator A - An to m(n_1), where n_1 denotes the
unique integer such that 1 < n_1 < p-1 and n(n_1) = 1 mod p. Define functions
from RSO0(G) and RO0(G) into Z by composing this homomorphism with sections
of the projections from R0(G) to RSO0(G) or RO0(G). Let da denote the integer
assigned to the virtual representation a by either map. The sections can not be
chosen to be homomorphisms, so the assignment of da to a will not be a
homomorphism from RSO0(G) or RO0(G) to the multiplicative monoid T. However,
the assignment of da to a does give a homomorphism from R0(G) to the group of
units (2/p) of 2/p and a homomorphism from RO0(G) to the quotient
(2/p) /{±1} of (2/p) . For later convenience, we select our sections so that d0 is 1.
Stong's description of the additive structure of HJ3S0 can now be translated
into the Mackey functor language of section one.
THEOREM 2.3. If p is odd, then
< 0
> 0
H£S° = < (Z) if |a| ^ 0 and |aG| = 0
if |a| > 0, |aG| < 0, and |aG| is an even integer
if |a| < 0, |aGl > 1, and laG| is an odd integer
otherwise
if |a| = |aG| =
if |a| = 0 and
if |a| = 0 and
if |a| ^ 0 and
0
|aG
|aG
|aG

66
As in the case p = 2, HGS is best visualized by plotting it on a coordinate
plane whose horizontal and vertical axes specify la I and |a| respectively. In this
plot, given as Table 2.4 below, the zero values of HGS° are indicated by blanks. The
vertical and horizontal coordinates of all the nonzero values, except the (Z/p) values
in the fourth quadrant, are even. Notice in the plots for both the odd primes and 2
that the vanishing of HGS° on the vertical line la 1 = 1 (for |a| ^ 0 if p = 2) is
unlike its behavior on the vertical lines corresponding to the other odd positive values
for |a |. These unusual zeroes for HGS° are the key to our freeness and projectivity
results. When G = Z/pn for n > 1, the corresponding values are not zero, so our
techniques do not extend to these groups.
Hereafter, we will often describe elements in HGS° by their position in these
plots. For example, we may refer to the torsion in the fourth quadrant or the copies
of (Z) on the positive vertical axis.
(Z/p) (Z/p) (Z/p)
(Z/p) (Z/p) (Z/p)
(Z/p) (Z/p) (Z/p)
R R
A[d0]
<Z)
(1)
<Z)
L ]
(Z/P)
(Z/P)
(Z/P)
: l
(Z/P)
(Z/P)
(Z/P)
TABLE 2.4. H*GS° for p odd.
Recall, from Examples 1.1(f), the new Mackey functor MG which can be
derived from any Mackey functor M, and the observation that AG = L(ZP) = R(Z ).
It is easy to check that HGG + is HG(S°)G , and from this, to compute HGG+.

67
COROLLARY 2.5. For any prime p,
Proposition 4.12 tells us that HqG+ is an RO(G)-graded projective module
over HqS , and that any map
f: H*GG+ -► M*
of RO(G)-graded modules over HqS0 is completely determined by the image of
(1,0,0,...,0) € ZP = H£(G+)(e) in M°(e).
A generalized G-cell complex X is a G-space X together with an increasing
sequence of subspaces Xn of X such that X0 is a single orbit, X = UXn, X has the
colimit (or weak) topology from the Xn , and Xn+1 is formed from Xn by attaching
G-cells. We will allow two types of G-cells. If V is a G-representation and DV and
SV are the unit disk and sphere of V, then the first type of allowed cell is a copy of
DV attached to Xn by a G-map from SV to Xn. The second type of cell is a copy of
G x em, where em is the unit m-disk with trivial G action, attached to Xn by a
G-map from G x Sm_1 to X„. For each n, we let Jn+1 denote the set of cells added
to Xn to form Xn+1. Regard a cell DV of the first type as even-dimensional if |V|
and | V | are even. Regard a cell Gxe™ as even dimensional if m is even.
THEOREM 2.6. Let X be a generalized G-cell complex with only even-dimensional
cells.
(a) Assume that X0 = * and all the cells of X are of the first type; that is,
disks DV of G-representations V. Assume also that |VG| > |WG| whenever DV € Jn,
DWeJfc, l<k<n, and |V| > |W|. Then R*CX+ is a free RO(G)-graded module
over HqS with one generator in dimension 0 and one generator in dimension V for
each DVeJ„, li > 1. The homology HGX+ of X is also a free RO(G)-graded
module over HqS with generators in the same dimensions.
(b) If X contains cells of both types and all the cells of X of the first type
satisfy the condition in part (a), then HqX+ is a projective RO(G)-graded module
over H^S°. Moreover, K*GX+ is the sum of one copy of K*gXq , which is KqS° or
HqG , in dimension 0, one copy of HJ-S0 in dimension V for each DV £ Jn, and one
copy of HqG+ in dimension 2k for each G x e2k € Jn , n > 1. The homology H?X+
of X is also a projective RO(G)-graded module over HqS0 and decomposes into the
same summands.
PROOF. Abusing notation, we let Jn+1 denote both the set of cells to be added to
Xn and the space consisting of the disjoint union of those cells. Let <9J„+1 denote the
space consisting of the disjoint union of the boundaries of the cells in Jn+1.
Associated to the cofibre sequence
if |a| = 0
otherwise

68
Xn -* Xn + 1 -> in + i/din + i,
we have the long exact sequences
... -» H?x:+1 -» H?(J„+1/5Jn+1) 3 HliX+ -» ...
and
...- aaGK+i - nsxi - n£+1(Jn+1/ajn+1) - ....
The space J„+i/<9J„+1 is a wedge of one copy of S for each DV € Jn+1 and one
copy of G + aS2* for each Gxe2*€Jn+1. Thus, H*G(Jn+1/<9Jn+1) and
ii?(Jn+i/5Jn+1) are projective modules over R*GS° with generators in dimensions
corresponding to the cells added to Xn to form Xn+1. Moreover, if Jn+J contains
only cells of the first type, then KG(Jn+1/d3n+1) and H, (Jn+i/<9Jn+1) are free
modules over HqS°. The space X0 is either a point or the free orbit G, so HqXq and
H» Xj are projective, and perhaps free, modules over HJ3S0 generated by single
elements in dimension 0.
We will show inductively that the boundary maps d in both long exact
sequences are zero. The long exact sequences must then break up into short exact
sequences which split by the projectivity of H» (Jn+i/^n+i) and H^Xn. Thus, by
induction, HqXh and H» X« are free or projective, as appropriate, over HqS0, with
the indicated generators. It follows by the usual colimit argument that H» X+ is free,
or projective, with the appropriate generators. Since the map
H£X;+1 -+ 1[£X+
is always a surjection, the appropriate lim1 term vanishes, and the cohomology of X,
being the limit of the cohomologies of the Xn, is free (or projective) with the
appropriate generators.
The graded Mackey functors RG(3n+1/d3n+1), H?(Jn+1/5Jn+1), R'GX+
and H* Xq are sums of copies of RGS° and R*GG + in various dimensions. By
induction, we may assume that HqXr and H» Xn are also of this form. To show
that the maps d are zero, it therefore suffices to show that they are zero from each
summand of the domain to each summand of the range. For the cohomology
sequence, the four possibilities for the summands and the map between them are:

69
H*-2*G+ ^ ^(G+AS2fc) - HG+1(G+AS2m) = H*G+1"2mG+
HG"WS0 = HGSW _» HG+1(G+AS2m) = HG+1"2mG+
E*Q2kG+ = Hg(G+aS2*) -» HG+1SV = HG+1_VS°
and
TT»-WcO ~ TT* CW _^ TT* + lcV ~ TT* + l-VqO
Here, we use Hg(G + aS2 ) and HGS to denote summands of HGXn isomorphic to
HGG+ in dimension 2k or HGS° in dimension W. The four maps above are all maps
of RO(G)-graded modules over HGS°. Any such map out of HGS° is determined by
the image of 1 € A(l) = HG(S°)(1). By Proposition 4.12, such a map out of HGG+
is determined by the image of (1,0,0, ... ,0) € 2P = HG(G+)(e). Thus, to show that
the four maps are zero, it suffices to show that the groups HG + ~ m(G )(e),
Hw+1-2m(G+)(1)) fl«+i-v(So)(e)i and flw+i-v(So)(1) are zero_ The integers
|2k+l-2m| and |W+l-2m| are odd and HGG+ vanishes whenever |a| is odd, so the
first two groups are zero. The integer |2k+l-V| is odd and HG(S°)(e) vanishes when
|a| is odd, so the third group is zero. For the fourth group, if |V|<|W|, then
^W+l-Vgo is zero because |WG + 1_VG| is odd and |W + 1_V| is p0Sitive. Otherwise,
|VG| > |WG|, and H^y+1~VS° is zero because |wG + l-VG| is at most one. An
analogous proof shows that the map d in the homology sequence is zero. Note that if
|V|>|W| and |VG|=|WG|, then the vanishing of H^/+1_VS° is a result of the
anomalous zeroes on the la | = 1 line in the graph of HGS°.
In order to compute the ring structure of the equivariant cohomology of X,
we must compare it with more familiar objects, such as the nonequivariant ordinary
cohomology of X and X . If X is a generalized G-cell complex satisfying the
conditions of either part of Theorem 2.6, then so is XG. Thus, Examples 1.1(h)
describes HG(X ) in terms of the nonequivariant cohomology of X . Since the
group HG(X+)(e) is just the nonequivariant ordinary cohomology of X with Z
coefficients, the map
per -. nG(x+)(i) -» nG(x+)(e)enG((xG)+)(i)

70
offers a comparison between Hq(X+)(1) and two more easily understood cohomology
rings. This map does not detect the torsion in Hg(X+)(1) coming from the fourth
quadrant torsion in HqS0. Moreover, the torsion in Hg((XG) + )(1) makes it hard to
compute the image of p © i*. These difficulties suggest that we also consider the
image of H*G(X+)(l)/torskm in (H£(X+)(e) ©H£((XG)+)(l))/torsion. Since
HQ(X+)(e) contains no torsion, in the range we are only collapsing out the torsion in
H*G((XG)+)(1). The most useful comparison map is produced by also collapsing out
the image of the transfer map r from HQ((XG)+)(e). The quotient
H^((XG)+)(l)/(torsion© im r)
consists of copies of 1 in various dimensions; there is one 1 in the quotient for each
A[d] or (1) which appears in Hq((Xg)+)(1).
For many spaces, including complex projective spaces with linear actions, the
cells can be ordered so that |V| > |W| whenever DV£j„, DWeJfc, and k < n.
When the cells can be so ordered, there is no torsion in Hq(X+)(1) in the dimensions
of the generators of HJ~X+ as a module over HqS0. Therefore, the collapsing we have
done causes a minimal loss of information. The following result describes the extent
to which Hg(X+)(1) »s detected by p © i*.
COROLLARY 2.7. Let X be a generalized G-cell complex satisfying the conditions
of either part of Theorem 2.6 and let i: XG -* X be the inclusion of the fixed point
set. Then, for any a £ RO(G) with |a| even, the map
per : HG(X+)(1) -+ HS(X+)(e)©H£((XG)+)(l)
is a monomorphism. Moreover, for any a £ RO(G), the map
P®i': (H^(X+)(l))/torsion - Hg(X+)(e) © (H^((XG)+)(l))/(torsion © im r)
is a monomorphism.
PROOF. Since the equivariant cohomology of X is the limit of the cohomologies of
the Xn, it suffices to show that the result holds for every X„. It is easy to check the
second part for X0. Assume the second part for Xn , and let x be an element of
H^(X^+1)(l)/torsion vanishing under the map into
H£(X;t+1)(e) © (H£((XG ,)+)(1))/(^011 © im r)
induced by p © i*. We must show that x is zero. The group Ii§(X++1)(l) is the

71
direct sum of the groups HG(Jn+i/SJn+1)(l) and H£(Xn)(l), and this decomposition
is respected by the map p © i*. Thus, x is the sum of classes y and z in
HG(Jn+i/<9Jn+j)(l)/torsion and HG(Xn)(l)/torsion, respectively, which vanish
under the analogous maps. By our inductive hypothesis, z is zero. Since in+i/din+1
is a wedge of copies of S and G+aS2A: for various V and k, y vanishes by our remark
about X0. Thus, x is zero. The proof of the first part is similar. For this part, we
must assume that |a| is even because the map p © i* does not detect the torsion in
the fourth quadrant of Hq(S°)(1).
3. THE COHOMOLOGY OF COMPLEX PROJECIVE SPACES. As an application
of the results from section two, we show that the cohomology of a complex projective
space with a linear action is free over H^S°. Let V be a finite or countably infinite
dimensional complex G-representation and let C* be C-{0}. The complex projective
space P(V) with linear G-action associated to V is the quotient G-space (V-{0})/C*.
Note that if W C V, then P(W) may be regarded as a subspace of P(V). If V is
infinite dimensional, then we topologize V as the colimit of its finite dimensional
subspaces W; the quotient topology on P(V) is then the same as the colimit topology
from the associated subspaces P(W). To describe the cohomology of P(V), we must
n
write V as the sum ]P <f>i of irreducible complex representations (including possibly
i=o
the trivial complex representation). Of course, if V is infinite dimensional, then
li = oo. Points in P(V) will be described by homogeneous coordinates of the form
\x0' Xli X2> ••• ) X"/! Xi € <j>i
with the conventions that not all of the x{ are zero, and if V is infinite dimensional,
that all but finitely many of the x,- are zero. Each element of the group G acts on
each homogeneous coordinate of P(V) by multiplication by a complex number.
Therefore, if all the irreducibles in V are isomorphic, then the action of G on P(V) is
trivial. Moreover, if r/ is any irreducible complex representation, then P(V) and
P(r?V) are isomorphic G-spaces. If t\ and <f> are irreducible complex representations,
then P(r]) is just a point and Y{rj © <j>) is G-homeomorphic to the one-point
compactification of either 77-1 <j> or 77 <f>~1.
Since we have selected a colimit topology on P(V) when V is infinite, to show
that P(V) is a generalized G-cell complex for any G-representation V, it suffices to
k-l
show this when V is finite dimensional. Let Vk be the representation ^2 <p{ and let
»=0
W be the representation <j>nl Vn-1. Describe points in the unit disk DW by complex
coordinates (x0, xx, ... ,xn_1), with x,- € <j>nl <j>t. Define a map f: DW -* P(V) by
n-l 2
f((x0; xl> ••• Jxn-l)) = \x0' xl> x2) ••• 1 xn-l> 1 _ 2-, |xj| )•
t = 0
The tensor product with <j>nl is inserted in the definition of W to ensure that the map
f is equivariant. The image of SW in P(V) lies in the subspace P(Vn-1) of P(V), and
f is a homeomorphism from DW^SW to its image in P(V). Thus P(V) is formed

72
from P(V„_j) by adjoining the G-cell DW along the map f | SW : SW -► P(Vn_!).
Working backwards through the sequence of representations Vk , we conclude that
P(V) is a generalized G-cell complex with cells the unit disks of the representations
<t>ZlVk for 1 < k < n.
In order to show that the equivariant cohomology of P(V) is free over HqS ,
we must show that the cells of P(V) can be attached in an order satisfying the
condition in Theorem 2.6(a). This proper ordering of cells is derived from a careful
ordering of the set $ of irreducible summands of V. Since the remainder of our
discussion focuses on $, we write P($) for P(V). An ordering <f>0, <f>1, <fi2, ••• of the
elements of $ is said to be proper if the number of irreducibles in the set {<^,-}o<»<*-i
isomorphic to <j>k is a nondecreasing function of k. For example, if <f> and rj are
distinct complex irreducibles and $ consists of two copies of <f> and one of rj, then
77, <j>, 4> and ¢, 77, <f> are proper orderings of ¢, but ¢, ¢, 77 is not. The dimension of
the fixed subrepresentation of the representation <j>k J2 </>,- is the number of
t' = 0
irreducibles in the set {0,-}o<i<*-i isomorphic to <f>k. Thus, if $ is properly ordered,
then the cell structure described above satisfies the conditions of Theorem 2.6.(a).
PROPOSITION 3.1. If 4>0, <t>x, 4>2, ... is any ordering of the elements of a set $ of
irreducible representations, then P($) is a generalized G-cell complex with cells the
unit disks of the G-represeutations <pk Yl '/'i; f°r k > 1. Moreover, HgP($) and
H. P($) are free RO(G)-graded modules over E*GS°. If the ordering of $ is proper,
then the homology and cohomology of P($) are each generated by one element in
dimension zero and one in each of the dimensions <f>k ]T tpi, for k > 1.
i = 0
The G-fixed subspace of P($) is a disjoint union of complex projective
spaces, one for each isomorphism class of irreducibles in $. The (complex) dimension
of the complex projective space in P($) associated to the irreducible <j> is one less
than the multiplicity of <j> in $. Thus, the effect of properly ordering the irreducibles
is that the maximal dimension of the components of the G-fixed subspace of
P({0,-}o<t<*) increases as slowly as possible with increasing k.
REMARKS 3.2. Our description of the cohomology of P($) contains one apparent
anomaly. Suppose that ¢, r], and <j> are distinct complex irreducible representations
and $ = {C, //, 4>}- If we assign the proper ordering (, r/, <j> to ¢, then we find that
the generators of HqP($) are in dimensions 0, rj'1 (, and ^-1(C©??). However, if
we select the proper ordering ¢, (, ?;, we, find that the generators are in dimensions 0,
(~l ¢, and tj-1 (4> (B (). In particular, the cohomology in dimension 77-1 C must be
A © (I) (B (I) if we use the first set of generators, and A[d] © (I) © (T) if we use the
second, where d is the integer associated to the element r]~l(-(~l<j> of RO0(G).
There is no contradiction in these two claims about the cohomology in dimension

73
f) ( because these two Mackey functors are isomorphic by Examples 1.1.(d). The
apparent difficulties in all the other dimensions are resolved in exactly the same way.
This example illustrates the latitude that one has in selecting the dimensions
of the generators of the cohomology of P($) for almost any set $ of irreducibles.
This latitude is necessary because, for most $, there are many proper orderings and a
choice of a proper ordering corresponds to a selection of the dimensions of the
generators.
It would be nice to have some simple cohomology invariants of P($) which
could be used for problems like comparing the cohomology of projective spaces with
different G-actions. The fact that the dimensions for the cohomology generators
don't provide such an invariant is a disappointment. However, one invariant related
to the dimensions of the generators is readily available. Select a proper ordering of $
and plot the dimensions a of the resulting set of generators of HqP($) on a
coordinate plane whose horizontal and vertical axes indicate la I and I a I,
respectively. The dimensions lie on a stair-step pattern whose foot is at the origin.
This plot is an invariant of P($). The height of the steps in the plot decreases, or
remains constant, as one goes up the steps (that is, moves in the direction of
increasing \a I and |a|). The height remains constant only if irreducible types
appearing in $ have equal multiplicity. The step-like structure of the plot reflects a
filtration on $ which plays an important role in our discussion of the ring structure
of HqP($) • An increasing filtration
0 = $(0), $(1), $(2), ..., $(r), ...
of the set $ is said to be proper if, for every r and every complex irreducible <j>, the
number of irreducibles in $(r) isomorphic to <j> is the lesser of r and the number of
irreducibles in $ isomorphic to <j>. Any two proper filtrations of $ differ only by an
interchange of isomorphic irreducible complex representations, so there is essentially
only one proper filtration of $. The steps in the plot of the dimensions of the
generators are in a one-to-one correspondence with the stages in the filtration of ¢.
The height of the step corresponding to filtration level r is the number of elements in
$(r)-$(r-l).
4. CUP PRODUCTS IN H^S°. Here we describe the multiplicative structure of
HqS . We begin with the case p = 2, which is due to Stong.
DEFINITIONS 4.1. Let ( be the real one-dimensional sign representation of
G = 1/2. The identity element 1 in A(l) = H^(S°)(1) is the identity element of the
RO(G)-graded Mackey functor ring R*CS°. Let k € H^(S°)(1) be 2-rp(l). Observe
that k~ = 2k. Let e £ HG(S°)(1) be the Euler class; that is, the image of
1 € HG(S°)(1) under the map induced by the inclusion S C S . Select a

74
nonequivariant identication of S with S1 and let i1_(- € HG (S°)(e) —HG(S )(e)
and tc_! GUcT^Xe^Hci^Xe) be the images of p(l) € HG(S°)(e) =Hj3(S1)(e)
under the maps induced by this identification. Let £ € HG (S )(1) be the unique
element with /?(£) = t?_x. The elements 1 and k generate the abelian group
HG(S°)(1) and the Mackey functor K°GS°. Each of the elements em, £m, and emC,
for m, n > 1, generates the abelian group HG(S°)(1) and the Mackey functor HGS° in
the appropriate dimension a. For m > 1, the element i^ or l^1_1 generates the
abelian group HG(S°)(e) in the appropriate dimension and i^ generates the Mackey
functor HGS° in the appropriate dimension. For m > 2, r(t^^) generates the
abelian group HG(S°)(1) in the appropriate dimension.
LEMMA 4.2. The class k € HG(S°)(1) and, for n > 1, the classes
2n + K _ (2n + l)(l-0,oCK
rid) GUci (su)(i)
are infinitely divisible by € € HG(S°)(1); that is, for m > 1, there are unique
elements
rmK e HGmC(s°)(i)
such that
e Hh-C )£J1G (a )(ij
em(e-mK) = K and em{e-mr{i{n_l1)) = r(t^+x).
Moreover, each of the elements e~mK or e~mr(t n+ ) generates the abelian group
HG(S°)(1) and the Mackey functor HGS° in its dimension.
THEOREM 4.3. The elements
e GHS(S°)(1)
^-(^¾ (Su)(e)
*<_! e HG_i(Su)(e)
and
£ € H2Gc_2(S°)(i)
CmK e HGmC(S°)(l)( for m > 1,
e-mT{L\n_+l) e HGn+1_(2n+m+,K(S0)(l)( for m, n > 1,

75
generate H^S as an RO(G)-graded Mackey functor algebra over the Burnside
Mackey functor ring A. The only relations among these elements, other than those
forced by the Frobenius relations or the vanishing of HqS0 in various dimensions, are
generated by the relations
pW
Ai-C i(-i
T(h-c)
^2CT)
r(&
T(l?-()TtfH)
Pit)
2ei
p(e-mK)
e(e~mK)
(e-mK)(e-nK)
It T(l1_(. )
P^rtil?))
f(f-mr(^))
Ttil?)Kc-'K)
= 0
= /0(1)
= 0
= 0,
= nm,
( o,
"I 2r(C?)>
= 0
= 0,
= €l~mK,
= 2eHm+n)K,
= 0,
= 0,
_,l-m / 2n + l\
-(■ i-(i!_c ),
= o,
for m > 0,
for m > 1,
if m or n is odd,
if m and n are even,
for m > 0,
for m > 1,
for m, n > 0,
for m > 0 and n > 1,
for m > 0 and n > 1,
for m, n > 1,
for m, q > 0 and n > 1,
and
£ (£~m r{t\n_\X)) = Cm T{i\n_-'), for m > 0 and n > 2.
REMARKS 4.4. (a) The last relation indicates that, for m > 0 and n > 1,
€-mr(i1™^ ) is infinitely divisible by £. Thus, we can think of all the elements in
the fourth quadrant of the graph of Hg(S°) as being derived from r(tj_^) via division
by powers of e and £. One mnemonic for the effect of e and £ on the elements in the
fourth quadrant is to denote the nonzero element in £^^^^)(1), for m > 2
and n > 1, by e~m £~n w, where w is regarded as a fictitious element in dimension 1.
The reason for selecting a fictitious element in dimension 1, instead of the actual
element in dimension 3^3(, is discussed in Remarks 4.10(b).
(b) For p = 2, the elements ±(l-rp(l)) in A(l) are units, and l-rp(l)
appears in the formula describing the anticommutativity of cup products. For any
G-space X, if a € H'G+KX+ and b € Hg+"CX+, then

76
ab = (-l),'m(l-rp(l))inba.
The generators i1_(-, i(_lt e, e~nn, and e~m t(l\™^1) are in dimensions where the
behavior of this nontrivial unit matters. Of course, since e~m t(l1"^ ) has order 2,
any unit acts trivially on it. It is easy to check that
(l-rp(l))i1_< = -i1-c and (1 - rp(l)) t(_: = -ic_x.
This action of 1 -rp{\) on lx_(- and i^_1 never affects cup products in HqS because
it is always balanced by the (~-l)lm term in the commutativity formula. However,
there are algebras over HqS0 where the effects of this unit on ix_^ and t^_x are
visible. The unit 1 -rp{\) acts trivially on e and e~nK. This shows up dramatically
in H,qS°. The elements e and e~2n+1K are odd-dimensional, so our intuition about
graded algebras from the nonequivariant context suggests that their squares should
vanish, or at least be 2-torsion. In fact, the squares are not torsion elements, an
apparent anomaly possible only because the action of 1 - tp(\) is trivial. The overall
effect of the actions of the units of A on the generators of HqS is that HJ3S is
commutative in both the graded and the ungraded sense.
When p is odd, several complications in the multiplicative structure of HqS
arise from the greater complexity of RO(G). The most obvious are a host of sign
problems coming from the identification of representations with their complex
conjugates. Initially, we resolve these sign problems by grading HqS0 on RSO(G)
instead of RO(G). In Remark 4.11, we explain steps which must be taken to pass
back to an RO(G)-grading. The most serious complication arises from the
misbehavior of the integers da associated to the virtual representations a in
RSO0(G). One way to deal with this complication is to avoid it. This can be done
very nicely if one is only interested in HqS°. Because of the intuition this approach
offers, we outline it as an introduction to the odd primes case.
The stable homotopy groups 2.«S°, for j3 € RSO0(G), have been studied
extensively by torn Dieck and Petrie [tDP], and the stable Hurewicz map
is an isomorphism [LEI] if /? G RSO0(G). Thus, many of their results can be applied
to homology in the appropriate dimensions. They have shown that the
multiplication map
is an isomorphism for any /? G RSO0(G) and any j G RSO(G). By similar
reasoning, the multiplication map
H£S°DIL1S0 _» ii£+7S0
is an isomorphism under the same conditions on (3 and 7. Thus, to understand all of
HqS , it suffices to understand the part of HqS which torn Dieck and Petrie have
already described and the part indexed on some subset of RSO(G) complementary to
RSO0(G). Recall that A is the irreducible complex representation that takes the

77
standard generator of 2/p to e2wi/,p. Let RSOA(G) be the additive subgroup of
RSO(G) generated by 1 and A. As an additive group, RSO(G) is the internal direct
sum of RSO0(G) and RSOA(G). To complete our description of HqS0, it suffices to
describe that part of it indexed on RSOA(G). This part is almost identical to H^S°
for G = 2/2. Consider the description given above of that part of HqS for p = 2
indexed on the additive subgroup of RO(2/2) generated by 1 and 2(. Replace 2( by
A and all the other 2's by p's. The result is a description of the part of HqS° for p
odd indexed on RSOA(G). This approach describes HqS° as the graded box product
of two subrings indexed on complementary subsets of RSO(G). The unpleasant
behavior of the integers da is buried in the computations of the box products.
Unfortunately, because of peculiarities in the dimensions of the algebra
generators of HqP(V) , this description of HqS0 as the box product of two subrings
can not be used to describe the ring structure of the cohomology of complex
projective spaces. Thus, we offer an alternative description of the ring structure of
E*GS° for p odd. In section 2, we defined a function from RO0(G) to I using a
section of the projection from R0(G) to RO0(G). Since we are now working with
RSO0(G) instead of RO0(G), we define an analogous function from RSO0(G) to 1
using a section s: RSO0(G) -* R0(G) of the projection from R0(G) to RO0(G). We
insist that s(0) = 0 and that our original section RO0(G) -* R0(G) factor through s.
DEFINITIONS 4.5. (a) If a € RSO0(G) and s(a) = £ <£,--»?,•, then we wish to
define an equivariant map na: S n* -* S ' with nonequivariant degree da. If
a = \m - An with 0 < m, n < p and n-1 is the unique integer such that 1 < n-1 < p - 1
and nn- = 1 mod p, then fia is the extension to one-point compactifications of the
complex power map z -> zm(n \ for z € C. In general, we identify S"" l and S n%
with AS * and AS ', respectively, and take the smash product of the appropriate
i i
complex power maps to obtain the equivariant map \ia from S ' to S n* with
nonequivariant degree da. Also denote by na the image of this map in Hq(S°)(1)
under the Hurewicz map. Clearly, if the 4>i and the r\{ were paired off in a different
order, then a different map from S ' to S '! would be obtained. However, the
maps coming from the two pairings would be equivariantly homotopic and so would
give the same element in H£(S°)(l).
(b) Let a be an element of RSO(G) with |a| = 0. Then a must be
represented by a sum ^24>i~ViJ where the <j>{ and r\i are irreducible complex
i
representations, some of which may be trivial. Since the <f>i and rji are complex
representations, they have canonical nonequivariant orientations. Combine these to
produce a nonequivariant identification ia of S * with S ' which is unique up to

78
homotopy. Let ia also denote the image of this identification in Hq(S )(e). The
resulting cohomology classes ia are then independent of the ordering of the </>,- and
the 77,-. The abelian group H£(S°)(e) is generated by ta. If |aG| > 0, then r(ta)
generates the abelian group Hq(S°)(1) and ta generates the Mackey functor H^S0.
(c) If a € RSO0(G), then in H£S°,
p(fia) = data and pr(ia) = Pit*.
We have already asserted that H^S0 is A[d«]. Under this identification, pa and
r(ia) become the elements p and r of A[da](l) and ia becomes 1 £ 2 = A[da](e).
There is a unique integer ba such that d-»da + bap = 1. Let Ka = p pa-dar(ia)
and a a = d-a pa + b0T(ta). Then, a a and Ka form an alternative 2-basis for
nS(s°)(i).
(d) Let (3 be an element of RSO(G) with |/?| > 0 and \(3G\ = 0. There exist
an a in RSO0(G) and a G-representation V such that V = 0 and /3 = a + V. Let
tp & Hg(S°)(1) be the image of pa € Hg(S°)(l) under the map from Hg(S°)(l) to
Hg(S°)(1) induced by the inclusion S° C SV. In Lemma A.11, it is shown that this
Euler class e » is independent of the choice of the decomposition of /? into the sum of
the representation V and the element a of RSO0(G). The class e^ generates the
abelian group Hq(S°)(1) and the Mackey functor HGS°.
(e) If |a| = 0 and |aG| < 0, let £a be the unique element of Hg(S0)(l) with
p(£a) = i«; this class generates the abelian group Hq(S°)(1) and the Mackey functor
H£S°.
When p is odd, it is harder to pick a multiplicative basis for the torsion in
the fourth quadrant of the graph of HqS0. In each dimension there is a choice of
p - 1 generators, instead of a single nonzero element. Moreover, since these torsion
elements are not tied by an Euler class to elements on the positive horizontal axis,
there is no way to base the choice of a generator on choices already made for the axis.
The following lemma justifies the procedure we employ to select multiplicative
generators for the fourth quadrant.
LEMMA 4.6. Let /3 be an element of RSO0(G) and let a, j, and 8 be elements of
RSO(G) such that

79
tnd
\6\ = |7G| = 0,
|a|, \6G\ < 0,
|7| > 0,
laG| > 3,
la I is odd.
If x is any nonzero element in Hq(S )(1), then /z»x is a generator in HG (S )(1).
Moreover, x and /z»x are uniquely divisible by both £7 and £s.
DEFINITIONS 4.7. Select a generator in HG_2A(S°)(1) and denote it by i/3_2A. If
a = 1 -m(A-2) -nA, for m, n> 1, then let va be the unique element in Hq(S°)(1)
such that
€{n-l)\^{m-l){\-2)Va = l/3-2A'
For any a £ RSO(G), there are unique integers m, n, and q such that q = 0 or 1
and
a^[q^m(A^2)^nA] € RO0(G).
Denote by <a> the element q-m(A-2)-nA associated to a by these conditions. If
a e RSO(G) with |a| < 0, |aG| > 3, |aGl odd, and a ^ <a>, then define
ya e H£(S°)(1) by
Va = Ha~<a> v<a> ■
The element va generates the abelian group Hg(S°)(1) and the Mackey functor
H£S°.
LEMMA 4.8. If a € RSO0(G), then Ka e H£(S°)(1) is divisible by ep, for any
/3 € RSO(G) with \/3\ > 0 and \j3 \ = 0; that is, there is a unique element
such that
£ a {£ ft Kqc) — ^a-
The element e^1 Ka generates the abelian group Hq (S°)(l) and the Mackey functor

80
RaG-pS°.
THEOREM 4.9. The elements
/ia € HS(S°)(1),
ta € H§(S0)(e),
€A € H^(S°)(1)
£A_2 € H£f 2(S°)(1)
*2_A € H2G-A(S°)(e)
4*oG HamA(S°)(l),
for a = ±(An^A), with 1 < n < p,
for a = ±(An^A), with 1 < n < p,
and
va e H£(Su)(i)
for in > 1,
for a = 1-m(A-2)-n A, with m, n>l,
generate HqS0 as an RSO(G)-graded Mackey functor algebra over the Burnside
Mackey functor ring A. All of relations among the elements of HqS0, other than
those forced by the Frobenius relations or the vanishing of HqS in various
dimensions, are generated by the relations
P(p-a) = d«ta ,
Ha Up = Ha + f} +
/>M = o,
taCp = ea + p,
PaCp = (a + p,
p(£a) = la,
T {<■<*) = p£a,
£«£/? = £» + /?>
Pain = daZa + 8 '
dadp- da + f}
for a € RSO0(G);
-(ia+/?), for a, H e RSO0(G);
for |/?|>0 and|/?G| = 0;
for |a|, |/?|>0 and
|aG|=|/?G| = 0;
for a G RSO0(G), |/?| >0,
and |/?G| = 0;
for |q| = 0 and laGl < 0;
for |a| = 0 and |aG| < 0;
for |a| = |/?| = 0 and
la I, \(3 1 < 0;
for a € RSO0(G), |/?| = 0,
and |/?G| < 0;

81
PCgta = 0,
tgtia — df-a £7 £j,
£q Ka — ¢7 Kg >
pitp1 Ka) = 0,
A* 7 \£/} Ka) = €0 Ka + i 1
£p {tp Ka) = Ka,
£-r(ep Ka) = tp„-,Ka,
(f/Ka)!^1^) = Y>ep + 1Ka + 6,
pi/a = 0,
P{va) = 0,
HpVc = Va + p,
for |a| = |/?G| = 0, |aG|<0,
and |/?|>0:
for |a| = |<5| = |/?G| = |7G| = 0,
|aG|J*Gl<0, |/?|,|7|>0,
and a + /3 = 7 + 8;
for a, 5 € RSO0(G),
|/?G| = |7G| = o,
|/?|, |7|>0, and
a + 7 = /? + <5;
for a € RSO0(G), |/?G| = 0,
and |/?| > 0;
for a,7€ RSO0(G),|/?G| = 0,
and |/?|>0;
for a € RSO0(G), |/?G| = 0,
and |/?|>0;
for a £ RSO0(G),
|/?G| = |7G| = 0, and
|/?|>|7|>0;
for a, 5 € RSO0(G),
|/?G| = |7G| = o,
and |/?|,|7|>0;
for |a| < 0, laG| > 3, and
I Gl ,,
la I odd;
for |a|<0, laG| > 3, and
la I odd;
for/? € RSO0(G),|al<0,
la I > 3, and |a I odd;

82
for |a + /?|<0, |aG|>3,
|aG| odd, |/?|>0, and
|/?G| = 0;
for|a|<0, |aG + /?G|>3,
|aG| odd, |/?| = 0, and
|/?G|<0;
for 7 € RSO0(G), |a|<0,
la I > 3, la I odd,
|/?G|= 0, and |/?|>0;
for |a|=|/?| = 0.
REMARKS 4.10. (a) For p odd, the only units in A(l) are ±1. The only
generators in odd dimensions are the va ■ Since va v » is zero for any a and /?, no
sign problems occur in commuting products in HqS . Thus, HqS is commutative in
both the graded and ungraded senses.
(b) As an alternative to using the va as a basis in the fourth quadrant, one
may define elements e^^w in HQ_a_/?(S°)(l), for |a| = |/?G| = 0, |aG| < 0, and
|/?|>0, by
Here, w is regarded as a fictitious element in dimension 1 which is divisible by any
product £a eg . We employ a fictitious element because there is no canonical choice
for the dimension of an actual element. The relations satisfied by the elements
(p1^1^ are
for |a| = |/?G| = |7G| = 0,
|/?|>|7|>0,
and la I< 0;
for |a| = |7| = |/?G| = 0,
|aGl<|7G|<0,
and |/?|>0;
tpVa — Va + p,
ipVa — d<p>-[}1/a + p ,
(€p Ky)l>a = 0,
t-t{tBlZalu) =
e/3i7^a1w,
^(ffl^w)
j £a—fU/,

83
/i7 {ej&w) = £pl7^a1w, for 7 € RSO0(G),
|a| = |/?G| = 0, laG|<0,
and |/?|>0;
/^7(^^^01 w) = d<7>_7e^1^I7w, for 7 € RSO0(G),
|a| = |/?G| = 0, |aG|<0,
and |/?|>0.
The one difficulty with this alternative basis is that if a + /3 = 7 + 6, then ejl£aluj
and ejl£^luj are in the same dimension, but they need not be equal. In fact,
((3 £a w = da_7_<a_7> £,5 £7 w.
(c) Observe that in the formulas for the product of fia with any of £^,
Cpl k1 , or vp there is no da, but there is such a constant in the formula for the
product Ha ig- On the other hand, cra£g = £,a+g, but there is a d-a in the formula
for the product of <ra with any of £^, e~Zl K-y, or v*. This difference in the behavior
of the elements \ia and cra of Hq(S°)(1) reflects the fact that there is a conjugacy
class of subgroups of G associated to any well chosen element of any G-Mackey
functor M for any finite group G. This association is based on the splitting of M
which occurs when M is localized away from the order of G. This splitting can not
be observed directly before localization, but it can be seen indirectly in the association
of subgroups to well chosen elements in the Mackey functor. The elements \ia , £p,
eZlK7, and Vg are all associated to the subgroup G of G, and products of pairs of
them behave nicely. The elements aa and £» are associated to the trivial subgroup,
and their product is nice. However, the product of elements associated to two
different subgroups will either be zero or involve some fudge factor like a da ■ We
have introduced both /j,a and aa so that, when one of these elements is needed in our
description of the relations in HqP(V) , we can always choose the one that will give
us the simpler formula.
REMARKS 4.11. In order to explain the passage from an RSO(G) grading on R*GS°
to an RO(G) grading, we must first clarify what is meant by the assertion that HqS°
is RO(G)-graded. The assertion does not mean that, for a € RO(G), H§S° can be
described without reference to a choice of a representative for a. Rather it means
that if Vx - Wj and V2 - W2 are two representatives for a and H1 and H2 are the
values of HqS obtained using these representatives, then we can construct an
isomorphism between H1 and H2 in a natural way from any isomorphism
f: V2 © Wx -► Vx © W2 of representations illustrating the equivalence of W1 - Wj
and V2 - W2 in RO(G). This is exactly what we mean when we say that
nonequivariant homology is 1 graded. To define the nonequivariant homology group
HnX, we must pick a standard n-simplex. Different choices of the n-simplex lead to

84
different groups, as anyone who has been embarrassed by an orientation mistake
knows all too well.
Let /? = V2 0 Wj - Vx © W2 and let f denote the image of f in H^(S0)(1).
Then the isomorphism from Ji1 to H2 is just multiplication by f. To provide a
means of computing the effect of this isomorphism, we write f in terms of the
standard generators of Hq(S°)(1). The map f induces a map f6 between the fixed
point subspaces of the representations. If nonequivariant orientations are choose for
their domains and ranges, then the maps f and V3 have well-defined nonequivariant
degrees. It follows from Lemma A. 12 that
f (a & , (deg f)-(deg f0^
f = (deg f°)^ + p t{lp).
The structure of HqG as an algebra over HqS° follows easily from our
results on HqS and the description of the additive structure of RqG+ given in
section 2.
PROPOSITION 4.12. As an RO(G)-graded module over JLIS0, H^G+ is generated
by the single element 4> = (1, 0, 0, ... , 0) of U^(G+)(e) = 2P. Mor eover, for any
RO(G)-graded module M* over HqS°, there is a one-to-one correspondence between
maps f: R*GG+ -+ M* of RO(G)-graded modul es over HqS0 and elements in M°(e).
This correspondence associates the map f with the element f(e)(^>) of M°(e). Thus,
HqG4" is a projective RO(G)-graded module over HqS0.
PROOF. Unless |a| = 0, H£(G+) = 0. If la| = 0, then ia ^ generates HgG+ as a
module over A. Thus, 4> generates HqG+ as an RO(G)-graded module over HqS0,
and any RO(G)-graded module map f: HqG+ -* M* is determined by f(^). On the
other hand, recall the observation from Examples 1.1(f) that a map from AG to any
Mackey functor N can be specified by giving the image of (1, 0, 0, ... , 0) € AG(e) in
N(e). Let m be an element of M (e). For each a € RO(G) with lal = 0, tam is in
Ma(e) and there is a unique map f°: HqG+ -* Ma of Mackey functors sending
ia^ € Hg(G+)(b) to iam € Ma(e). These maps fit together to form a map
f: HqG+ -+ M* of RO(G)-graded modules over JL^S0. The projectivity of R*CG+
follows immediately.

85
5. THE MULTIPLICATIVE STRUCTURE OF HqP(V)+. We assume that there
are at least two distinct isomorphism classes of irreducibles in V; otherwise, the
multiplicative structure of HqP(V) is completely described in Examples 1.1.(h). As
in section 3, we take $ to be the set of irreducible summands of the complex
representation V. Let ¢(0), ¢(1), ¢(2), ... be a proper filtration of ¢. Then ¢(1)
consists of exactly one representative of each of the isomorphism classes of
irreducibles that appears in ¢. Let <j>0, (f>1, </>2, ... , 4>m be an enumeration of the
elements in ¢(1), and let n{ be the number of elements of ¢ isomorphic to 4>t (with
n; = oo allowed). Arrange the enumeration of the elements of ¢(1) so that
n0 > nx > ... > nm. Extend the ordering of ¢(1) to ¢ by selecting the unique proper
ordering of ¢ which is consistent with the filtration and in which, for each r > 1, the
ordering of the representations in <&(t+ l)-$(r) is the same as the ordering of the
corresponding representations in ¢(1). If the irreducibles which appear in ¢ appear
with equal multiplicity, then, regarded as an ordered set, f is a sequence of blocks,
each of which is a copy of ¢(1). If the multiplicities are not equal, then ¢ is still a
sequence of blocks, but each block after the first will be either a copy of ¢(1) or of an
initial segment of ¢(1). The lengths of the initial segments in the sequence can not
increase. We will abuse notation by writing <j>t £ ¢^ +1)-¢(1^) to mean that
¢(1^+1)-^r) contains an irreducible representation isomorphic to 4>{. We say that
two sets of irreducible representations are equivalent if they contain the same number
of irreducibles in each isomorphism class. Moreover, we sometimes identify
equivalent sets of irreducibles.
Corollary 2.7 will be used to derive the multiplicative structure of HqP(V)
from the multiplicative structures of E"G(P(V) + )(e) and H^((P(V)G)+)(1). The
group Hq(P(V) )(e) is isomorphic to the nonequivariant cohomology group
H"(P(V) ;2), and we will think of the restriction map p as a map from
Hg(P(V)+)(l) to Hla'(P(V)+;2). Select an algebra generator x € H2(P(V) + ;Z) for
H*(P(V) ;2). The fixed point space of P(V) is the disjoint union of the spaces
P(nj 4>t) = P(nt). Let q{ denote both the inclusion of the subspace P(nt-) into P(V)
and the map H^(P(V)+)(1) -► H^(P(n1)+)(l) induced by this inclusion. By
Examples 1.1.(h), HqP^,)* is a truncated polynomial algebra over HqS0 generated
by an element x; in HG(P(ni) )(1)- Let
q,-: H^(P(V)+)(1) -+ H*G(P(nI)+)(l)/(torsion © im p)
denote the composition of q; and the projection onto the quotient. If y is in
■H(3(P(n;)+)(l)> then M denotes its image in H^(P(n;)+)(l)/(torsion © im p).
Throughout this section, we will index HqP(V) on RSO(G) to simplify the
selection of the integers da. The comments in Remarks 4.11 on the passage from
RSO(G)-grading to RO(G)~grading for H^S° apply equally well to H^P(V) + . Recall
that A is the irreducible complex representation that sends the standard generator of

86
2/p to e2wl'p and that C is the real one-dimensional sign representation of 2/2. If p
is 2, then A, regarded as a real representation, is just 2£.
We begin with the case p = 2. Any complex irreducible representation is
isomorphic to either the complex one-dimensional trivial representation or the
complex one-dimensional sign representation A. Since P(V) and P(AV) are
G-homeomorphic, we may assume that there are at least as many copies of the trivial
representation in $ as there are copies of the sign representation. Thus, we may take
4>q to be the trivial representation and <j>1 to be the sign representation.
By Theorem 3.1, H^P(V)+, regarded as a module over HqS0, has one
generator in each of the dimensions
2k + 2k( and 2k + 2(k + 1)(,
for 0 < k < n1, and one in each of the dimensions
2k + 2niC,
for nj < k < n0 . If one assumes n0 = n1, or ignores the generators special to the case
n0>n1, then one might guess that, as an algebra, HqP(V) had an exterior
generator in dimension 2£ and a truncated polynomial generator in dimension
2(1 + ¢). Except for the fact that the generator in dimension 2 C is not quite an
exterior generator and for some difficulties in the higher dimensions when n0>nj,
this guess is a good description of HqP(V) . However, in order to describe the
behavior in the higher dimensions as simply as possible, we adopt a notation that
does not immediately suggest this.
THEOREM 5.1. (a) As an algebra over H^S°, H^P(V)+ is generated by an
element c of H^(P(V)+)(1) in dimension 2( and elements C(k) of Hg(P(V)+)(1) in
dimensions 2k + 2min(k, nx) (, for 1 < k < n0 .
(b) For any positive integer k, let k denote the minimum of k and nx. Then
the generators c and C(k) are uniquely determined by
q0(c) = [o]
qi(c) = M
p(c) = x € H2(P(V)+; 2)
q0(C(k)) = [e"x*]
^(0(10) = [e"x*]
and
p(C(k)) = xk+\
Moreover,

87
q0(c) = £x0 € H2GC(P(n0)+)(l)
q,(c) = e2 + £x, € HGC(P(ni) + )(l)
q0(C(k)) = 4(e2 + tx0)~k € aT+lQ(P(n0)+)(l)
and
qi(C(k)) = x* (e2 + {xj* € Hr+KU(P(ni) + )(l).
If iij is finite, then xt-' = 0 and some of the terms in the last two sums above may
vanish.
(c) The generators c and C(k) satisfy the relations
c2 = e2c + ¢0(1),
cC(k) = £C(k + l), fork>ni,
and
C(j)C(k)
for j +k <nj,
i = 0 v '
In these relations, we take C(i) to be zero if i > n0 ,
REMARKS 5.2. (a) By iteratively applying the third relation, we obtain
C(k) = (C(l))*, for k<ni,
so that below the dimensions where we run short of copies of the sign representation,
HqP(V) is generated by c and C(l). Moreover, in these dimensions, C(l) acts like a
polynomial generator.
(b) If n0=nl5 then HqP(V)+ is generated by c and C(l). The only
relations satisfied by these two generators are the relation
c2 = e2c + ¢0(1)
and, if n0 < oo, the relation
C(1)"0 = 0.
REMARKS 5.3. Notice that the maps q0 and qj behave differently on the generator
c. The element c=c+£2-kc of RG P(V)+ may be used as a generator in the
place of c and its behavior with respect to q0 and qj is exactly the reverse of the
behavior of c. To understand the geometric relation between these elements, observe
that c and c can be detected in the cohomology of any subspace P(l + A) of P(V)
arising from an inclusion 1 + A C V. The space P(l + A) is G-homeomorphic to S ,
but unlike S , it lacks a canonical basepoint. Either choice for the basepoint of
P(l + A) determines a splitting of HqP(1 + A) into the direct sum of one copy of

88
HqS0 and one copy of HqSa. The canonical generator of HqS in dimension 2 ( is
identified with c by one of the two splittings and with c by the other.
When p is 2, the multiplicative structure of HqP(V) does not really exhibit
any complexities beyond those one might experience in a 2-graded ring. However,
when p is odd, there are quirks in the multiplicative structure of HqP(V) which are
only possible because of the RSO(G)-grading. For the odd prime case, recall the
stairstep diagram obtained by plotting the dimensions a of the generators of
HqP(V) in terms of |a| and |aG|. Looking at this diagram in the special case where
the irreducibles appearing in V appear with equal multiplicity, one might guess that
,H.qP(V) was generated by two truncated polynomial generators, one in a dimension
a with |a| = 2 and [aG[ = 0 and one in a dimension /? with |/?| = 2m + 2 and
\(3 | = 2. Unfortunately, such a guess would badly underestimate the complexity of
HqP(V) . The set of dimensions for a full set of additive generators must generate a
larger additive subgroup of RSO(G) than can be accounted for by a pair of truncated
polynomial generators. For example, recall that the first two additive generators of
HqP(V) are "in dimensions <j>yl<j>0 and ¢^(¢0 + <fii)- W the additive generator in
dimension <j>il<j>0 were to serve as a truncated polynomial generator, then the additive
generator in the next higher dimension would need to be in dimension 2<j>'[l<j)0
instead of ¢2 (4>o + 4>i)- Any replacement of these two generators by an element
and its square requires the introduction of further generators in some other
dimensions inconsistent with a simple truncated polynomial structure. To provide a
better feeling for the multiplicative structure of HqP(V) , we give two sets of
multiplicative generators. The first is a natural set with a great deal of symmetry.
It does not exhibit a preference for any one ordering of $. Unfortunately, this set is
much too large. By selecting an ordering on $, we are able to construct a much
smaller, but very asymmetrical, set of algebra generators.
In order to describe the effect of the maps q,- on our algebra generators, we
must introduce more notation related to the integers da.
DEFINITIONS 5.4. (a) For any two distinct integers i and j with 0 < i, j < m, let
Pij denote the irreducible representation 4>~l<f>:, and let dri denote the integer da ,
for a = (3 ■■ - prs- Note that d^j is 1 for any pair of distinct integers i and j. For any
integer i and any distinct pair of integers r and s such that 0 < i, r, s < m, let dlrls be
zero. The integers dri satisfy the relations
dridu„ = dui mod p,
d'i + dJrs = drs mod p,
and
,ij ,tu _ tu ij
drsdvw = arsdvw mod p.

89
(b) If fa G $(r+l)-$(r), then let a^r) denote the representation
<fil E <t>i and let drij be d„ , for a = a,-(r) - a;.(r). Note that, if fa € $(r + 1) - $(r),
<t> e *(r)
d;t- = l. If either fa or ¢- is not in $(r+l)-$(r), then let d[;- be zero. If fa, fas,
and fa. are in $(r + l)~$(r), then the integers dj'- satisfy the relations
3^=¾ modp
and, if i ^ j,
0<yfc<m
where afc is the multiplicity of fa. in $(r).
THEOREM 5.5. (a) If i and j are distinct integers with 0 < i, j < m, then there is a
unique element ctj in E£;'(P(V)+)(1) such that
qfc(ci;)
d,, e,
'»J
for 0 < k < m,
md
P(C,-;) = X.
Hcj(r)(P(V)+)(l) such that
If r>0 and </>■ £ $(r+l)-$(r), then there is a unique element C;(r) in
that
q*(C,-(r)) =
1^^-(^)1/2
d^Co-i(r)-rXt
for 0 < k < m,
ind
p(C;.(r)) = x1
The elements c,-,-, for 0<i,j<m and i ^ j, and the elements (^(r), for r>l and
fa. e $(r+ 1)- $(r), generate H^P(V)+ as an algebra over R*GS°.
(b) For 0 < i, j, k < m and i zfz j,
(c) For r > 1 and fa. € $(r + 1) - $(r),

90
qt(Ct(r)) = x[
If <j>j€ $(r+l)-$(r) andj ^k, then
n (<*„ + *. a*t)
izfzk
<u
(c;w) = 4(d^;£/3;fc+^_2x
•e*C) J pji z
+
d^^(d^)rn (d")
«^J,*
%(r)-rXfc'
If 4>k ¢. $(r + 1)- $(r), then qyt(C;(r)) is zero.
(d) For 1 <j <m, let 7- be the representation 4>jl ]P ¢,- and let D; be the
i — Q
element Ylcji m 2G (P(V) )(1). Then the elements D-, for 1 <j <m, the elements
»=0
C0(r), for r>l and <j>0 £ $(r +1) - $(r), and the elements DC(r), for r>l and
<j>j £ $(1-+1) -$(r), generate H^P(V)+ as an algebra over H^S°.
REMARKS 5.6. In order to simplify our indexing, we define D0 and Cj(0), for
0<j<m, to be 1 G H^(P(V)+)(1). We also define 7o and a;-(0) to be 0. Our
second set of generators for HqP(V) is then just the set of elements D-C.-(r), for
r > 0 and <$>■ £ $(r+ 1) - $(r). This set of elements of H^(P(V)+)(1) is also a set of
additive generators of HqP(V) as a module over HqS . One might hope that a set
of multiplicative generators could be much smaller than a set of additive generators,
but if the various irreducibles in $ appear with very different multiplicities, then
small sets of multiplicative generators do not exist.
We will order the set of generators D -C;(r) by the dictionary order on r and
then j. On the stairstep plot of the dimensions of these generators, moving in the
direction of increasing order corresponds to moving up and to the right.
REMARKS 5.7. Nothing that has been said in the discussion of the odd prime case
actually depends on p being odd; rather, mod 2 arithmetic is so simple that most of
the technicalities necessary when p is odd are unnecessary when p = 2. The elements
c and c in the case p = 2 are c10 and c01. The element C(j) is C0(j).

91
In order to describe the relations among the generators in HqP(V) in a
palatable form, we must introduce one more ba tch of elements in H^(P(V)+)(1).
DEFINITION 5.8. Observe that, for 1 <j <m, /cD;- is divisible by e7 ,. Moreover,
p^./cD-) = 0, and ;
i-i
e R°G(P(nk4>k)+)/(tovslon®im t).
q*(s «D;) =
;-i
pnd*;
;-i
Since f] dk' is zero if k<j and 1 if k=j, the coefficients p f] dfc! which appear in
»=o ;t t=0 Jt
the qk(e~lnDj) form a matrix which is p times an upper triangular matrix with l's
on the main diagonal. Applying the obvious analog of the process for diagonalizing
an upper triangular matrix to the elements e~1/cD,- produces elements k • of
j
Hq(P(V) )(1) characterized by the conditions
P(*j) = o,
and
r [p], ifk=j,
q*(«i) = \
(. 0, otherwise.
These elements can be described inductively by the equations
*m = S1 KDm
m
and, for 1 < j < m,
1 k=J+ly i=0 J '
0 + m
Define k0 £ Hq(P(V) )(1) to be k-^2 it,-. The equations above characterizing /c,-
; = i
for j zfz 0 then also characterize k0 . Moreover,
r P, ifk=j,
I 0, otherwise.
For r>l and ^ € $(r+l)-$(r), define K;(r) € H^Vc^X1) to be
K;C;(r). These elements /c -(r) are characterized by the equations
p(*/(0) = o,
and
f l"p£ (r)_rxjn, ifk=j,
q*(«i(0) = ^ L J
t 0, otherwise.
Moreover,

92
f P£a(r).rXi. ifk=j,
q*(«iW) =
( 0, otherwise.
For convenience, we define /c ,-(0) to be /c •. Observe that, for r > 1, the elements
K,-(r) can also be constructed from the elements /cD -Cj(r) in the same way that the
elements k ■ are constructed from the kD •.
We begin our list of relations with the relation between any two of the ctj-
and the relation between any two of the C,-(r).
PROPOSITION 5.9. (a) Let i, j, r, and s be integers with 0 < i, j, r, s < m and
i zfzj, r zfzs. Then
,*> A»i Ars Aks
si ,-^ "if ~dji ~°-a drj
Ctj = <?PirPrsCrs + dljipi, + Z — ^.«f
(b) Let r > 1 and let i and j be integers such that <j>i and ¢^ are in
$(r + l)-$(r). Then
j r _ j r 1 r
C,-(r) = '0,.(r)-«,(r)C,-(r) + E, *' p^ "X,.(r)-at(r)«*(r)-
An obvious initial response to this result is to assume that HgP(V) Can be
generated as an algebra over HqS by any one of the c,-,- and, for each r with
$(r+1) - $(r) nonemepty, any one of the C,-(r). The kk and «j.(r) in the formulas
spoil this simplification, especially since they are defined in terms of precisely the
generators one would hope to omit. Solving this by taking the elements kk and kk(j)
as part of a generating set is hardly satisfactory since, from a Mackey functor point of
view, these are torsion elements (because p(kk) and p(kk(v)) are zero).
The remaining results in this section describe the products of pairs of
elements from either of the generating sets in terms of the smaller generating set. All
of the relations in HqP(V) follow from the relations in Proposition 5.9 and the
relations below. If V is finite, then some of the elements appearing on the right hand
side of these relations may not appear in the list of generators of HqP(V) . Any
such element is to be regarded as zero. We begin with the products which land in
dimensions where there is no torsion. These are easily computed using the maps qk
and p.
PROPOSITION 5.10. (a) Let i,j, r, and s be integers with 0 < i,j, r,s < m and
i zfz j, r zfz s. If m > 2, then
cijCrs = dtjdrsep^+f}rs + (do-dr.-d^.dr.je^.^^^cjo + <xaD2 +
Aki Aks A°J A°3 (AlJAU A°>A°S\Ak0 Ak° A*1 A
m^ df j d„ - d{j dr, - (djj dri - dtj dra)d10 - d20 d2I d_a
2^ _ . __ ~ f +R„„Kk.
k = 2 ^i]
+ Prs

93
where a= /3{j + prs - j2 .
If m = 1, then
+ (^:^^ Cl0 +
£ , c0(i).
(b) Let i, j, and r be integers with 0 < i, j < m, i ^ j, and 1 < r < m. Then
ci;-Dr= d[;j£/^Dr + <xaDr+1 +
3 = 0 3=:0 '_
P /3,-jH
where a = ptj + 7,--7,.+1-
'»;
E
"k >
(c) Let i, j be integers with 0 < i, j < m and i ^ j. Then
c,.jDm=dy£ Dm + £ C0(l).
(d) Let i, j, r, and s be integers with 0<i,j, s<m, i ^ j, r>l, and
<j>, e $(r+l)-$(r). If 4>x e $(r +1)- $(r), then
CijC,(r) -d°;'dSi£/3..+ai(r)_ao(r)C0(r) + ^D^r) +
^ di j ^L - di j d£, dfc0 - d*0° d^ d-q „
^ pi e0ij + <*,<.r)-ak(r)Kk\*)>
4>k i *(r + l)-*(r)
where a = /?,j + a,(r) ~7i - aj(r).
If ^i £ $(r+l)-$(r), then
<^C,(r) =d?/dS,e/J.jC0(r) + ^..+0o(r)_0o(r+1)C0(r+l).
(e) Let i, j, r, and s be integers with 0 < i, j, s < m, i ^ j, r > 1, and
4>s e $(r+ 1) - $(r). If 4>s + l e $(r + 1) - $(r), then
CijD.Cfr) = d'je D,C8(r) + <r„ Ds + 1 C,+1(r) +
E
<(<M,')'n<iiHu.d-«fK:1
<=o
t=0
Jfc>*+1
^ e <P(r+l)-<P(r)
e«t«t(r).

94
where a = Ptj + y, + as(T)-ys+1 - aa + l(r) and 6k = ^^ + y, + aa(v)-ak(i).
If^ + i £ $(r+l)-$(r), then
cijD,Cs(r) = d^.D^r) + ^.i+7i+o,(r)-O0(r+1)C0(r+l).
(f) Let r, s > 1 and assume that 1 <j < m. If the irreducibles that appear in
$(r+ s) appear with equal multiplicities, then
C;(r)C;.(s) = C^r + s) + £ ^^f^ ^a ,r+,)-ok(r+,)^+s).
<t>k e *(r + *+l)-*(r + »)
Moreover, the integers d^t" may be selected to be the products d£-d"L so that the
Kj.(r + s) correction terms are not needed.
Since the elements kk(v) appear in so many formulas, we include a
description of products involving them.
LEMMA 5.11. Let i, j, k, r, and s be integers with 0<i,j,k<m, r, s > 0, and
4>k e $(s+1)-$(s).
(a) If i zfz j, then
Cij«*(S) = ¢¢0.^(8).
(b) If 4>j e $(r +1)- ¢(0 and <j>k € $(r + s + 1) - $(r + s), then
Cj(r)/Ct(s) = Kj*Qj{r) + Qk{T)-ak{r + .)kk(?+*)
and
D;- C^A^s) = drkj
In the formula for C/rJK^s), replace cc ^) + 0^,)-0^+,) bv
^aj(r) + ok(,)-ok(r+,) if |aj(r) + ak(s) -a*(r+s)| is ^ro.
(c) If <t>j e $(r + 1)- ¢(0 and <j>k 0 $(r + s + 1) - $(r + s), then C/r) kk(s)
and D;C;-(r) kk(s) are zero.
To complete our description of the multiplicative structure of HqP(V) we
need to describe the products of various pairs made from elements of the types Cj(r),
^-1 ,
nd;
(-,,, Kj.(r + s).
-rj + aj(r) + ak(,)-ak(r + ,)

95
D;-C;(r), and Dfc . If we use the convention that D0 = C;(0) = 1, then the products
we must describe are all special cases of the general product (D.,C,(r)) (D .,C,-(s)),
where r, s > 0, $} €_ $(r+l)-$(r), <f>z € $(s+l)-$(s), i' is 0 or i, and j' is 0 or
j. We may assume that i' > j'. Recall the formula given in Theorem 5.1(c) for the
product C(j)C(k) when p = 2 and j + k>nj. Observe that this formula may be
obtained from the binomial expansion of (e + £x);+ "l by replacing the powers of
x by various generators C(t). The formula for our general product is related in a
n
similar way to the expansion of an expression of the form n(a>' + b;x). The
i = 0
summands in this expansion are indexed on the subsets of the set {0, 1, ... ,n}. The
summand corresponding to the subset I is
where III denotes the number of elements in I. To describe the analogous part of our
formula for (D.,C,-(r))(D .,(L(s)), we must specify the indexing set which replaces
{0, 1, ... ,n}, the factors which replace Yla» and El^ti an^ the procedure for
replacing the powers of x by the appropriate DA.Cyt(t).
In the p = 2 case, describing how the powers of x are to be replaced by the
generators C(j) is very simple because, if j >nj, then the next generator after C(j) is
always C(j + 1). However, when p is odd, the generator after D^C^r) may be either
DA.+1 Cfc+1(r) or C0(r+1), To handle this complication, we introduce two functions f
and g from the nonnegative integers to the nonnegative integers. These functions are
to be chosen so that, for any i > 0, CUi+1 Jg(i+1)) is the generator immediately
following Cf^,(g(i)) in our stairstep ordering. If Cffn,(g(n)) is the last generator in
HgP($) ! then we define f(i) = 0 and g(i) = g(n) + i-n for i>n and use the
convention that D-C.-(r) is to be regarded as zero if it does not appear in the list of
generators of HqP($) • Each time we use this notation, the initial values, f(0) and
g(0), of the functions will be specified to suit the particular application.
The indexing set which replaces the set {0, 1, ... ,n} is related to the
difference in dimension between the product (D.,C,-(r)) (D .,C,-(s)) and the lowest
dimensional generator D.,Cj(r + s) which should appear in its description. If r > 0

96
and 0 < j < m, then define the subset $j(r) of $(r+ 1) by
$;-(r) = $(r) U {¢,-: i<j and ¢i € $(r + 1) - $(r)}.
Let $., (r) U $.,(s) denote the disjoint union of the sets $., (r) and $ .,(s). Our
replacement for the set {0,1, ...,n} is the set ty obtained by deleting from
$.,(r) U $ .,(s) a subset equivalent to the set $.,(r + s). We abuse notation by
writing $ as $.,(r) U $ .,(s) - $.,(r + s). Observe that $ .,(s) is equivalent to the
disjoint union of $ and $.,(r + s) - $.,(r). Let u be |ty|-l and number the elements
of $ from 0 to u. Let h be a function from the set {0, 1, ... , u} to the set
{0, 1, ... , m} such that the i element of ty is isomorphic to the irreducible
representation ¢^,...
One of the coefficients appearing in our formula is determined by a certain
element a of RSO(G) with |a| = 0 and la I < 0. This coefficient will be £ if
la I < 0 or aa if |a 1=0. To simplify our notation, we write \ for either of these,
relying on |a I to indicate whether £ or <rn is intended. Another coefficient will
depend on a certain element /3 of RSO(G) with \(3 | = 0 and \(3\ > 0. This coefficient
will be £p if |/?j > 0 and /j,0 if |/?| = 0. We write 80 for either of these, relying on |/?|
to indicate which is intended.
PROPOSITION 5.12. Let i, i', j, j', r, and s be integers with r,s>0,
4>i €$(r+l)-$(r), 0j €$(s + l)-$(s), i'= 0 or i, j'= 0 or j, and i'>j'. Let
VP = $,.,(r) U $ .,(s)-$.,(r + s). Initialize the functions f and g by
f(0)
i', if ¢., e $(r + s+l),
0, otherwise,
and
r + s, if ¢^ £ $(r + s+l),
r + s+1, otherwise.
g(0) =
Let u — |ty| - 1 and number the elements of ^1 from 0 to u. Let Acf and let s' and

97
s" be the number of elements isomorphic to 4>i m A and $ ,(r + s) - ^./(r),
respectively. If the subset A of f contains the elements numbered j0 , jn ... , jw,
with j0 <jj < ... <jw, then let
ft Ait-VMit)
Hit)*}
11 djfc
ft «
<=o 0jMit)
Hit)*i
Hit)=i
md
where
^ = Xc
<=° . ^£4>.,(r-M)-<f,,(r) '
<*o,j
E ^
+ [(s' + s")^-^o]^0 +
2s - ¢-1
f(UI)
E ^
The tag j^O on the bracket about the (s' + s") 4>ll4>0 indicates that this term is
present only if j ^0, The 2s term in a indicates 2s copies of the real one-dimensional
trivial representation. If a € RSO0(G), then let
d4= d .
If A = 0, then let d^ , eA , d A , and \ be 1.
A
If i'<k <m and <f>k € $(r + s+1) - $(r + s), let
where

98
and let Ak be
P = a,-(r) + ocj(s) + y., + j ., - afc(r + s),
^:,.( n<)( nd*J-E
1CS
Ul=«-
Then
(D^CWJCD.,0^(8)) = £/^^-^¾^)¾^1^) +
£ At0t/ct(r + s).
^j. e *(r+j+l)-*(r+i)
REMARKS 5.13. (a) Let r > 1. If $(r) contains r copies of every irreducible
complex G-representation, then a,-(r) is independent of i and it is easy to see that
C,-(r)=C-(r) for every i and j such that <pt, <j> • € $(r + l)-$(r). Moreover,
C (r) = C-(l)r. Thus, if $ contains every irreducible complex G-representation and
these representations appear with equal multiplicities in $, then C,(r) generates a
polynomial, or truncated polynomial, subalgebra of HqP($) • In this case, the
elements D-, for l<j<m, and C^l), for any i, generate HqP($) + as an algebra
over H*GS°.
(b) If p = 3, then we may choose the integers da so that da = ± 1 for every
a in RSOq(G). When this is done, the assignment of da to a is a homomorphism
from the additive group of RSO0(G) to the multiplicative group {±1}. With this
choice of the integers d<,, all the relations among the dri and the dj- given in
Definitions 5.4, except the one involving a sum, hold in 2 as well as in 2/3. If r > 1
and 4>t, 4>- e $(r+l), then
CiW = ffa,-(r)-0j.(r)Cj(r)-
Thus, the only elements of the form C^(r) needed to generate HqP($) + as an algebra
over HqS are the elements C0(r) for r > 1. Also, a pair of elements c,- • and cr> will
generate Dj and D2 if q^c,- -0,.5) is nonzero for only one value of k. In particular, c01
and c10 generate Dl and D2. When all three irreducible complex G-representations of
2/3 appear in $ with equal multiplicities, c01, c10 , and C0(l) generate HqP($) as
an algebra over HqS0-
6. PROOFS. The results stated in section 5 are proved here. As indicated in
Remark 5.7, our results for p = 2 are a special case of the results asserted for odd

99
primes. They have been presented separately only because they can be stated so
simply. The proofs given here are independent of whether p is 2 or odd. We begin
by construct the elements c{ • and C;(r). We then show that they generate HqP(^)
as an algebra over HqS . Finally, the relations stated at the end of section 5 are
verified. Throughout this section, $ is a set of irreducible complex representations of
2/p and $(0), $(1), ... is a proper filtration of $. We order the elements of $ in the
standard proper ordering introduced in section 5. Recall the maps q; and q; and the
cohomology classes x and x^ from the introductory remarks in section 5 and the
representations a^r), /?,-■, and 7, from Definitions 5.4 and Theorem 5.5(d). If
Ac*, then x also denotes the image of x € HG(P($)+)(e) in HG(P(A)+)(e); thus,
the powers of x are thought of as the standard additive generators for the
nonequivariant cohomology of all the sub-projective spaces of P(^). For each integer
j with 0<j<m, let P7($) be the component of the fixed point space of P($)
associated to the irreducible representation <f>,.
The classes c-• and C (r) are constructed by defining them on the smallest
possible projective space and then inductively lifting them to larger projective spaces.
CONSTRUCTION 6.1. (a) Let i and j be distinct integers with 0 < i, j < m. The
space P({(/>7-}) is just a point and the space P({(f>i ,(f> A) is G-homeomorphic to S '3,
The inclusion of P({</>,}) into P({^i,^;}) induces the cofibre sequence
P({^})+ - P({^;})+ - S^.
Let Cij e Eg J(P({0t-,^})+)(l) be the image of 1 € A(l) =11^(3^)(1) under w*.
Then q^(c^) = 0 by exactness and q,-(Cj •) = e^. . by the commutativity of the
diagram
P({^})+ -3^ P({*<,*;}) +
These are the correct values for q,-(Cj,-) and q-(c^ ■) because xt and x- are zero. Since
the map 7r*:HG!J(S !J)(e) -* HGIJ(P({<^j, 4>j}) )(e) is an isomorphism in dimension
Piji P(c«j)=x-
Let * be a subset of $ which properly contains the set {4>i,4>j} and assume
that, for every proper subset A of * containing {$i;<j>j}, c,-• has been defined in
Hg (P(A) )(1) and has the proper images under the maps qk and p. Pick an
irreducible representation <j>t which appears in * at least as often as any other
irreducible. If no irreducible appears more than once in ¢, then we may also insist

100
that t^i, j. Let A = 12 - {<j>t}, and let V be the representation <pt Y. $■ The
inclusion of A into ty induces the cofibre sequences
P(A)+ I P(*)+ -» Sv
and
P,(A)+ U P((*)+ -» SyG.
We will lift the class ctj € HG j(P(A)+)(l) along the map
0*(1):H£J(P(*) + )(D -» Hg°'(P(^)+)(1)
induced by ¢. To distinguish the class c- and its lifting, we will denote the class in
HG,J(P(A)+)(1) by cir The maps qk, for k ^t, factor through 0*(1), so any lifting
of c,-• along #*(1) will have the right image under qk, for k^t. Moreover, since
#*(e) is an isomorphism in dimension /3^-, any lifting of ci;- will also have the right
image under p.
It remains to show that we can choose a lifting of c^ • with the correct image
under q^. We have chosen t so that the long exact cohomology sequences associated
to our cofibre sequences have zero boundary maps. If|vG|>2, then flG''J'(Sv)(1) = 0
and we take c^ to be the unique lifting of c^ •. If |V |>2, then 8t induces a
cohomology isomorphism in dimension /?(- • and this lifting of c^ ■ along #*(1) must
have the correct image under q^. If |V I = 2, then the short exact sequence
o - Hg"'s2 - 4!}ptm+ 5 4JJp<(A)+ - o
splits. The end terms are
Hc'S2 = R and RPGllPt(A)+ = (I).
The image of 1 € 2 = R(l) in H^P^ty) is £0. ._2xt- By our induction hypothesis,
^(l)q,(cij)=q,(c,J)=d;j£ .
Since p(cij)=xJ pqt(cjj) is the generator of 11^(^^) )(e)* **' follows that
,Jii
/Gl
If |V 1=0, then no irreducible appears more than once in ^ and we have
selected <f>t so that t ^i, j. In the diagram
0 -» HciSV -» Hg,jP(*)+ -» Hg JP(A)+ -» 0
0 -» HG!'->S° -» HGijPt(*)+ - 0

101
comparing the cohomology sequences of our two cofibre sequences, we have that
Hq'-'S and HGtJS° are (2) and the map ev is multiplication by p. Thus, if z is a
lifting of c^ , then by adding elements from the image of Hq'^S to z, we can adjust
%(z) by any multiple of p. It now suffices to show that there is a lifting z with
q<(z)=d-^£ mod p. The lifting problems for P(\P) and P({<^, ¢,-, <fit}) can be
J Dij J
compared via the cohomology maps induced by the inclusion of {¢^, ¢,-, <pt} into ¢.
This comparison indicates that it suffices to show that the lifting problem can be
solved when ty = {<f>i, ¢ ■, (f>t). In this case, consider the diagram
0 -» Hg jSv -» IIG°'P(*)+ £ HciP(A)+ -» 0
I £ | q | qj
0 _ j^V"' Z HGIJP({^})+ q-i HG"P({^})+ - 0
comparing the cohomology exact sequences for the pairs (P(^), P(A)) and
(P({^-,^}), P({^j})). Let a = PirPtr If z is a lifting of ctJ along #*(1), then
q7-(z) = q7-q(z) = 0. Thus, q(z) = 7(y) for some y £ HG'J'(S ^)(1). Since pq(z) is
the generator x of B.q^{P{{4>j , <j>t}) )(e)> P(v) must generate HG'JS *J(e), and y
must be aa + ntta for some integer n. The diagram
nj's"'' I Hg*'j'P({^,^})+
| c | q<
4*>S° i Hg°'P({^«})+
commutes and gives that q*(z) = qq*(z) = e(y) = f(ir«) mod p. By the definition of
<ra, £(cr«) =diU .
P i j
(b) Let r > 1 and let ¢- £ $(r+l). The cofibre sequence associated to the
inclusion of P($(r)) into P($(r) U {¢^-}) is
P($(r))+ -+ P($(r)U{^-}r 5 S^'(''5.
Define C^r) € EGj(r)(P($(r) U {¢^)+)(1) to be the image under tt*(1) of
l£A(l)=HG (S j )(1)- Since n* is an isomorphism in dimension a7(r),
|ai(r)|/2
p(C (r)) = x . The cohomology diagram in dimension a -(r) induced by the
diagram

102
P7($(r)U{^})
+ *i
P($(r)U{^})+
,",-(«•)
indicates that q^C^r)) = €a .(r)_r xj . If k ^ j, qyt(CJ-(r)) = 0 for dimensional
reasons. As we did with the definition of c; in part (a), we extend the definition of
C (r) to HqP($)+ by working inductively along a sequence of subsets of $ between
$(r) U {4>A and $. The only difference between the argument given for ci;- and the
one which should be used for C (r) is that the liftings of C At) should be chosen to
behave properly with respect to p and qk instead of p and qk. This change is
necessary because qyt(CJ(r)) is more complicated than qyt(cij). The behavior of the
C -(r) with respect to the maps qk is established in the lemma below.
LEMMA 6.2. Let r > 1 and <bk e $(r+l)-$(r). Then
q*(ct(r)) = x
n.s^ + t. _axt)
<t>i( ¢(0
izfzk
If <Pj e $(r + l)-$(r) and j ^ k, then
qt(CJ.(r))=xl(d*i£/J.fc + ^ _2xk
'jf
i^j,k
+
d[
,kjs
kj-Wi) n (d;
*,-f*(r)
*K
i^j.k
£°<j(r)-~X
k ■
If ^ £ $(r+ 1) - $(r), then qfc(Cy(r)) is zero.
PROOF. If (f>k ¢. $(r + l) - $(r), then qfc(Cj(r)) vanishes for dimensional reasons.
Therefore, assume that <j> ■, <j>k € $(r+ 1) - $(r). Let
* = $(r) U {^: <£ € $ - $(r) and <£ =4>k}.
The image of the class C^(r) in HqP($) under the map
HGP($)+-HGP(*u{^})+
may be computed using the maps p and q,-. It is the class Cj(r) in HqP^ U {<Pj}) •
The image of this class under the map
H*GP(*u{^.})+^li*GP(*)+

103
is the class <ra _{r)_ak{r)Ck{r), Thus,
<u(C>(r)) = qfcKj.(r)_0jt(r)CJt(r)) = ff0j.(r)_0jt(r)qt(Ct(r)),
since P,t($) = P*(W) and the map qfc for P($) factors as the composite of the map
H^P($)+ -f HgP(*)+ and the map qk for ¢. Observe that
<T«;('*)-a*:('*) = ^/3 .._«.. )r FI %.._«,. + aK« ,-(^)-0^(11)
for some integer a. With this description of &ajr\_a (>■>> it is easy to derive the
formula for qyt(CJ-(r)) from the formula for qA.(Cyt(r)). The formula for q4(Cj.(r) is
derived using an iterative procedure. Let s>r and pick <j>t £ $ with t^k. The
image of Ct(s) € H£(P(*)+)(1) under the map H^P(*)+ -» H£P(* - {«U) + is
£«. C*(s) + ^.
^^
'*<
-2
Ct(8+1).
Iterating this process to eliminate from $ all the irreducible representations not
isomorphic to <j>k, we move from HqP(#)+ to E*cP(nk$k)+ =H^Pfc(*)+ and from
Cfc(r) to the expansion of
n (^,- + ^
*J
On the other hand, the image of Cfc(r) under this sequence of transformations must
be qt(Ct(r)).
Now that we have defined the classes c,-,- and C,-(r), we must show that they
generate HqP($) as an algebra over HqS0.
PROPOSITION 6.3. The classes ctj, for fa, ^ € $(1), and the classes C,(r), for
r > 1 and <f> - € $(r + 1) - $(r), generate HqP($)+ as an algebra over RqS°.
PROOF. If $ is infinite, then, by the proof of Theorem 2.6, H£P($) + is the limit of
the HqP(A) where A runs over the finite subsets of $. Thus, it suffices to prove
the result for $ finite. Recall the functions f and g and the subsets $,(r) of $
defined in the remarks preceding Proposition 5.12. For this proof, initialize f and g
by f(0) = 0 and g(0) = 0. We will show, by induction on n, that the classes ctJ- and
C (r) which are defined in HGP($f/n%(g(n))) generate that Mackey functor as an
algebra over HqS0. The result is obvious for n = 1, since $ffl,(g(l)) = {fi0} and

104
P({^o}) is a point. Assume the result for n. Denote af,+1,(g(n+l)) + 7f/„+1) by
a. The boundary map is zero in the cohornology long exact sequence associated to
the cofibre sequence
P(^f(„)(g(n)))+ - P($f(n+1)(g(n+l)))+ -+ S°.
Thus, we have a split short exact sequence
0 - R*GS° -+ H^P($f(n + 1)(g(n+l)))+ ^ H^P($f(n)(g(n)))+ -+ 0.
All of the classes ctj and C;(r) which are defined in H^P($f(n)(g(n))) are also
defined in HQP($ffn+1)(g(n+l)))+. Moreover, 9* takes these classes in
■H.GP($f(n+i)(g(n+1)))+ t0 the corresponding classes in H.GP($f(n)(g(n)))+. Thus> to
generate HGP($f7n+1}(g(n+l))) as an algebra over HqS0, it suffices to add to these
classes the image z of the canonical generator of A(l) =Hg(Sw)(1). Clearly, p(z) is
the generator of H£(P($f(n+1)(g(n+l))) + )(e). Moreover, for k^f(n-fl), qfc(z)=0
since q^ factors through JiQP($f,-)(g(n))) . Finally,
qf(n+1)(z) - - i- \
since the diagram
£a-g(n + l) VXf(n + i:
Pf(n + l)($f(n + l)(g(n+1)))+ ^^ P(^f(n + l)(g(n+l))) +
g9(" + l) e , g«
commutes. The elements z and Df(n+1, Cf/n 1^(g(n+l)) must be equal since they
have the same image under the maps qj. and p.
The equations in Propositions 5.9 and 5.10 describe elements in dimensions
where there is no torsion. As a result, these equations can be checked easily by
applying the maps p and qk to both sides. The equations in Lemma 5.11 are easily
checked using the maps p and qk because the images of the classes k:(r) under the
maps qk are so simple. However, the formula in Proposition 5.12 is more difficult to
verify.
PROOF OF PROPOSITION 5.12. We may assume that |$| >|$.,(r)| + 1$ ,(s)| so

105
that all of the Df(<A< Cf,iA,Ag(\A\)) on the right hand side of the equation are
nonzero. If |$| is too small, then form a sufficiently large set $' by adding enough
copies of 4>0 to $. The proof below applies to $'; the result for $ is obtained using
the cohomology map induced by the inclusion of $ into $'. We show the equality of
the images of the two sides of the equation under the maps p and q^.. Since the map
p preserves products, p(D., C,-(r) D ., C .(s)) is the generator of Hq(P($) )(e) in the
appropriate dimension. The only term on the right hand side of the equation in
Proposition 5.12 which is not in the kernel of p is the summand corresponding to $
regarded as a subset of itself. This term is \ Dffu, Cf(- ,(g(u)) and its image under p
is the generator of Hq(P($) )(e) in the same dimension. Thus, the expressions on
the two sides of the equation have the same image under p.
Let k be an integer with 0 < k < m. If <f>k ¢. $(r + s + l) - $(r + s), then both
sides of the equation vanish under qk . If <j>k £ $(r + s+ 1) - $(r + s), then expand the
polynomial obtained by applying qk to D.;C,-(r)D .;C,-(s). Each term in the
expansion consists of the product of an integer, a power of xk , and an element of the
form €.,( , or ea£ from Hr-S°. We classify these terms according to the factor
pa pa °
from HqS • There is exactly one term with a f ; its integer coefficient is one. There
is exactly one term with an e • its integer coefficient may be zero. This term is
exactly the part of qk which is detected by q*.. There may be any number, including
zero, of terms containing a product e.£ ■ These terms are all torsion elements of
p a
order p.
Expand the polynomial obtained by applying qk to the right hand side of the
equation and observe that the same three types of terms appear. The summand
indexed on ^ regarded as a subset of itself is the only source of a f . It is easy to
see that this £a term exactly matches the corresponding term from the left hand side
of the equation. If i' > k, then the expansion of the image of the right hand side
under qk will contain no e^ term. In this case, q^D.,) is zero and the image of the
left hand side under qk also lacks an ep term. If i' < k, then numerous summands
contribute to the e^ term of the left hand side, but the coefficient of the kk(c + s)
term is explicitly designed to ensure that the ep terms of the expansions of both sides
match. The only problem here is that it is not obvious that the coefficient Afc of
kk(v + s) is an integer. To show that Ak is an integer, it suffices to show that,
modulo p, the image under qk of the left hand side is equal to the image of the part
of the right hand side indexed on the subsets of ¢. Since the e^ f terms are all
torsion of order p and the «^(r + s) summands on the right hand side contribute
nothing to them, proving the equation
q^C^D.,^)) ^ qt(E#d,_4^_4X4Df(U|)Cf(U|)(g(|A|))) mod p
also shows that the e f terms of the two sides agree and so completes the proof of
the proposition.
We prove this equation modulo p by transforming the right hand side into

106
the left. In Theorem 5.5(c), qyt(C,(r)) is described as a sum of two terms when j ^ k.
The second term can be ignored in this transformation process because it vanishes
modulo p. Recall that each \A is a x f°r some virtual representation a. We
accomplish our transformation by writing a as a sum of differences Tj-<f> of
irreducible complex representations. We then rewrite \ = \ as the product of the
a a
elements \ . To see that such a rewriting is justified, recall that if /? and 7 in
i-<t>
RSO(G) are chosen so that the elements below are defined, then in H£(SU)(1)
£ £
P 1
= £
P + 1
i k.
p 7
= 0
£„ K
P 1
p<f
P+1
and
<x a ■
p 7
<x + A /c„ ,
P+i P+i
where A is some integer depending on /3 and 7. Now observe that every summand in
the expansion of q^fD^,^,. Cffi^i|.(g(l A|))) contains either an e or a f . Thus, the
k error terms that might arise in the rewriting of \ as the product of the \
P+i a i — 4>
are killed by the ep and f from qfeP^^C^i^gflAl))).
We perform our transformation of the left hand side in four stages. During
the first three stages, we think of the left hand side as a sum indexed on the subsets
of ^ and work on each summand separately. Therefore, fix a subset A of $ and let
a be the virtual representation such that y = y . Recall that s' and s" are the
a a
number of elements isomorphic to ¢- in A and $.,(r + s) - $.,(r), respectively. Recall
that u = |ty|-l, that the elements of \& are numbered from 0 to u, and that h is a
function from the set {0, 1, ... , u} to the set {0, 1, ... , m} such that the i* element
in $ is isomorphic to ^m- Assume that the elements of \P numbered
JO 1 Jl ! ■ • ■ > JtO ;
with j0 <jj <...<>, are in A and that the elements numbered i0 ,
i1, ... , i„, with i0 < i1 < ... <iv , are in ty - A. For any integers q and t, with
0<q)t<mi abbreviate e and f
Pat Pat~2
a2, and a3 of RSO(G) by qt
by eat and £gt. Define the elements ar,
6f(Ui)
£ tt
tjif(\A\),i,k
Jf(\A\)'
■a
*f(UI),fc
[(r + 8) {¢.J ^f(U|)- $f(\M) 4>k)~]f(lA
)jti,k

107
^'^-^(UD^U
> > f > *
L^^Jfc-Sji^U
l),fc
«2 = (tf/-0f(U|
t>t e*.,(r+s)- *.;(r)
<54f(U|),j,*
[(s-s'-s'O^J1^-^1^)!^,-,* +
[s"^1^-^!)^)]^^
I), It
and
[^^(Uir^cUl)**)]^!)*;.*
a3 = a-aj-aj,
/here
5 =
1, ifi'>f(|A|),
0, otherwise.
In the first stage of our transformation, y is used to convert
d*-^-^MDf(UI)Cf(UI)(g(|A
into the product of
d» ac» .X qt D.,Cf(r)
and

108
g(\A\)-r-d'
n
dkt. , €.., ... + i
*^#f(UI)(g('zil)^*i'(r)
<*f(U|),fc
d*,f(UI)£ f x A9(UI)-r-«
f(UI),t f(UI),t f(ui),fc *
f(UI)**
f(UI),fc
i'>f(UI)>* or •
f(UI)>*>t'
Here, 5 is as in the definition of ar and
(- 1, ifi'>f(|A|), k;
I. 0, otherwise.
In the second stage of the transformation, \ is used to convert this product into
the product of d*_ £*_ \ q;JD.,C,(r) J with the three factors
x,.
n
t>t £ <P./(r + s)- ¢./()-)
d*1 e . + £ x,
j« j« jt L
d^ + v*
i*k
g(\A\)-r-i-6'
ii V f(ui),f(*)£f(Ub,f(o + f(ui),f(<)X*
f(<)^f(Ui'),fc
and
d , , e , , + c Xi.
f(UI),fc f(U|),fc f(U|),jb *•
g(\A\)-r-s-6
K\A\)itk
f(\A\),k
i'>f(UI)>* or
f(\A\)>k>i'
Observe that the d" , e" , factor has been transformed into a dk , <A" , factor.
1° £°
*-A f-A
<t-A <f-A
This is accomplished by the (s - s' - s") (^71 ^fc — ^J1<^o) - summand in a2 . If
k =0, then obviously no such transformation is needed. If j = 0, then there will not
be any elements of ^ isomorphic to ¢,, and the value of d* Af-'L_A will not depend
on k. In the description of the factor above indexed on t, for 0<t<w, and
throughout the third stage of the transformation, the set $f,,- (g(|A|)) - $.,(r + s) is

109
identified with the set {4>f(t) ■ 0<t<w}. By this identification, constructions that
would naturally be indexed on ¢.,, , (g(|A|)) - $./(r+ s) may be indexed on t. The
description of the set {4>f(t) '• 0 < t < w} involves our usual abuse of notation in that,
whenever q ^ t and f(q) = f(t), the representations <f>-fi„, and 4>f/t) are intended to be
distinct, but isomorphic, elements of the set.
The factor
qt D,C,-(r) xj
n
t>t eiytr+s)-* .,(r)
jk ]k jk
j*k
appears in every summand of the transformation of the right hand side of the
equation. We therefore factor it out of the sum and ignore it for the rest of the
transformation. Observe that this factor consists of qj D.,Cj(r) J and that part of
qjD ., C (s)) which is associated with the set $.,(r + s) - $.,(r) when $., (s) is
regarded as the disjoint union of ty and $.,(r + s) - $.,(r). Thus, we must transform
what remains of the sum after this factor is removed into the part of qJD ,, C (s) J
coming from ¢.
In the third stage of the transformation, \ is used to transform the
a3
remaining part of the A summand into
<f-A <t-A
n
d***> e
t=o v iHit) iHh) jMit)
Hit)*}
n
t = 0
Ik
+ L
For the fourth stage of the transformation, consider the subsets A of $ that
contain the last element 4>^ru) of ¢. The summands indexed on A and A- {¢^, }
contain the common factor
Tr1 J(H-t)MH)
}=[ diMU)
t = 0
v-l
v-l
t = 0 3K

110
tu-1
n
Akm c
;_0 ^ iMh) i.h(ii)
iMit)
n
< = 0
d*fV
"y*
+ £
y*
which we have written down using the it and jt numbering of the elements in ¢- A
and A. Each of the two summands contains exactly one term not in this common
factor. If h(u) ^j, then these terms are
,f(u>),h(u) ,k,f(w)
di,h(u) £i,h(«) + djMu)cjMu)
If h(u) = j, then these terms are
y.h(u)
xt = d €
i,h(u) i,h(u)
,h(«)
,f(w),y
:y,*
d . . £
= d
*.y
j.* j,* 'j,jt * y>* y>* y,fc
In either case, the result is independent of A and may be factored out of the sum.
Moreover, this factor is exactly the contribution that <t>Mu-, should make to
qk( D ,,C (s) j when <^hfu-) is regarded as an element of $ .,(s) under the identification
of $ .,(s) with the disjoint union of ^ and $.,(r + s) - $,(r).
The sum that remains after the factor associated to ¢^/^, is removed may be
regarded as one indexed on the subsets A of ¢- {<j>h(u)}- We now pair the summand
indexed on a subset A containing the last element <^(u_d °f ^~{^hfu)} w^ ^e
summand indexed on A-{</>h, } to obtain the factor of qj D ., C -(s) J associated
to 0f,(u-iy R-ePeating this process until the elements of ^ are exhausted, we recover
the part of qJ D ., Cj(s) J associated with ¢.
APPENDIX. Computing R*GS°. Here, we outline the calculation of H^S0. The
computation of the additive structure and, for G = 2/2 or 2/3, the computation of
the multiplicative structure are unpublished work of Stong.
Three cofibre sequences suffice for the computation of the additive structure
of Hq(S ). Recall that ( is the real 1-dimensional sign representation of 2/2. Let 77
be a nontrivial irreducible complex representation of G = 2/p, for any prime p. Let
G + -» Srj+ be the inclusion of an orbit and let Sr;+-» S° and S( + -» S° be the maps
collapsing the unit spheres S77 and S( to the non-basepoint in S°. The cofibre
sequences associated to these maps are
G+->St7+-> EG+
Sr/+
,0 €
s"

111
and
G+ = S(+->S0-^ Sc.
The first step in the computation is obtaining the values of H* Sr;+ and
HqS77+ from the first cofibre sequence.
LEMMA A.l. For any nontrivial irreducible complex representation 77 of G,
if |a| = 0 and la I is even,
if |a| = 0 and la I is odd,
Ha Sr7+ = <^ R, if lal = 1 and |aG| is odd,
if |a| = 1 and |a | is even,
oth
erwise,
if |a| = 0 and |a | is even,
if la| = 0 and la I is odd,
H£St7+ = \L, if lal = 1 and |aG| is odd,
if lal = 1 and la I is even,
otherwise.
PROOF. The next map EG+ -> EG+ in the first cofibre sequence is 1 - g, the
difference of the identity map and the multiplication by g map, for some element g of
G which depends on 77. The homology and cohomology long exact sequences
associated to the first cofibre sequence have the form
...-»Il2G+-»IlSG+-»Il£s»7+->IIa->G+-»II°_,G+-»...
and
... ^ H£-'G+^ ■&£"'G+^ 2gV ^ H£G+^ H£G+^ ... .
The Mackey functor Ha G+ may be identified with the Mackey functor (Ha S°)G
defined in Examples 1.1(f). The difference 1-g may be regarded as a map in B(G).
Under the identification of Ha G+ with (Ha S°)G , the first map in the part of the
homology long exact sequence displayed above becomes the map from (Ha S°)G to
(HGS°)G induced by the map 1 -g in B(G). It follows that the cokernel of the map
(l-g),:IaG+->i!»G+ is the Mackey functor L(HG(S°)(e)) defined in Examples
1.1(e). Similar observations reduce the homology and cohomology long exact
sequences of the first cofibre sequence to the short exact sequences
0 -> L(H?(S°)(e)) ^HaSr?+ -> R(HG_,(S°)(e)) -> 0
and
0 -> L(HG-'(S°)(e)) -+H£St?+ -> R(HG(S°)(e)) -> 0.

112
Since n£(S°)(e)=H|a|(S°;Z), L(Ha(S°)(e)) is zero if |a|^0. If |a| = 0, then
L(Ha(S°)(e)) is L(2) for some action of G on 1. This action is the sign action of 2/2
on Z when p = 2 and a contains an odd number of copies of (; otherwise, the action
is trivial. Similar remarks apply to L(H£_1(S°)(e)), R(H°_,(S°)(e)), and
R(H£(S°)(e)).
Notice the frequency with which HqSt?4" and RaSr]+ vanish. From the
dimension axiom, we also obtain that RqG + = RaG+ = 0 if |a|^0. These
vanishing results determine most of the homological and cohomological behavior of
the maps e in our second and the third cofibre sequences.
LEMMA A.2. Let a e RSO(G).
(a) The map £*: R^S0 = E£(S") -» E£(S°)
f mono for |a| ^ 1, 2,
is < epi for |a| ^0, 1,
I iso for lal^O, 1, 2.
(b) If p = 2, then the map e*: Rq~($° = H£(SC) -» Hg(S°)
{mono for )a| ^ 1,
epi for |a| ^0,
iso for |a| ^0, 1.
The divisibility results involving Euler classes in Lemmas 4.2, 4.6, and 4.8
follow from this lemma. Moreover, from this lemma and the vanishing of HqS0, for
n£ 1 and n ^0, one can derive all of the zeroes in the first and third quadrants of
our standard plot of HJ-S0.
LEMMA A.3. Let a G RSO(G). Then H£S° = 0 if |a| and |aG| are both positive or
both negative.
Lemma A.2 indicates that all of RqS° can be determined from the values of
HgS° for the a in RSO(G) with -2 < |a| < 2. If p = 2, it suffices to know H£S° for
the a in RSO(G) with -1 < |a| < 1. The next lemma describes HqS0 on the edges of
these two ranges of values for |a|.
LEMMA A.4. Let a £ RSO(G) and let rj be any nontrivial irreducible complex
representation of G.
(a) If |a| = 2, then
RaGS° - coker(r: ^6+-^-¾0).

113
(b) If |a| = -2, then
H£S° = keT(p:EaG+r>S°^JlaG+vG+).
(c) If p = 2 and \a\ = 1, then
H£S° = coker(r:IiG"CG+^IiG"Cs0)-
(d) If p = 2 and |a| = -1, then
HGS° = ker(p:I^+CS0^+CG+).
Moreover, in all four cases, Hq(S )(e) = 0.
PROOF. Part (d) follows immediately from the cohomology long exact sequence
associated to the third cofibre sequence. Part (c) follows via duality from the
homology long exact sequence associated to the third cofibre sequence. For part (b),
consider the diagram
0 -» EG$° -> Hg+"S° l RaG+t,Sri+
Ha ^
in which the row is from the cohomology exact sequence of the second cofibre
sequence and the vertical arrow comes from the inclusion of an orbit G into Srj.
Clearly, HqS0 = ker f. By our computation of B.GSrj+, the map h is mono, so
ker f = ker hf. The composite hf is just p. The proof for part (a) is similar, but
uses the homology long exact sequence to describe H-»S as the cokernel of the map
H,,_aG+ -»£,)-aS° induced by the collapse map G+ -> S . Dualizing the homology
Mackey functors to cohomology Mackey functors gives the result since the transfer is
the dual of the collapse map. In all four cases, the group RG(S )(e) is zero either
because r(e) is surjective or because p(e) is injective.
Most of the values of RGS° for |a| = 0 and la I ^0 follow immediately from
the cohomology long exact sequence of the second cofibre sequence and Lemmas A.l
and A. 3.
LEMMA A.5. Let a e RSO(G) with |a| = 0. Then
if |aG| < -2 and |aG| is even,
if |aG| < -1 and \aG\ is odd,
TTOrqO _ /
I L, if |c*G| > 2 and |aG| is even,
if |aG| > 3 and |aG| is odd.

114
PROOF. Let t] be any nontrivial irreducible complex representation. If la I < 0,
then consider the portion
iLr"S° = JigS" -» H£S° -» £^+ -» H£+1s" = Hg+1-"S°
of the cohomology long exact sequence of the second cofibre sequence. The left hand
term is zero by Lemma A.3 and the right hand term is zero by the same lemma
unless |aG| is -1. If |aG| = -1, then p = 2, a = ( - 1, H£St?+ is R_ by Lemma A.l,
and HG+ S° is (Z) by Lemma A.4. The last identification is based on the
observations that rj must be 2( and HGS° is A. By inspection, there are no
nontrivial maps from R. to (I). Thus, if |a | < 0, the middle arrow must be an
isomorphism.
If |a | > 2, then consider the portion
H^-V -> Hg+""1S»?+ -> HG+"S" =H£S° -> HG+"S°
of the cohomology long exact sequence for the second cofibre sequence. The left and
right hand terms in this portion of the sequence must be zero by Lemma A.3.
Therefore, the middle arrow is an isomorphism.
If p = 2, then the results above reduce the computation of HGS to the
determination of HGS°, which is A by the dimension axiom, and HG S , which is
given by the following lemma.
LEMMA A.6. If p = 2, then HG~CS° = R..
PROOF. Consider the portion
H°GS° -> H°GG+ -> HGSC=HG-CS° _ HiGs°
of the cohomology long exact sequence of the third cofibre sequence. By the
dimension axiom, the right hand term is zero and the first two terms from the left
are A and AG, respectively. The value ofHG S° follows by computation.
If p ^ 2, then we must still determine the value of HGS° when la| = ±1 or
a G RSO0(G). The next three lemmas dispose of the a with la| = ±1 which are not
already covered by Lemma A.3.
LEMMA A.7. Let M be a Mackey functor and f:L-»M be a map. If f(e) is a
monomorphism, then so is f.
PROOF. The composite f(e) p is a monomorphism and pf(l) = f(e) p.

115
LEMMA A.8. If p ± 2, a e RSO(G), |a| = 1, and |aG| < 0, then H£S° = 0.
PROOF. Consider the portion
Hg""S° = Eac$v -» HgS° -» Eac$r,+ I Rac+lS" = R^S0
of the cohomology long exact sequence associated to the second cofibre sequence. The
left hand term must be zero by Lemma A.3. By Lemma A.l, HqSt;+ = L. Since
|a +1-77(=0, HG+1~"(S°)(e) is 2. The map f: H£Sr?+ -» Hq+1~"S° is induced by
the geometric map S^ -> ESr;+ which identifies the points 0 and oo in S*. From this
description, it follows that f(e) is an isomorphism. By the lemma above, f is a
monomorphism. Therefore, HqS must be zero.
LEMMA A.9. Assume that p ^2, a £ RSO(G), la| = -1, and |aG|>0. Then for
any nontrivial irreducible complex representation 77,
iiGS = coker (HG S -t£G St? ).
Moreover, if la I > 1,
HGS° - (Z/p).
PROOF. Consider the portion
Hg+"-1S0 ^ HG+"_1S7?+ -» HG+"S" =H£S° -» RaG+I>S°
of the cohomology long exact sequence for the second cofibre sequence. The right
hand term must be zero by Lemma A.3. The first part of the lemma follows
immediately. By Lemma A.l, HG ^- Srj+ = R. The map h is induced by the
collapse map Srj+ -> S°. Since |a + 77 - l| = 0,
The map h(e) is an isomorphism by an obvious computation in nonequivariant
cohomology. If |aG| > 1, then by Lemma A.5, HG+,?_ S° = L. The only two maps h
from L to R with h(e) an isomorphism have cokernel (2/p).
If d ^ 0 mod p, then the only maps h : A[d] -> R with h(e) an isomorphism
are surjective. Therefore, once we have shown that RCS° is Afd^] when
P € RSO0(G), it will follow from the lemma above that HGS = 0 when |a| = -1 and
UG| = i.
Lemma 4.6 follows from Lemma A.9.
PROOF OF LEMMA 4.6. Let a and /? be elements of RSO(G) with |a| = -1,
]aG|>0, |/?| = 0, and |/?G|<0. Let 77 be a nontrivial irreducible complex

116
representation. Consider the diagram
R = Hg+"-1S° -> H£S° - 0
1 1
R£flaG+"+,"1S11 -» Hg+/?S° -» 0
in which the vertical arrows are given by multiplication by f ^ or /i^ . The rows of
this diagram are exact by the proof of Lemma A.9. Let y € HG (S )(1) be a
generator and let x G H£(S°)(1) be its image. Since p preserves products, p(£py)
must be a generator. Thus, f^y must be a generator and so must ^x. Similarly,
p(ppy) is dp times a generator, so p^y is dp times a generator. It follows that ppX
is a generator. This proves Lemma 4.6 in the special case where |a| = -1 and
|aG| > 0. The general case follows from the special case and Lemma A.2.
Let a be an element of RSO0(G). The main difficulty in identifying H^S
with A[da] is that we must select a representative for a in R0(G) in order to define
pa and da- To circumvent this difficulty, we work primarily with elements of R0(G)
instead of elements of RSO0(G) in the remainder of our discussion of the additive
structure of HqS0. If a is in R0(G), we write H£S° for the cohomology Mackey
functor associated to the image of a in RSO(G). To work with elements of R0(G),
we must introduce variants of Definitions 4.5(a) and 4.5(d).
DEFINITION A.10. Observe that the procedure used to produce the element pa in
Definitions 4.5(a) actually associates a map p: S ''-»S 'to any element Y^^i^Vi
of R0(G). If a is a nonzero element of R0(G), denote this map, and its image in
H£(S°)(1), by pa. Let p0 denote the identity map of S° and 1 € Hq(S°)(1). If <t> is
a nontrivial irreducible complex representation, then let ea ^: S ^ -* S l denote
the smash product of the map e: S° -> S and the map pa ■ We also use ea ^ to
denote the corresponding element in Hq (S°)(l).
If a and /3 are elements in R0(G) which represent the same element in
RSO0(G), then pa and pp need not be the same class in Rq(S )(1). However, the
class ta^ in Hq (S°)(l) is uniquely determined by the sum a + <fi in RSO(G). This
uniqueness can be exploited to resolve the problems caused by dependence of pa on
a.
LEMMA A.11. Let a and /3 be in R0(G) and let <j> and rj be nontrivial irreducible
complex representations such that a + <j> and /3 + 7? represent the same element in
RSO(G). Then the cohomology classes ea ^ and e^ in H.q (S°)(l) are equal.

117
PROOF. We establish the result for three special cases and then argue that the
general case follows from them. Let 77, 77^ rj2, <f>, ^, and 4>2 t>e nontrivial
irreducible complex representations and let c: S 1 2-»S 2 1 be the switch map.
Regard al = <f>l - 77, a2 = <j>2 ~ *?) and a = ¢1 + ¢2 ^2 77 as elements of R0(G). Let
e: S° -> S^ be the usual Euler class. The two maps 1 a £ and £ a 1 from S to S are
obviously equivariantly nomotopic. On the level of maps,
^2,^ = ^(^ ^ 1) and £ai>2 = c/ia(lA£).
Therefore, £^^ and c£„ll?i2 are equivariantly homotopic. Thus, £^,^ and eai>^2,
regarded as cohomology classes, are equal. Here, the map c is, of course, absorbed in
the passage to an RSO(G)-grading for HqS .
If 77 and (f>l are equal and e': S -► S 2 is the inclusion, then the trick used
above can also be used to show that 1 a e': S -> S l is equivariantly homotopic to
ea2,^ • Thus> if a3 = <t>i ~ <t>i € R0(G), then e' and £^^ are equal in HG2(S°)(1).
Regard /?i = (^1 - ^0 + (<^2 - ^2) and /?2 = (¢1-¾) + (¢2 ~ ^l) ^ elements
of R0(G). By three applications of the result just proved for ea , and ea ^ , it is
possible to show that e^ ^ and e» ^ are equal in Hq1 + (S )(1)-
If a and /3 are in R0(G) and </> and 77 are nontrivial irreducible complex
representations such that a + <f> and /3 + r/ represent the same element in RSO(G),
then we can convert the pair (a, <j>) into the pair (/3, 77) by some combination of the
three basic transformations for which the lemma has already been proved. Thus,
ea ^ and tp must be equal in Hq+ (S°)(l).
This lemma establishes that the element €„ of Definition 4.5(d) does not
depend on the choice of a and V used in its definition.
LEMMA A.12. If a € RSO0(G), then H£S0 = A[da]. Moreover, if 77 is any
nontrivial irreducible complex representation, then \xa is the unique element of
H§(S )(1) such that ev p.a = £a+r, and p(na) = da ta ■
PROOF. Recall the map s: RSO0(G) -> R0(G) introduced in section 2. Let
n
a € RSO0(G) and assume that s(a) = Y^(f>i^Tli- Let a0 be 0 £ R-o(G) and, for
»' = !

118
l<k<n, let. ak be the element £ ^i - 77 ,■ of R0(G). Denote by d(afc) the integer
» = 1
associated to ak by our homomorphism from R0(G) to 1. For 0 < k < n, let f3k be
the element ak + 4>k+i of RSO(G). We will show by induction on k that
i) E.QkS° is isomorphic to A[d(afc)],
ii) Jiak and r(tak) generate HG*(S°)(1),
ill) Hq S° is isomorphic to (2), and
iv) €^ generates RGkS .
By the dimension axiom and Lemma A.4, these statements are true for
k = 0. Consider the portion
flSt_1(S^+i)+ - H^S"*+1 = RGk+iS° -, RGkS° -» H^(Sr?fc+1) +
of the cohomology long exact sequence of the second cofibre sequence. By
Lemma A.l, The left hand term is isomorphic to L and the right hand term is zero.
By Lemma A.7, the left hand arrow is a monomorphism. Thus, we have a short
exact sequence
Assume that the assertions above hold for some integer k. The element nk+l in
Hg*+1(S°)(1) hits the generator ep in RGk{SQ){\) by Lemma A.ll. Since f(e) is an
isomorphism, we may assume that f(e) takes the generator 1 € 1 = L(e) to the
generator iak+l of KGk+l(S°)(e). It follows that, l>-ak+l and r(iak+l) generate
Hg*+1(S°)(1). Since
P(^«fc+i)=d(ayt+1)i„yt+1 and p r(iak+1) =piak+l,
KGk+l$° is isomorphic to A[d(afc+1)]. By Lemma A.4, RGk+lS° is isomorphic to {I)
and is generated by e„ . Since Jian = l^a and d(an) = d», HqS is isomorphic to
A[d0]. ^
Replacing ak+l by a. rjk+l by 17, and (3k by a + 77 in the cohomology long
exact sequence above, we obtain the short exact sequence
0 - L - RaGS° h Hg+"S° -» 0.
Our characterization of \xa in terms of eT, fj,a = h(/ia) and p{jJ.a) follows directly from
this sequence.
Two general observations suffice for the proofs of many of the multiplicative

119
relations. Any product involving at least one element in the image of the transfer
map t is easily computed using the Frobenius property
xi-(y) = r(p(x)y).
Any relation involving an element, like e~mn, obtained by divided some other
element by an Euler class may be checked by eliminating the division by the Euler
class and checking the resulting relation. The original relation then follows by
Lemma A.2.
PROOF OF THEOREM 4.1. We will describe the individual Mackey functors H£S°
of HqS by their positions in our standard plot of HqS • Since
Hq(S )(e) =H a (S ;2), it is easy to check that the elements tl_, and i^_1 generate
■H<3(-3°)(e) and satisfy no relations in HJ3(S°)(e) other than the obvious relation
1-1-(^(-1 = /°(1)> 1^ follows immediately from the structure of the Mackey functors
R_, L, and L. that the elements r(t"_A for n > 1, generate the part of Hq(S )(1) on
the positive horizontal axis. For any positive integer n, p(£n) = t?2i • Therefore, £n
must generate 11^^(8^(1). The relation r(i^_,) = 2 £m follows from the additive
structure. No other relations involving only £ and i,_1 are permitted by the additive
structure. Lemmas A.2 and A.4 ensure that the powers of e generate the part of
Hq(S )(1) on the positive vertical axis. These two lemmas also indicate that the
elements em £n, for m, n > 1, generate the part of Hq(S )(1) in the second quadrant.
The same two lemmas indicate that the elements (Tm k and the elements
€_mr(i1™^ ) generate the parts of Hq(S°)(1) on the negative vertical axis and in the
fourth quadrant, respectively. The relations not already verifed follow easily from the
additive structure of HqS0 or from our general observations. The additive structure
of HqS eliminates the possibility of any unlisted relations involving a single element.
Since we have described every possible nonzero product of a pair of generators in
terms of the generators, no further relations involving products are possible.
PROOF OF THEOREM 4.9. Again, we describe the individual Mackey functors
H£S° in terms of their positions in our plot of R*GS°. Since E£(S°)(e) = H|a|(S°; Z),
it is easy to check that the relation ia i„ = t,a+p holds for any a, /? € RSO(G) with
\a\ = |/?| = 0 and that no other relations in HJ3(S°)(e) hold among the ia ■ Therefore,
for any /3 € RSO(G) with |/?| = 0, t» can be written as a product of the ca included
in the proposed list of generators of HqS0. The elements t» , for 0 € RSO(G) with
\/3\ = 0, generate HQ(S°)(e) and the elements r(ip), for /3 € RSO(G) with \/3\ = 0 and
|/3 I > 0, generate the part of Hq(S )(1) on the positive horizontal axis.
Let a and /3 be in RSO0(G) and let y be an element of RSO(G) such that
|-7| > 0 and |-yG| = 0. The relation /ia£7 = £a+7 follows from Lemma A.11. The
relation

120
P»Pp = Pa+p + [(dad/3-da+/3)/p]r(ia+/?)
follows from our characterization in Lemma A. 12 of Pa+g as an element of
H^+/?(S°)(1). From this relation, it follows that all of the elements pa can be
constructed from the pg and t» in our proposed list of generators. By Lemma A. 12,
the elements pa and ia generate all of the HqS0 which are plotted at the origin. The
relation pa €7 = eff+7 indicates that we can construct all the elements £7 from our
proposed list of generators. By Lemmas A.2 and A.4, these elements generate all of
the HqS on the positive vertical axis.
Let a e RSO0(G) and 0,j € RSO(G) with \/3\ = \y\ = 0 and |/3G|, |-yG| < 0.
The element aa can be obtained from pa and ia. The relations
P(.P<* ip) = d« l-a + p = P(d« Za + p),
p(a°tp) = l-a+p = p(ta+p),
and
P(tpti) = '-p+-< = p(tp+-r)
follow from the fact that p is a ring homomorphism. They imply the relations
Ha£p=da£a+p, <7a£p = £a+p, and £^ = ^,^ since p is a monomorphism in
dimensions a + /3 and /3 + 7- These relations indicate that all of the elements £^ can
be produced from our proposed list of generators. These elements generate the part
of HqS0 on the negative horizontal axis. By Lemmas A.2 and A.4, the elements
e6 £g generate the part of HqS in the second quadrant.
The relations /i7 (¢^1 kq) = e^1 Ka+f} and eZl Ka = e^1 k6 , for a + y = /3 + 5,
may be checked by our general procedure for relations involving division by an Euler
class. Together, these relations indicate that our proposed set of generators suffices to
construct all of the elements e^1 Ka and therefore to generate the part of HqS on the
negative vertical axis.
Let /3 € RSO0(G) and let a £ RSO(G) with |a| < 0 and |aG| > 0. Recall the
class va and the virtual representation <a> from Definitions 4.7. By definition,
<a + j3> = <a>, and by the Frobenius relation, v<a> T{i-a+p) = 0- Therefore,
Pf} Va = Pp Pa-<a> v<a>
= lia + p-<a> l/<a>
= "a+p-
This relation indicates that our proposed set of generators suffices to produce all of
the elements va and therefore the part of HqS in the fourth quadrant.
We have now shown that our proposed set of generators does generate HqS°.
Seven of the relations we have not already established deserve comments. The
relation ea e« = €a+p follows easily from the definition of the Euler classes, the
Frobenius relation and the product relation for the classes p1. The relation
tp (,a = ds_a €7 £s, for a + /3 = y + 8, follows from the sequence of equations

121
— (-1 /J-l-a $<*
= ds-aeiis-
The relations Ka k6 = p Ka+6 and k1 va = 0 can be confirmed from the definitions,
the Frobenius property, and the relations which have already been established. Given
these equations, the relations
£7 (£p Ka) = £/J_7 "oi
and
(e^ /c7)i/a = 0
follow from our general procedure for checking relations involving classes divided by
Euler classes. For the relations c^ va — v'a+p and £^ va — d<f}>-f} va+p > observe
that £» can be written as Cg-<g> (,<g> and that e» can be written as /ix7enA, for
some 7 € RSO0(G) and some positive integer n. The relations now follow by
straightforward computations using the definitions, the Frobenius property, and the
previously established relations. All of the remaining relations in the theorem follow
directly from the definitions or the additive structure of HqS0. The additive
structure of HqS eliminates the possibility of any unlisted relations involving a
single element. Since we have described every possible nonzero product of a pair of
generators in terms of the generators, no further relations involving products are
possible.

122
REFERENCES
[tDP] T. torn Dieck and T. Petrie, Geometric modules over the Burnside ring.
Inventiones Math. 47 (1978), 273-287.
[DRE] A. Dress, Contributions to the theory of induced representations. Springer
Lecture Notes in Mathematics, vol. 342, 1973, 183-240.
[ILL] S. Illman, Equivariant singular homology and cohomology I. Memoirs
Amer. Math. Soc. vol. 156, 1975.
[LEI] L. G. Lewis, Jr., The equivariant Hurewicz map. Preprint.
[LE2] L. G. Lewis, Jr. An introduction to Mackey functors (in preparation).
[LMM] L. G. Lewis, Jr., J. P. May, and J. E. McClure, Ordinary RO(G)-graded
cohomology. Bull. Amer. Math. Soc. 4 (1981), 208-212.
[LMSM] L. G. Lewis, Jr., J. P. May, and M. Steinberger (with contributions by J. E.
McClure). Equivariant stable homotopy theory. Springer Lecture Notes in
Mathematics, vol. 1213, 1986.
[LIN] H. Lindner, A remark on Mackey functors. Manuscripta Math. 18 (1976),
273-278.
[LIU] A. Liulevicius, Characters do not lie. Transformation Groups. London
Math. Soc. Lecture Notes Series, vol. 26, 1976, 139-146.
[MAT] T. Matumoto, On G-CW complexes and a theorem of J. H. C. Whitehead.
J. Fac. Sci. Univ. Tokyo 18 (1971/72), 363-374.
[WIR] K. Wirthmuller, Equivariant homology and duality. Manuscripta Math. 11
(1974), 373-390.

THE EQUIVARIANT DEGREE
by
Wolfgang Luck
0. Introduction
Abstract. In this paper we study the possible values deg f ,
H c G for a G-map f : M -» N if M and N are compact smooth G-
manifolds and G a compact Lie group. We generalize results about
maps between spheres of G-representations. We give applications
to one-fixed point actions and G-surgery. We prove that the
unstable H-homotopy type of the sphere of the H-normal slice
H H
S Vs (M ,M) for x £ M is a G-homotopy invariant of M.
Survey. As an illustration we state a consequence of our main
result in a very special situation where it is easy to formulate.
Let G be finite. Consider a compact smooth G-manifold M such that
M is non-empty, connected and orientable for all H <= G. Assume
either that G is nilpotent or that dim MH > dim MK - 2 holds for
H c K, H, K € iso(M) = {G I x £ M}. Here and elsewhere G denotes
t X
the isotropy group {g £ G I gx = x} of x £ M. The set of finite G-
sets S with Iso(S) <= iso(M) is an abelian semi-group under disjoint
union. Let A(G,Iso(M)) be its Grothendieck group. The cartesian
product induces the structure of a commutative ring with unit on it.
Let Con(G) be the set of conjugacy classes of subgroups of G and
C(G) be the ring n a. Then A(G,Iso(M)) is a subring of C(G)
Con(G)
by identifying S with (card S I (H) £ Con(G)). For a G-selfmap
f : M -» M define DEG(f) £ C(G) by (deg fH I (H) £ Con(G)).
Theorem A.
a) DEG(f) £ A(G,Iso(M) <= C(G).
b) If H c G is a p-group then:
deg f a deg f mod p.

124
c) If G has odd order and deg f £ {±1} for each H c G, then we
have for all H <= G:
deg f = deg f . d
This theorem is well known for M as the one-point compactification
c c
V of a G-representation V. The proof for V uses the equivariant
Lefschetz index and Smith theory. These methods do not suffice
for M a G-manifold. Our main tool is quasi-transversality and the
notion of a local degree.
The notion of the degree is used to classify G-homotopy classes
of G-maps f : VC -» WC (see torn Dieck [6'], p. 213, Laitinen [14],
Tornehave [21]) and G-homotopy types of G-homotopy representations
(see torn Dieck-Petrie [8]). It plays also a role in equivariant
surgery theory (see for example Dovermann-Petrie [11], Lilck-
Madsen [17]). we give a survey over the various sections.
In section one we define the fibre transport tp of the tangent
bundle of a G-manifold and the notion of an 0(G)-transformation
*
ip : f tp -» tp for a G-map f : M -» N. Roughly speaking, ip
assigns to each point x in M a G-map (not necessarily a G -homo-
topy equivalence) TN? -» TMC such that certain compatibility
conditions hold. Using (p we get a one-to-one-correspondence between
H H
local orientations of M at x and N at fx for each H <= G and
u
x £ M . This enables us to define the equivariant degree
DEG(f,(p) e C(N) = rr n a in section two.
(H) £ Con(G) it (N )/WH
In section three the Burnsidering A(G,3r ) of a compact Lie group
with respect to a family? is treated. We identify [VC,VC] and

125
A(G,lso(V)) for a G-representation V. We introduce in section
four a multiplicative submonoid End^ c C(N) and prove
DEG(f,(p) £ End. for any f and ip in section five. We will see
tPN
that End. does not involve f and ip but depends only on the coropo-
PN
nent structure of N. The main idea of the proof is best explained
in the special case where G is finite and all N are non-empty
and connected. Then C(N) = C(G) n a and End. is A(G,Iso(N))
Con (G) PN
C -1
Choose y in N and make f quasi-transverse to y. Then f (y) is
-1
finite and for each x £ f (y) f looks like a (not necessarily
linear) norm-preserving G -map TM -» TNinaG -neighbourhood
-1
of x. Consider a G-orbit c of f (y). For each x in c we obtain
G -maps TM -» TNC by f and TNC -» TMC by (p. Their composition
TN°, -» TN°, defines an element in A(G ,Iso(TN )). Its image in
A(G,lso(N)) under the induction homomorphism for G c G is
independent of the choice of x and denoted by d(c). Let d be the sum
Id(c) running over c £ f (y)/G. Since the global degree can be
computed by local degrees, DEG(f,<p) £ C(G) is just d £ A(G,Iso(N)).
Roughly speaking, we have counted the local degrees orbitwise in
the Burnside ring to get the global degree.
Section six contains some examples to illustrate our results. We
give an elementary proof of the following known statement (see
Atiyah-Bott [1], Browder [4], Ewing-Stong [12]).
Corollary B. There is no closed G-manifold M with dim M > 1 such
H P
that each M is connected and orientable and M a single point
if G is the product of a p-group and a torus, n

126
It is of special interest to choose ip : f tp -» tp as an
0(G)-equivalence i. e. all Tn| -» TMC are G -homotopy
equivalences. Then another choice of <p would change the equivariant
degree only by a unit. Moreover, we have:
Theorem C. A normal G-rna£ f : M -» N can be changed into a G-
homotopy equivalence by equivariant surgery only if there is an
*
0 (G) -equivalence ip : tp -» f tpM with DEG(f,ip) = 1. □
The existence of an 0(G)-equivalence <p is related to the notion
of the equivariant first Stiefel Whitney class w of a G-mani-
fold. In section seven we relate tp und w„ and show that the
M M
*
existence of an 0 (G)-equivalence ip : f tp -» tp is equivalent
*
to f w = wM. We prove:
Theorem D. If f : M -» N is a G-homotopy equivalence we have
*
N M
This implies the unstable version of the stable result in Kawakubo
[13].
Corollary E. If f : M -» N is a G-homotopy equivalence, we get
for x £ M:
TMC <* TN^ . D
x G fx
x
Our setting and proofs would be much simpler if we supposed that
all fixed point sets are non-empty, connected and orientable.
Unfortunately, such conditions are unrealistic in the study of G-
manifolds. Hence we make no assumptions about the existence of

127
G-fixed points or about the connectivity or orientability of the
fixed point sets and do not demand it (f ) being bijective.
Our notion of the equivariant degree using 0(G)-transformations
has some advantages compared with the one using fundamental
classes. It is in this generality much easier to state
elementary properties like bordism invariance or the computation by
local degrees in our language. We have the global choice of ip
instead of the various choices of fundamental classes [M ] and
H H
[N ]. Notice that the choice of [M ] is independent of the one
of [MK] for (K) * (H) and [NK]. Hence in the case of fundamental
classes the interaction between the various fixed point sets are
not taken into account, what is done in our setting. It seems to
be difficult, or even impossible, to state some of our results
by means of fundamental classes. For example, the statement of
example 6.5 makes no sense if it is formulated with fundamental
classes and in example 6.3 there must appear signs because we
H H
can substitute [M ] by -[M ] and thus change the corresponding
degree by a sign. The advantages of our approach for the notion
of an equivariant normal map is worked out in Ltick-Madsen [17],
(see also theorem C above and example 2.8).
Conventions: We denote by G a compact Lie group unless it
explicitly is stated differently. Subgroups are assumed to be
closed. A G-representation is always real, A G-manifold M is a
compact smooth G-manifold with smooth G-action and possibly
non-empty boundary. We call a component C of M an isotropy
component if there is a x in C with isotropy group G = H. We
say that M fullfills condition (*) if it satisfies the conditions

128
i) and ii) or the conditions i) and iii) below.
i) C * {point} for all C £ i (MH) , H <= G.
o
H >H
ii) If C £ tr (M ) is an isotropy component, C is
>H H
{x e C | G 1 H} and H c G we have dim C + 2 < dim C •
iii) G is finite and nilpotent.
A G-map f : M -» N respects always the boundary and we assume
dim C = dim D for all C £ M (MH) , D £ M (NH) , H <= G with fH(C) <= D.
Acknowledgement. The author wishes to thank the topologists at
Arhus for their hospitality and support during 1985 - 1986 when
the main part of this paper was written. The author is indebted
to lb Madsen and Erkki Laitinen for their useful comments.
1. The fibre transport.
We organize the book-keeping of the components of the various fixed
point sets and their fundamental groups for a G-space as follows.
We recall that an object of the fundamental groupoid n (¥) of a
space y is a point in y and a morphism y -» y. is a homotopy
class of paths from y. to y , The orbit category 0(G) has the
homogenous spaces G/H as objects and G-maps as morphisms.
Definition 1.1. The fundamental 0(G)-groupoid n X of a G-space X
is the contravariant functor n X : O(G) -» {groupoids} sending
G/H to rr(xH) = n(map(G/H,x)G) . o
In general an 0(G)-category resp. 0(G)-groupoid is a contravariant

129
functor from 0(G) into the category of small categories resp.
groupoids. We recall that a groupoid is a category whose mor-
phisms are all isomorphisms. An 0(G)-functor F : C -» D between
0(G)-categories is a natural transformation. Let I be the
category of two objects 0 and 1 and three morphisms ID : 0 -» 0,
ID : 1 -» 1 and u : 0 -» 1 . We define an 0(G) -transformation
(P : F -» F between 0 (G)-functors F and F1 : C -» D as an
0(G)-functor ip : C>I -» D with C I i = F.. Given a second 0(G)-
transformation ¢: F. -» F., let the composition \fi » ip : FQ -» F2
be determined by i>-=- ip(id,u) = i);(id,u)» ip(id,u) : (x,0) -» (x,1)
for all x £ C. One should think of an 0(G)-functor F : C -» D as
a collection of functors F(G/H) : C(G/H) -» D(G/H) and of an
0(G)-transformation ip : F -» F as a collection of natural
transformations <p(G/H) : F (G/H) -> F (G/H) fitting nicely together.
An 0(G)-transformation ip : F -» F is called an 0(G) -equivalence
if there is an 0(G)-transformation i> : F^ -» F with both
compositions the identity.
c c c
A G-map f : X -» Y induces an 0(G)-functor n f : T7 X -» TT Y
whereas a G-homotopy h : X«I -» Y between f and g determines an
0(G)-equivalence n f -» n g.
A G-S -Hurewicz-fibration n-S-X is called locally linear if there
exists a G -neighbourhood U for each x in X such that U is G -
XX XX
fibre homotopy equivalent to U *SV for some G -representation
V . we call a locally linear G-Sn-Hurewicz-fibration briefly a
a G-S -fibration. An example is the fibrewise one-point compacti-
fication £c of a G-]Rn-bundle £. Denote by bfG (X) the category

130
of G-S -fibrations over X with G-fibre homotopy classes of
fibrewise G-maps as morphisms. We obtain an 0(G)-category bf
by letting X vary over all homogenous spaces. One should notice
that bfG (G/H) is equivalent to the category with spheres of H-
representations and H-homotopy classes of H-maps as morphisms.
We prefer bf_ (G/H) because of its better transformation be-
G , n
haviour in view of 0(G).
The fibre transport of a G-S -fibration n^X defines an 0(G)-
functor tp : n X -» bf analogously to the non-equivariant
case (see [19], p. 343). The functor tp(G/H) : n(XH) -» bfQ n<G/H)
H *
sends a point in X given by x : G/H -» X to x n. Let
h : G/H " I -» X be a G-homotopy from y to x representing a raor-
phism x -» y. Choose a solution "H of the G-homotopy lifting
problem
X T\ X 1
n
x n x I
-» x
hc (p x id)
Define x n
y n by the pull-back property and h
Definition 1.2. We call tp : it M -» bf„ the fibre transport
of n+X. The fibre transport tpM of a G-manifold M is tp c.
2. The equivariant degree.
We consider a G-map f : M -» N between G-manifolds an an 0(G)

131
tp„° ii f. We want
N
transformation ip ■, f tp, -» tp.. with f tp
N M N
to define its equivariant degree DEG(f,(p) lying in a certain ring
C(N).
We consider the case G = 1 and both M and N connected first.
Recall that we always assume dim M = dim N. Suppose that ip(x):
c c
TN -» TM is not nullhomotopic for one (and hence all) x £ M.
Otherwise define DEG(f,ip) £ TL to be zero. Let u be any loop in M
at x. By functoriality of ip we get (p(x)o tp (f= u) ^ tp (u) o (p(x) .
Since the first Stiefel-Whitney class w (M) £ H1 (M,E/2) = HOM(n1 (M) ,2/2)
sends u to deg tp (u) we have f w. (N) = w (M) . Let p : ft -» M be
the orientation covering if w.(M) is non-trivial and the identity
M = M -» M otherwise and define p : R -» N analogously. Then M
and N are orientable connected manifolds and we can choose a lift
f : M -» N. If f(M) <= 3N let DEG(f,(p) be zero. Otherwise choose a
^ ^ ^ ^^ ^ ^ ~ ~ ~c
point x £ M-OM with fx £ N-ON. Write x = px. Let c : M -» TM
and c : N -» TNf be the collaps maps uniquely determined up to
homotopy by the property that the differentials at x and fx are the
identity. Let d be the degree of the following endomorphism of S.
E = H (M,.3M)
-^ Hn(N,3N)
H (TM*)
n x
(Tpft)*
H (TMU) /-
n x' ~
<p(x).
H (TN§^)
n fx
W
<TP&> „
H (TIC )
n fx
A straightforward calculation shows that d is independent of the

132
choices of f and x. Now define DEG(f,ip) as 2d if w1 (M) = 0 and
w (N) * 0, and as d otherwise. The factor 2 in the case w1(M) = 0
and w (N) * 0 is due to the fact that then M is only one of the
two components of the pullback of the orientation covering of N.
The global degree has an easy description by local degrees. Let y
-1
be a point in N-ON. Assume that f (y) is finite and f looks in
a neighbourhood of x like a proper map k(x) : (TMx,0) -» (TN ,0)
-1
with k(x) (0) = 0 if we identify the tangent space with
neighbourhoods by an exponential map. Then:
Proposition 2.1
DEG(f,(p) = I , deg(k(x)Co (p(x)C : TN^ - TN^)
x e f (y) Y Y
Proof. Use [9], p. 267. o
As an illustration consider the example of a n-fold covering
p : M -» N between connected manifolds. Its differential induces
*
an 0(1)-transformation (p : p tp -» tPM- BY proposition 2.1
DEG(p,(p ) is n. This applies in particular to p : S m -» KP
Notice that S2m is orientable but KP2m not.
Now we treat the general case. Let Con(G) be the set of conjugacy
classes of subgroups of G. The set of isomorphism classes x of
objects x in a category C is denoted by C. Given an 0(G)-groupoid
"% , we write CON (6,) for n «6, (G/H)/WH and Cpft for the
0 6 (H) eCon(G) * *
ring of functions CON (¾.) -» TL. Let CON(X) and C(X) be C0N(n X)
p
and C(TT x) for a G-space X.

133
We will define DEG(f,<p) in C(N) by specifying integers
DEG(f,(p) (D,H) for all H <= G and D c ttq(NH). Let C.,,...,^ be the
H H
components of M with f (C.) <= D and f. : C. -» D be the map in-
H H H H
duced by f . Because of (TM I M ) = T (M ) we obtain from (p non-
*
equivariant transformations ip. : f, tp -» tp by restriction
and taking the H-fixed point sets. We have introduced DEG(f.,(p.)
above. Define:
DEG(f,(p) (D,H) = I DEG(f.,<p. )
i=1
The sum shall be zero for r = 0.
Definition 2.2. We call DEG(f,<p) in C(N) the equivariant degree
of f with respect to (p. □
Finally we state the elementary properties. Consider a G-map of
triads (F,f,f+) : (P,M,M+) - (q,n,N+) and O(G)-transformations
<P : f tpN -» tp and $ : F tp -» tp . Identifying TPx = TM^ 1R
using the inward normal we get tp I M = tPfrMffiTR')c and anal°g°uslY
tP~!N= t ,m„„^>c. We assume that * I N and ip fit together under
U p(TNS>iK)
these identifications
Let j : C(Q) -» C(N) be the ring homomorphism
given by composition with the obvious map CON(N) -» CON(Q). Then
the equivariant degree turns out to be a bordism invariant.
2.3 DEG(f,(p) = j*"pEG(F,*)
The equivariant degree is a honotopy invariant in the following
sense. Given a G-homotopy h : Hxi -» N between f and g we get

134
an 0(G)-equivalence *h : g tp -» f tpN by the fibre transport.
Then:
2.4 DEG(f,(p) = DEG(g,(p«= ^)
Consider G-maps f : L -» M and g : M -» N and 0 (G) -equivalences
<p : f tp -» tp and ty : g tp -» tp . Provided that
it (gH) : it (MH) -» it (NH) is bijective for all H c G, we obtain
the composition formula;
2.5 DEG(g«s f,ips f%) = DEG(g,*) • (g ) 1(DEG(f,cp))
The following examples illustrate our definitions.
Example 2.6. Let f : VC -» WC be a G-map for two G-representations
V and W with dim V , dim W > 1 . Any G-map $ : W -» V can be
*
interpreted as an 0(G)-transformation (p : f tp,7c -» tp c using
v w
the facts that TVC« (Vc x]r) = vc x (vs TR) holds and the suspension
[VC,WC]G -> [(V* K)c,(We K)C]G is bijective. Then DEG(f,(p) lies
in C(WC) = C(G) and DEG(f,<p)(H) is just deg(*Hc fH) for (H) £ Con(G). o
Example 2.7. Let M be a G-manifold such that the components of M*
are orientable for all H <= G. If f : M -» M is a G-map with
TTQ(fH) : ttq(MH) -» tiq(MH) the identity for all H <= G we can define
its degree DEG(f) £ C(M) by the collection (deg(fHIC : C -» C)
C £ it (M ) , H <= G) . The orientability condition ensures that we
*
get a well-defined 0 (G)-equivalence ip : f tp -» tp uniquely
determined by the propprty that
(p(G/H)(x)eH : tpM(G/H) (fx) - tpM(G/H)(x) is given by the fibre

135
H H
transport of the H-bundle TM|M along any path in M from x to
fx. One easily checks DEG(f) = DEG(f,<p). □
The following non-equivariant example indicates the advantage
of our notion of the degree with the one using fundamental
classes for surgery.
Example 2.8. Let M and N be closed orientable connected manifolds
with fundamental classes [M] and [N] and let M be M with -[M].
Consider a normal map f:M »N, f: TM 9 IR ► E, of degree one
taken with respect to the fundamental classes. If M is M + M + M
disjoint union gives a normal map of degree one g = f + f + f :
M ) N, g = f + f + f. The reader should figure out by himself that
it is impossible with these bundle data and orientations to convert
f by surgery into a normal map f : M+ ► n of degree one with
connected M . We can see this using our degree as follows. Fix an
0(1) - equivalence (p : tp tpTN 9 IRk. Let ¢^ : g tpN —-► tpM
be the 0(1) - equivalence uniquely determined by the property that
its suspension is ((p o tp.) : f*tpTN ffi IRk ■* tpTM 0 IRk.
Since "normally bordant" includes the bundle data, DEG (g,t|/^) is
a normal bordism invariant. But DEG (g *A) is - 1 by Proposition 3.1.

136
3. The Burnside ring of a compact Lie group.
The Burnside ring of a compact Lie group G was introduced and
examined by torn Dieck [5] and [6], p. 103 ff. Since we need
some modifications of this material and want to keep the paper
self-contained we make some remarks about it in this section.
A prefamily T is a subset of ,-(G) = {H|H <= G} closed under
conjugation. We call ? a family if it is also closed under
intersection and finite if {(H) £ Con(G) |He J } is finite.
The set of isotropy groups Iso(X) = {G | x e X) of a finite
G-CW-complex X is a finite prefamily. If X is a G-manifold
with connected fixed point sets, iso(X) is a finite family for
finite G, but not in general. A counterexample is the sphere
3 3
in the SO (3) -representation 1R <$ 1R if SOP) acts in the
obvious way on both summands. If ? is a prefamily and x denotes
the Buler characteristic let A(G,? ) be the set of equivalence
classes of finite G-CW-complexes X with Iso(X) cJ" under the
H H
equivalence relation X~Y»x(X)=x(Y) for all H c G. The
disjoint union defines an abelian group structure. Moreover,
the cartesian product induces the structure of an associative
commutative ring with unit if ? is a family containing G, We
can identify A(G) := A(G,S(G)) with the Burnside ring in [6] p. 103.
Let C(G,? ) be the ring of functions {(H) £ Con (G) |H e? } -» 2Z
and C(G) = C(G,S(G)). For each K c G we obtain a ring homomorphism
chK : A(G,fr) »22 [X] t—* X (*H) •
Since WH acts freely on G/H and WH contains a circle for infinite

137
WH we get ch„(G/H) = 0 for all K if WH is infinite. For any pre-
familyCf let ? _ be {H e ? | WH finite}. Using the ideas in [6]
p. 3, 4, 104, 119 one proves that ch is given by the product of
the ch -s:
K
Proposition 3.1. Let ^ be a finite prefamily. Then {[G/H] |H € <F }
is a E-base of A(G,? ). The homomorphism
Ch : A(G,7 ) -» C(G,? f)
is injective with a finite cokernel of order TT |WHI ,
{(H) I H e? f}
Moreover, each ch(G/H) i^s divisible by 'WHl and
{-j-~gy ch(G/H) I H e ? {) is a S-base for C(G,3if). d
Now we introduce the equivariant Lefschetz index following [14],
c c G
chapter 1 to produce a bijection [V ,V ] -» A(G,iso(V)) for
an appropriate G-representation V.
Consider a G-self map f : X -» X of a finite G-CW-complex X. Let
L(fn,rn) be the Lefschetz index of the self map (f ,f ) of the
H >H
pair of CW-complexes (X ,X ).
Definition 3.2. The equivariant Lefschetz index L (f) in A(G,Iso(X))
is defined as
LG(f) = I 1.. L(fH,f>H) . [G/H]
{(H)|Heiso(X)t} ,WH|
Since (XH,X>H) is WH-free, L(fH,f>H) is divisible by iWHf.
Proposition 1.8 in [14] extends to compact Lie groups:

138
Lemma 3.3. ch (LG(f)) = L(fK) for K <= G.
K
Proof. Since the Lefschetz index is additive ([9], p. 213) one
can reduce the problem by induction over the orbit bundles and
dimensions to the case X = 11 G/H x Dn/ll G/H x sn~ where one has
r r
to show with * the obvious base-point:
1 L(fH,*) • X(G/HK) if WH is finite
IWHI
L(fK,*)
otherwise
The second case follows from the fact that WH acts freely relative
* on X and X and contains a circle. The canonical inclusions and
projections of the wedge X yield a pair of inverse isomorphisms
between H„(X,*) and « H^ ((G/H x Sn) / (G/H x *) , *) where * denotes
r
the various base points. Now an easy homological computation
reduces the proof of the first case to X = (G/H x s )/(G/H x *) with
WH finite. Then fH is a self-map of (WH x sn)/(WH x *). The Kunneth
formula and the obvious map G/H x (WH x sn)/(WH x *) -» x
induce a chain homotopy equivalence such that the following
diagram commutes up to homotopy
C(G/HK) ® C(WH x Sn/WH x *,*) > C(XK,*)
SWH
ld ®2ZWH C(fH'*)
N"*
C(fK,*)
C(G/HK) ®avffl C(WH x sn/WH x *,*) > C(XK,*)
Notice that C(WH x s /WH x *,*) is concentrated in dimension n and
is SWH there. Let la • w e SWH be the element determined by C(f ,*)

139
Then L(fH,*) is IWHl-a1 and L(fK,*) is x(G/HK) • a1 since C(G/HK)
is EWH-free. This finishes the proof. □
A G-homotopy representation X of G is a finite-dimensional G-com-
plex of finite orbit type such that for each subgroup H of G the
fixed point set X is an n(H)-dimensional CW-complex homotopy
equivalent to S . If dim X > 1 and Iso(X) is a family, we equip
p
[X,X] and A(G,Iso(X)) with the monoid structure given by compo-
c c
sition and multiplication. If 1 denotes [G/G] and x (X)'= L (idv)
A
p
we have the unit x (X) - 1 in A(G,Iso(X)) and maps
X : [X,X] » A(G,Iso(X)) [f] » (LG(f) - 1) • (XG(X) -1)
DEG : [X,X]G —— C(G) [f] } {deg fH I (H) f Ton(G) }
The main result of this section is:
Theorem 3.4. Let X be a G-homotopy representation faith dim X > 1
satisfying condition (*) defined in the introduction.
a) L -1 : lX,X]G -» A(G,Iso(X)) is bijective.
b) If_Iso(X) is a family the monoid map X : [X,X]G -» A(G,lso(X))
is bijective and ch o X = DEG. a
Theorem 3.4 follows from proposition 3.1, lemma 3.3 and the equi-
variant Hopf theorem 3.5 below. For its proof and further
explanations we refer to [6] p. 213, [7] H.4., [14] , [18] and [21].
Theorem 3.5. Let X and Y be G-homotopy representations with
H H
dim X = dim Y for all H c G satisfying condition (*). Choose

140
fundamental classes for X and Y such that deg f for a G-map
X -» Y is defined.
Then [X,Y] is non-empty. Elements [f] are determined by the set
H H
{deg f I H £ Iso(Y)f}. The degree deg f is modulo IWHl determined
K K
by the deg f , K => H, and fixing these degrees deg f the possible
H *
deg f fill the whole residue class mod IWHI . d
We end with some remarks about induction and restriction for an
inclusion j : H —* G of compact Lie groups.
Let ? be a prefamily for H. Then j + ? = {g j(K)g \ g £ G,K €? }
is a prefamily for G. We want to define an abelian group homomor-
phism
indj : A(H,3" ) - A(G,j#3" )
by sending [X] to f°G x. x]. The following formula and proposition
3.1 show that this is well-defined.
3.6 X((G*.X)K) = E x(XqKq ) for K c G, WK finite.
3 gH £ G/HK
Notice that G/H has only finitely many WK-orbits ([2], p. 87)
and is therefore finite if WK is finite.
Given a prefamily Tf for G, we have the prefamily
* — 1
j «T = {j (K) | K £J } for H. We obtain an abelian group homo-
mo rph ism

141
res. : A(G,T ) - A(H,j T )
by restriction: [X] -» [res, X]. If 7* is a family containing
H and G then j 5* is a family with H £ ]f and res. is a ring
homomorphism.
4. The monoid of endomorphisms of the fibre transport.
If we want to examine the dependency of DEG(f,(p) on (p we have
to compute in view of the composition formula 2.5 the
0(G)-transformations (p : tp -» tp and the possible values DEG (ID,(p) in
C(N).
More generally we consider the monoid End(tp) of
0(G)-transformations (p : tp -» tp of any 0(G)-functor tp : & -» bf . The
group of invertible elements End(tp) consists of the
0(G)-equivalences (p : tp -» tp.
Consider C("& ) as monoid by its multiplicative structure. The
monoid map
DEG : End(tp) -» CC^)
maps ip to DEG(ip) specified by the following function CON(ti) -» TL.
For H <= g and x in t (G/H) we get a G-fibre map ip(G/H) (x) . Let
DEG(<p) (x,H) be the degree of the induced self map on the H-fixed
point set tp(G/H)(x)HH of the fibre over eH. Recall that tp(G/H)(x)eH
is H-homotopic to SV for some H-representation V. We want to show
that DEG : End(tp) -» C(d».) is an embedding of monoids and describe
its image.

142
We say that an 0(g)-transformation tp : % -» bf_ satisfies
<.J \j f n —————^^—
condition (*) if for any H c G and x £ t> (G/H) tp(G/H) (x) does
and has an H-fixed point. If furthermore Iso(tp(G/H)(x) „) is a
family we call tp admissible. Consider a G-manifold N satisfying
condition (*). Then tp„ satisfies condition (*) and is even ad-
N
missible if G is finite. If G is finite nilpotent and N a G-mani-
u
fold such that no component of N is a point for H <= G then tp„
is admissible.
We recall the noticn of the homotopy colircit r (ȣ. ) (see [20]
p. 1625). Objects are pairs (x,H) with x £ 6 (G/H) and H c G. A
morphism (a,u) : (x,H) -» (y,K) consists of a G-map
*
a : G/H -» G/K and a morphism u : x -» ay with
*
a = "S (a) : ^(G/K) -» -6, (G/H) . Composition is defined by the
"semi-direct product formula" (t,v)e(a,u) = (t c a,a"v<»u). Notice
ate-
that r (¾ ) is Con (6) (see section 2). The fundamental group c
gory of a G-space X appearing in [7] p. 57 and [15] is r (ti X). We
now introduce contravariant functors A , C^ and E and relate
their inverse limits to End(tp) and C(t,). The contravariant
functor into the category of monoids
Etp : r (/f ^ "* M0N0
maps (x,H) to ttp(G/H)(x)eH' fcP(G/H)(x) H] . Given a morphism
(a,u) : (x,H) -» (y,K) choose g in G with a(eH) = gK so that we
obtain a group homomorphism c(g) : H -» K h -» g hg. If
-1 -1
l(g ) is multiplication with g we get a H-homotopy equivalence
a : tp(G/H)(x)eH - resc ( . tp (G/K) (y) gK by 1 (g_1) e tp(G/H) (u) ^.

143
[tp(G/f}(x)eH,
Define E (a,u) : [tp(G/K)(y)eR, tp(G/K)(y)eR]
tp(G/H)(x) ] by restriction with c(q) and conjugation with a.
This is well defined since conjugation withon H-self-equivalence
induces the identity on [X,X] for a G-homotopy representation X
(theorem 3.4).
The contravariant functors
Atp : r (% ) ~" M0N0
C ^ : r (Cs ) -> MONO
send (x,H) to A(H,Iso(tp(G/H)(x) )) and C(H). Given a morphism
(a,u) : (x,H) -» (y,K) let g £ G and c(g) : H -» K be as above.
Define A. (a,u) and C^,(a,u) as the restriction with c(g).
Let the transformation
D : Etp -> Cf
A : E
tp
tp
CH
tp
be induced by the degree and the maps of section three
X : [tp(G/H)(x)eH,tp(G/H)(x)eH]H - A(H,Iso(tp(G/H)(x)eR))
ch : A(H,Iso(tp(G/H)(x)eH))
C(H)
The inverse limit of a contravariant functor F : C
MONO is
the submonoid inv F of n F(x) consisting of those elements
x e c

144
(a I x £ C) such that F(f)(a ) = a holds for any morphism
f : y -» x.
We define a monoid map
(3 : inv lim Cf
c(-a )
n a
tTST
r*
as follows. Let pr„ : C(H) -» 7L be the projection onto the
factor belonging to (H) £ Con(H). An element in the inverse
limit given by {u(x,H) £ C(H) | (x,H) £ 1-½)}^ is sent to
{pr„(u(x,H)) £ Z5 ! (x,H) £ r(e )},
Let a(x,H) : End(tp)
E (x,H) be the monoid map sending
ip to (p(G/H) (x) . We obtain a homomorphism of monoids
a : End(tp) -» inv lim E
tp
Theorem 4.1 .
a) If tp : €, -» bf fullf ills condition (*) , the following
diagram of monoids commutes. All maps are injective and a
is bijective.
End(tp)
DEG
C(-A.)
-> inv lim E
tp
inv lim D
( inv lim Cj
S 7

145
b) lf_ tp _is admissible the following diagram of monoids commutes.
All maps are injective and inv lim A is bijective.
2.T1V
lim E
/ tp
inv lim D
^ 4'"
inv lim C.
inv lim A
. v inv lim A
'> CP
,-'' inv lim CH
Proof. Everything follows directly from theorem 3.4 and the
definitions. □
Let * : MONO -» GROUPS be the functor "invertible elements".
*
Since the inverse limit is compatible with * and End(tp) is
the group Aut(tp) of 0(G)-equivalences tp -» tp we conclude:
Corollary 4.2. For admissible tp the following diagram of abelian
groups commutes. The maps a and inv lim A* are bijective the
others injective.

146
Aut(tp)
V
DEG
C(-fi) <-
1
-^ inv lim E
IK
S,
tp
inv lim A
inv lim A,
tp
inv lim CH
-* Inv lim C
$
Corollary 4.3. Let N be a connected G-manifold satisfying
condition (*) .
a) If N i§ connected and non-empty for all H <= g and Iso(N) a
family then:
End(tpN) = A(G,Iso(N)) c C(N) = C(G)
b) Let G be a torus. Assume that any component of N contains a
G-fixed point for H <= G. Then we have for y £ N the bijection
End(tpN) -» a (p -» deq((p(G/1) (y))
c) If G is finite of odd order we get for y £ N an isomorphism.
Aut(tpN)* - { + 1} cp - deq((p(G/1) (y))
Proof:
a) If x is a G-fixed point, we have for any object (y,H) in r (ti X)
a morphism (y,H) -» (x,G). Two such morphisms define the saine

147
map E (x,G) -» E (y,H) and inv lim E (x,G) is End(tp ).
PN PN PN
Hence End(tpN) = Efc (x,G) = A(G,Iso(N)).
b) If X is a G-homotopy representation of the torus G with
dim X > 1 then [X,X] -» TL [f] -» deg f is bijective by
proposition 3.1 and theorem 3.4.
& *
c) ch1 = A(G) -» {+1} is a isomorphism by [6] p. 8 if G has odd
order. □
If N is a G-manifold and tp is admissible, End(tp ) a- inv lim A
depends only on the component structure of N and the sets Iso(TNx)f
for all x £ N which can be read off from the dimension function.
5. The degree relations.
In this section we state the central result of this paper. In the
following we identify End(tp) with its image in C(N) under the
embedding DEG.
Theorem 5.1. Let f : M -» N be a G-map of n-dimensional G-mani-
folds satisfying condition (*) and (p : f tp -» tpM be an 0(G)-
transformation. Then:
DEG
(f,(p) £ End(tpN) c C(N) d
The rest of this section deals with its proof. Examples to
illustrate its meaning are given in the next section. The most
important ingredients are the concept of quasi-transversality which
we will extend to compact Lie groups (see [10], chapter 3 for
finite G) and local degrees.

148
We call the G-map f : M -» N of G-manifolds quasi-transverse to
y in N if the following is true.
i) The preimage f (y) consists of finitely many orbits G /H
with all WG H finite.
Y
ii) Equip the G-normal bundle ^ (f~ (G/G ) ,M) and \) (G/G ,N)
with equivariant metrics. There is a norm preserving G-
fibre map
-1 k
(f (G/G ),M) > (G/G ,N)
f (G/Gy) > G/Gy
such that f looks like k in a tubular neighbourhood.
Lemma 5.2. We can change f up_ to G-homotopy such that f _is quasi-
transverse to y. (see also [10, ch. 3] ).
Proof. Let K..,K2,...,K be a complete system of representatives
of conjugacy classes (K) of subgroups of G with K occuring as
isotropy group in M and K <= G . We construct inductively an open
K K
G-set U. containing M 1,...,M i such that i) and ii) hold if one
substitutes f (y) and f (G/G ) by their intersections with U..
We can assume (K.) <= (k .) «► i > j .
The induction begin i = 0 is trivial: U = 0. In the induction step
from i - 1 to i write U = U._1, K = K.. By possibly shrinking U we
can suppose the existence of a closed G-set V with int(V) => clos(U)

149
and f 1 (G/G ) n V-* U = 0. Let M be MK ^ (u n MK) . By induction
y kj
v
hypothesis WK acts freely on M . If f is f IM consider the non-
equivariant map
(f x id)/WK : M /WK -» (NK x M )/WK
We can change it homotopically relative V n M /WK into f. such
that f. is transverse to (G/G xM )/WK, By a cofibration argument
we can assume (f x id)/WK = f.. Now G/G is a finite disjoint
r ^
union of WK-orbits 11 WK«(g-G ) (see [2], p. 87). One easily
-1i=1 X Y -1
checks dim(fQ/WK) (WK.g^G ) = dim(f x id/WK) (WK-giG xMq/WK) =
-dim NK n G„ y/K. Hence (f /WK)" (WK-g^^G ) consists of finitely
many points if NK n g^G /K is finite and is empty otherwise. In
other words f (y) n GM consists of finitely many orbits G /K.
such that WG K. = NK. n G /K. is finite. We can treat any such orbit
y J J y J
separately.
Consider any x in GM with f(x) = y so that f maps G/Gx -» G/G
by the projection. We identify v*(G/G ,M) with a tubular G-neigh-
bourhood of G/G and analogously for y. We have
dim i>(G/G ,M)L <, dim ^ (G/G ,N)L for all L <= G so that we can
xx y y g x
extend any non-equivariant map Sv>(G/G ,M) -» S\> (G/G ,N) x
to a G -map S,> (G/G ,M) -» S \>(G/G ,N) . Since v> (G/G ,M) =
x r 'xx y y x
G x >(G/GY,M) , >(G/G„,N) = G x >(G/G„,N) and (K) = (Gv)
"jj x x y g y y x
holds we can construct a norm preserving fibre map
CMG/G .M) * sMG/G „,N)
x y
I
G/G ^ G/G
x ' y

150
such that the restriction of k to the K-fixed point set agrees
with f . By a cofibration argument we can change f in a small
G-neighbourhood of G/G relative to M such that k coincides
with f on all £ (G/G ,M). Now one easily enlarges U to the
desired U.. This finishes the proof of lemma 5.2. d
Proof of theorem 5.1. We have to construct A £ End(tp„) such that
DEG : End(tpN) -» C(nGN) sends A to DEG(f,<p). Let
e(y,H) £ r (y,H) := C(H) for (y,H) £ r(nGN) be defined by
Con(H) »Z (K) ► DEG(f ,ip) (y,K) . One checks directly that
{e(y,H)|(y,H) £ r(n N)} determines an element in inv lim C r
G n N
mapped by 3 : inv lim C -» C(n N) to DEG(f,(p). Suppose that
n N G
we can construct for each (y,H) £ r(n N) a H-self-map 6(y,H) of
tp(G/H)(y) „ = TN* such that D(y,H) : E (y,H) - C (y,H)
N eH y tpN nGN
sends 6(y,H) to e(y,H). Then {<5 (y ,H) I (y ,H) £ rflT N)} defines an element A'
in inv lim E. . By theorem 4.1 there is A £ End(tp„) such that
FN
a : End(tp ) » inv lim E. maps A to A', and A has the de-
PN
sired property.
Let ch' : A(H,Iso(TN )) -» C(H) be the composition of the
inclusion A(H,Iso(TN )) -» A(H), multiplication with the unit
XH(TN^)-1 : A(H) -» A(H) and ch : A(H) -» C(H). The map
LH-1 : [TN^,TN^]H -» A(H,lso(TN )) is a bijection (theorem 3.4.)
and ch' o (L -1) = DEG. Hence theorem 5.1. is true if we can
n
construct for any (y,H) £ r(n N) an element d £ A(H,Iso(TN ))
satisfying:
5.3. ch'(d) = DEG(f ,(p) (y,K) for all K <= H.
K

151
Now we construct d. We can assume in view of 2.4 and lemma 5.2
that f is quasi-transverse to y. Furthermore we can suppose
H = G . We want to assign to each H-orbit c in f (y) an
element d(c) in A(H,lso(TN )). Choose x in c. Then TM -
Y x
= (TG/G ) £ vN-(G/G ,M) and TN = (TG/G ) ® vN-(G/G ,N) . Split
Tpx : (TG/Gx)x -» (TG/G ) induced from the projection p as
0 ¢5 q : (TG /G ) * V -» (TG/G ) with q a G -linear isomor-
^x y x x y y x x
phism. Checking the dimensions and using elementary obstruction
theory we can extend the G -map k : v1 (G/G ,M) -» ^(G/G ,N)
« X XX JL JL
coming from k appearing in the definition of quasi-transverse to
a norm preserving G -map k' : \> (G/G ,M) © (TG /G ) -» vv(G/G ,N) .
r ^ x ^ x xx yxx y y
Since k' 05 q : TM -» TN is norm-preserving we obtain a G -self
map 1 : TNC -» TNC by (k' * q )C o (p(G/G ) „ . As H/GK contains
^ x y y J x ^x *' ' x eG x
X K
only finitely many W K-orbits (see [2], p. 87) H/G is finite and
(TG /G )* = {0} for K <= H with finite W K. Hence
yxx H
5.4. lx = (Tpx<3Dkx)C o (p(G/Gx)(x)^G if WRK is finite.
x
G G
Denote the image of 1 under L x-1 : [TNC,TNC] x -» A(G ,Iso(TN ))
x y y x y
by d(x) and the image of d(x) under ind" : A(G ,Iso(TN ))-^
\j x y
x 2
A(H,Iso(TN )) by d(c). For u £ A(H) and v £ A(G ) one easily checks
H H H
ind (resr (u)-v) = u-ind., (v). We obtain from 3.6. and 5.4. and
XX X
res" (XH(TN^)-1) = X X(TNt=)-1
gx y y
5.5. ch^ e ind" (d(x)) =
x
-1 -1 -1
t- j i /m hKh _ , hKh . C ,„ ,„ > , > hKh >
I deg((Tp e k ) o <p(G/G ) (x) _ ) =
hG £H/GK xx x eGx
X X
I deg((Tp* ® k*)C o (p(G/G_)(z)* ) if W„K is finite,
z £ cK z z z eGz H

152
This shows in particular using proposition 3.1. that d(c) does
not depend on the choice of x. Define d = I_- d(c).
cef (y)/H
If fK : MK -» NK and (pK : fK*tpNK -+ tpMK are induced by f and
K K
DEG(f ,(p ) is the non-equivariant degree we have by definition
DEG(f ,(p) (y,K) = DEG(fK,(pK) (y) and by proposition 2.1.:
5.6. DEG(f ,(p) (y,K) = I deg ( (Tp* <S k*) °cp(G/G)(z)*
ze (fK) n(y)
if W K is finite.
H
Combining 5.5. and 5.-6. gives
5.7. ch'(d) = DEG(f ,(p) (y,K) if W„K is finite.
Let K c H be any subgroup. We can find a bigger subgroup K' with
K <= K' c H such that K'/K is a torus T, W„K' is finite and
H
ch„ = ch.., and hence ch' = ch', holds (see [6] p. 113). If we can
show DEG(f ,(p) (y,K) = DEG (f ,(p) (y ,K') the assertion 5.3 is a conse-
K K '
quence of 5.7. Since T acts on N with fixed point set N this
follows from:
Lemma 5.9. Let g : P -» Q be a T-map between T-manifolds and
*
4> : g tp -» tp an0(T) -transformation. Then we get
DEG(g,i(j) (y,T) = DEG(g,*) (y,1) for a T-fixed point y.
Proof. We can assume that q is quasi-transverse to y. Since WL
for L c T is finite only for L = T the preimage f (y) is a
finite set of T-fixed points x1,..x . By proposition 2.1 we obtain

153
for certain T-self maps 1. : TN -» TN that
^ l y y
DEG(f,*)(y,T) = I deg(lT) and DEG(f ,i|j) (y ,1) = I degdi). Now
apply proposition 3.1 and theorem 3.4. This finishes the proof
of lemma 5.9 and of theorem 5.1. □
6. Some examples
The theorem 5.1 is very general so that it is necessary to give
some examples to explain its meaning. The general problem is to
calculate inv lim A. as a subring of C(6> ). This can be done in
special cases where T(6 ) is rather simple or A(H,? ) c C(H) is
well understood for all subgroups H of G. We recall that DEG(f,(p)
lies in C(N) n n (NH)/WH for a G-map f : M -» N and a
Con(G) ° „
0(G)-transformation (p : f tp„ -» tp„ and DEG(f ,w) (z ,H) is the
N M
TT U
integer belonging to the component of N containing z £ N .
In the following we always assume that N fulfills condition (*)
and is connected.
Example 6.1. Assume that N is non-empty and connected for all
H <= G. Suppose that iso(N) is a family. This follows already
from our assumption if G is finite. Then we have C(N) = C(G)
and by corollary 4.3 and theorem 5.1.
DEG(f,(p) £ A(G,Iso(N)) <= C(G)
Hence we obtain the same relations as in the special case
M = N = V with V a G-representation (see theorem 3.4.). The
assumption N f 0 is essential. If N is connected and free we
get End(tpN) = A(1) = C(N) = Z. Indeed, each integer d can be

154
realized as the degree of a self-map of some connected free
orientable G-manifold N. Take any connected free orientable
G-manifold N and a map f : S -♦ S of degree d then
o
1 1
iA x f ; N x S -» N x S is an example. However, if N is the
sphere of a free G-representation the degree of f is 1 modulo IGl
for finite G and 1 for infinite G. One
explanation for this phenomenon is that the suspension of a
manifold is not a manifold in general but the suspension of a homo-
topy representation is again a homotopy representation. □
Example 6.2. Let G be a torus T and assume that each component
of N contains a G-fixed point for H <= G.
We get from corollary 4.3 and theorem 5.1
DEG(f ,(p) (Z,H) = DEG(f ,(p) (y,1)
for all H <= G and z £ N . d
Example 6.3. If H is a p-group the homomorphism ch- and
chH : A(H) -» a fulfill ch1 ■ ch mod p. If H is a torus ch1
and chH agree. Hence we get for each (z,H) by theorem 5.1 (see
5.3).
DEG(f ,(p) (z,H) = DEG(f ,(p) (z,1) mod p, if H is a p-group
DEG(f ,(p) (z,H) = DEG(f ,(p) (z,1) , if H is a torus
If G is itself a p-group we obtain for all (z,H) (see also [3],
[11] p.- 10).

155
DEG(f ,(p) (z,H) ■ DEG(f ,(p) (z,1 ) modp □
Remark 6.4. Now we give the proof of corollary B stated in the
introduction. Assume the existence of M. Since M has finite
orbit type (see [6] p. 121) we can find a finite p-group L c G
P T
with M = M . Hence we can suppose that G itself is a finite p-
group. We use induction over iGl. The induction begin G = Z/p
is done in [1] or by the following argument reflecting the re-
suits of example 6.3. Let c : M -» TM be the collaps map. If
— 1 G ~1
<=» is the point at infinity c (<*>) n M is empty. Since c (°°)
is contained in the free part of M we can use non-equivariant
transversality to change c up to G-homotopy such that c is
transverse to °° in the non-equivariant sense and still
c (<=) n m = 0 holds. We can assume that G acts orientation
preserving, otherwise consider M xm. Hence the local degree of
c at x and gx for x £ c (°°) and g £ G agree. Each orbit in the
finite set c (°°) consists of p elements. Therefore the degree
of c must be divisible by p. A contradiction, since computing
deg c by its local degrees at O £ TM yields one. In the induction
c p
step choose a central subgroup C in G with C = Z/p. If M f M
we get a contradiction to the induction hypothesis. Namely, con-
c c c
sider the G/C-action on M . But M = M is impossible by the
induction begin applied to the C-action on M. This finishes the
proof of corollary B.
If one drops the assumption in corollary B that all M are
connected the result remains true for G an abelian p-group and
is false for G a non-abelian p-group provided that p is odd
(see [4], [12]). A complete classification of compact Lie groups

156
with one-fixed point actions on (orientable) closed G-manifolds
is given in [12]. d
Example 6.5. Let G be a finite group of odd order. If DEG(f ,ip)
lies in C(N) we get from corollary 4.3. and theorem 5.1.
DEG(f,(p) (z,H) = DEG(f,(p) (y,1)
for all (z,H). d
Remark 6.6. All the relations we get for DEG(f ,(p) also hold in
the case of an endomorphism f s M -* M of a G-manifold with
H H H
ti (f ) : it (M ) -» tt (M ) the identity and all components of
u
M orientable for H <= G. Then we can define DEG(f) without
specifying ip (see theorem A in the introduction and example 2.7.). a
Finally we mention a consequence of theorem 4.1. and theorem 5.1.
and the composition formula 2.5.
*
Corollary 6.7. If DEG(f,ip) lies in C(N) for some
Q(G)-transformation ip then there is an O(G)-equivalence i|j with
DEG(f,*) = 1 □
7. The fibre transport and the first equivariant Stiefel-Whitney
class.
In this section we analyze the fibre transport from a bundle
theoretic point of view. We relate it and the question when an
*
0(G)-equivalence ip : f tpw -» tp exists to the equivariant

157
analogue of the first Stiefel-Whitney class.
We have introduced the notion of an 0(G)-groupoid, 0(G)-functor
and 0(G)-transformation in section one. We call two 0(G)-func-
G)-
tors F and F., : ti -» "6 homotopic if there exists an 0(
equivalence (p : F -» F . Let [6 ,^-,] be the set of homo-
topy classes of 0(G)-functors ^ -» ^.. A G-map f : X -» Y
PC P
induces an 0(G)-functor it f : n X -♦ it Y. A G-homotopy
p p
h : X x i -» y defines an 0 (G)-equivalence it h -> n h., . Hence
we get a well-defined map [X,Y]G - [tiGX,tiGY]0(G) [f] - [TiGf ] .
Let n = n(G,n) + BF(G,n) be the classifying G-Sn-fibration. It is
p
characterized by the property that the map [X,BF(G,n)] -» bfQ n(X)
*
sending [f] to the G-fibre homotopy class of f n is bijective.
Definition 7.1 ■ Let £ + X be a G-Sn-fibration and f : X -» BF(G,n)
P C C O (P)
be a classifying map. We call w,. = [it f] e [ti X,tt BF(G,n)] the
first equivariant Stiefel-Whitney class of £. Let w be w c for a
G-manifold M. □
This notion reduces for G = 1 to the ordinary definition of the
first Stiefel-Whitney class w1 (M) £ H1(M,Z/2) = Horn (it., (M) ,it1 (BF(n))
It is related to the fibre transport by:
Proposition 7.2. Let tp : it B(G,n) -» bf be the fibre
transport of the universal G-Sn~fibration.
a) For each H <= G tp (G/H) : it(B(G,n)H) -» bf„ (G/H) is an
rn G,n
equivalence of categories.

158
b) For any G-complex X we get a bisection
(tpn), : [TiGX,TiGB(G,n)]0(G) -> [itGX,bfGfn]°(G).
c) If £ +X is a G-Sn-fibration (tp ) sends wr to [tpr].
d) Let C-i and £_ be G-Sn-f ibrations over the same one-dimensional
G-complex x. Then C- and £2 are G-fibre homotopy equivalent if
and only if wf = wr holds.
Proof:
a) We must show that tp (G/H) induces a bijection between the sets
of isomorphism classes of objects and for any object x e n(B(G,n) )
an isomorphism Aut(x) -» Aut(tp (G/H) (x) ) u •* tp (G/H) (u) . The
first assertion follows directly from the universal property. The
second follows from the observation that H-fibre homotopy classes
of H-S -fibrations over S equipped with the trivial H-action are
in one to one correspondence to H-homotopy classes of self H-maps
of the fibre by the fibre transport, b) and d) follow directly
from [16] whereas c) is obvious, d
Given two G-S -fibrations £ and n over X we want to analyze when
H H
w^ = w holds. If w. (C ) and w.. (n ) are the (non-equivariant) first
1 H
Stiefel-Whitney classes in H (X ,Z/2) we have the following obvious
conditions for w„ = w :
£ n
i) ?x -G nx for each x e X.
x
ii) w^C ) = w1 (nH) for H <= G.
The following example shows that they are not sufficient in general.
If 1 is the trivial and K_ the non-trivial S/2 representation and

159
1 1c 1c
S/2 acts freely on S , consider £ = S x jr and n = S x ]R_.
Under certain conditions, however, i) and ii) are sufficient.
Theorem
7.3. Let £ and n be G-S -fibrations over X. Then w£ = w
holds if one of the following conditions is satisfied.
H H H
i) X is connected and w-(£ ) = w1(n ) for all H c G. There is a
x in XG with Cx ~ nx-
x
ii) The group G is finite of odd order and £x ^G n for all x £ M.
x
We have after forgetting the group action w.. (£) = w^ (n) .
Proof: We have to specify for each H <= G and x £ X a G-fibre ho-
motopy equivalence ip(G/H) (x) : tp (G/H) (x) -» tp (G/H) (x) . We do
this by determining a H-homotopy equivalence (p(G/H)(x) „:
C -» n between the fibres over eH. The independence of the
choice of the path u below follows from the assumptions about the
first Stiefel-Whitney classes.
i) Fix y in X and a G-homotopy equivalence
n . Define
ip(G/H)(x) by requiring that the following diagram commutes
up to H-homotopy for a path u from y to x in X .
tp (G/H)(u)
eH
(p(G/H) (x)eH
tp (G/H)(u)
eH
ii) Without loss of generality we can suppose that X is connected.
Fix a point y in X and a non-equivariant homotopy equivalence
$ : £ -» n . By assumption there is a H-homotopy equivalence

160
n and there are only two up to homotopy because of
A(H)* = A(H,IS0(n ))*
x
(+1} (see [6], p. 8) and theorem 3.4.
Let ip(G/H)(x) „ be the one making the following non-equi-
variant diagram commutative up to homotopy for a path u
between y and x.
tp(G/1)(u)
(p(G/H) (x)
eH
-¾^
tp(G/1)(U)
-h n.
Now we examine whether w„ is a homotopy invariant. If f : M -» N
is a G-homotopy equivalence f (TN & V)c and (TM £• V)c are G-fibre
homotopy equivalent for appropriate V by [13], theorem 2.3. If
this could be destabilized to f TNC ^„ TMC we would in particular
*
obtain f w„ = w . Unfortunately, this is not possible in general,
theorem 3.4. gives counterexamples over a point. Now we prove the
*
unstable result f w„, = w...
N M
Theorem 7.4. Let f : M -» N be a G-map between G-manifolds such
that TTQ(fH) : ttq(MH) -+ "o^^ Is- bijective for all H <= G. Suppose
for C £ ttq(MH) and D e ttq(NH) with fH(C) <= D that (fH|C)*w1(D) is
w1 (C) . Then the (non-equivariant) degree deg (fH IC : C -» D) £S/{±1}
is defined. Assume deg (f I C) = ±1 and that M and N fulfill condition
(*).
Then there is an 0 (G) -equivalence to: f tp„ -» tp uniquely
determined by the property that DEG(f ,u>) a 1 .

161
H H
Proof. Denote by M (x) the component of M containing x for
H <= G, x e MH. Let MH(x) be MH(x) if w1(MH(x)) is zero and the
A H
orientation covering otherwise so that M (x) is an orientable
^ H
connected manifold. Choose a lift x £ M (x) of x and a lift
fH : MH(x) -» NH(fx) of f|MH(x) : MH(x) -> NH(fx). Write
y = fH(x) and y = fH(x). Let *(x,H) : (TN^)H - (TM^)H be a
(non-equivariant) map for x £ M , H c G making the following
diagram commutative where c denotes the collaps map and p the
H H
projection and n is dim M (x) = dim N (fx).
Hn(MH(x)),3MH(x))
H (TMH(x)5):
(Tpx)
ll<,
Hn((TMx)H) /_
-!—!>, Hn(NH(y)) ,3NH(y))
Hn(TNH(y)?);
US
(TPy)
Hn((TN=)H)
One easily checks that the homotopy class of i(j(x,H) depends only
on (x,H) but not on the choice of x and f .
Lemma 7.5. There is up to G -homotopy exactly one G -map
^x)
TN*L -» TMC for each x in M such that ¥ (x)H and i()(x,H)
'fx
are non-equivariantly homotopic for all H c G , Each f(x) is a
G -homotopy equivalence.
Let m be the product tt(Wg H( running over {(H) e Con(Gx) | H e ISO(TN ),
'x
W H finite}. We get from [7] p. 173 + 174 the existence of G-maps

162
co : TN -» TM and co' : TM -» TN with deg((co' ° co) ) = 1 mod m
y x x y ^
for all H c G. Because of the equivariant Hopf theorem 3.5. and
theorem 3.4. lemma 7.5. follows from:
7.6. The element d = {deg (co o $ (x,H) ) | (H) £ Con(G )} £ C(G )
G xx
lies in the image of DEG : [TN ,TN ] x -» C(G ) .
u
We firstly give the proof of 7.6. under the assumption that N
is connected for all H c G . Consider f : M -» N as a G -map so
that x is a G -fixed point in M. As in the proof of theorem 7.3.
*
ii) we get aiO(G)-transformation cp : f tp -» tp uniquely
determined by the property that cp(G /G ) (x) is just co. Note that
H H H
for any z £ M , H c G the case w1 (M (z)) = 0 and w1 (N (z)) * 0
H H
never occurs because of deg(f |M (z)) = ±1. Now one checks
directly that DEG(f,cp) £ C(G ) is just d. By theorem 5.1. we have
G
d £ image(DEG : [TN°,TN°] x -» C(G )).
y y x
In the general case one has the problem that co does not determine
an 0(G )-transformation cp : f tp„ -» tp„ if there is a non-connected
u
N for some H <= G . But we can restrict everything to the 0(G )-
Gx Gx Gx H
subgroupoid n M of n M with n M(G /H) := n(M (x)) so that we con-
u
sider only the component of M containing x. Then we get an 0(G )-
„ G G
transformation cp : f tpN | nxxM -» tp I n XM by co as before. As in
G
section two we can at least define DEG(f,cp) in C(n N) = C(G ) and
get d = DEG(f ,cp) . The same argument as in the proof of theorem 5.1.
G
gives d £ image (DEG : [TN ,TNC] x -» C(G )). This shows 7.6. and
finishes the proof of Lemma 7.5.

163
Let (p : f tp -» tpM be defined by the property that
(p(G/H) (x) : TN^ -» TMC is the restriction from G to H of
cn IX X X
i|i(x) . We leave it to the reader to check that ip is an well-
defined 0(G)-equivalence. By construction DEG(f,<p) a 1 holds.
The uniqueness follows from theorem 4.1. This finishes the
proof of theorem 7,4. o
We obtain as a corollary the homotopy invariance of the first
equivariant Stiefel-Whitney class and the unstable version of
the result in [13], corollary 2.4.
Corollary 7.7. Let f : M >N be a G-map satisfying the assumption
of Theorem 7.4. Then we have f w = w and the spheres of the normal
slices of M at x and N at fx are G„ - homotopy equivalent for all
x £ M.
¥r C
Proof We derive f w„ = w„ and TM a* „ TN, from theorem 7.4.
===== N M x G fx
x
Now apply theorem 3.5. d
If G is connected TM and TN,- are even isomorphic as G -represen-
X X X X x
tations (see [22]). For finite G there are non-isomorphic G-represen-
tations V and W with VC a* r WC (see [6], p. 249).
Now we give a necessary condition for converting a G-map
f ; M -» N between G-manifolds into a G-homotopy equivalence by
surgery. Notice that this would imply the existence of a bordism
appearing below. Theorem 7.8. motivates the approach to
equivariant surgery given in [17].

164
Theorem 7.8. Let (f,f,f+) : (P,M,M+) -» (Q,N,N+) be a G-map
between G-triads of G-manifolds such that f+ is a G-homotopy
H H
equivalence. Assume that M -» P is 1-connected for H c G.
Then we can find an 0 (G) -equivalence ip : f tp„ -» tpM with
DEG(f ,(p) = 1 .
Proof. We get the existence of an 0(G)-equivalence
*
<P+ : f+fcPN "* fcPM with DEG(f ,(p+) = 1 by corollary 7.7. Using
the inward normal we get identifications tp I M+ = tp,„MSj.,c
and tp I M = tp(TM <felP)c and analogously for Q. For two G-repre-
sentations V and N with (V t£ TR) C =*_ (W * TR) C and dim V = dim W > 1
G
the suspension [v°,WC] -» [v«iC,W*lC) is bijective (theorem
*
3.5.). Hence we obtain O(G)-equivalences * : F tp -» tp and
*
(p : f tp -» tpM such that $|M corresponds to ip and $|M+ to (p+
under the identification above. Now apply the bordism invariance
2.3. d
References
[1] Atiyah, M. F. and Bott, R.: Lefschetz fixed point formula
for elliptic complexes II. Applications. Ann. Math. 88,
451 - 491 (1968).
[2] Bredon, G. E.: Introduction to compact transformation groups,
Academic Press, New York-London (1972).
[3] Bredon, G. E.: Fixed point sets of actions on Poincare
duality spaces. Topology 12 (1973), 159 - 175.

165
[4] Browder, W.: Pulling back fixed points, inv, math,
87, 331 - 342 (1987).
[5] torn Dieck, T.: The Burnside ring of a compact Lie group I.
Math. Ann. 215, 235 - 250 (1975).
[6] torn Dieck, T.: Transformation groups and representation
theory. Lect. notes in math. 766, Springer Verlag, Berlin-
Heidelberg-New York (1979).
[7] torn Dieck, T.: Transformation groups, de Gruyter (1987).
[8] torn Dieck, T. and T. Petrie: Homotopy representations of
finite groups, Publ. Math. IHES 56 (1982), 337 - 377.
[9] Dold, A.: Lectures on algebraic topology, Springer Verlag,
Berlin-Heidelberg-New York (1972).
[10] Dovermann, K. H.: Addition of equivariant surgery obstructions,
algebraic topology, Waterloo (1978), lecture notes in math.
741, Springer Verlag (1979), 244 - 271.
[11] Dovermann, K. H. and T. Petrie: G surqery II. Mem. of the AMS
Vol. 37, no. 260 (1982) .
[12] Ewing, J. and R. Stong: Group actions having one fixed point.
Math. Zeitschrift 191, 159 - 164 (1986).
[13] Kawakubo, K.: Compact Lie group actions and fibre homotopy
type. J. Math. Soc. Japan, Vol. 33, no. 2, 295 - 321 (1981).
[14] Laitinen, E.: Unstable homotopy theory of homotopy
representations, in "transformation groups", Poznan, lect. notes in
math., vol. 1217 (1986).
[15] Luck, W.: Seminarbericht "Transformationsqruppen und alqe-
braische K-Theorie", Gottingen 1983.

166
[16] Luck, W.: Equivariant Eilenberg-MacLane spaces K(S,y,1)
with possibly non-connected or empty fixed point sets,
manuscr. math. 58, 67 - 75 (1987)
[17] Luck, W. and Madsen, I.: Equivariant L-theory, Aarhus
preprint, (198 8) .
[18] Rubinsztein, R. L.: On the equivariant homotopy of spheres,
preprint, Polish Academy of Science (1973).
[19] Switzer, R. M.: Algebraic topology - homology and homotopy,
Springer Verlag, Berlin-Heidelberg-New York (1975).
[20] Thomason, R. w.: First quadrant spectral sequences in
algebraic K-theory via homotopy colimit, Comm. in Algebra 10
(15), 1589 - 1668 (1982).
[21] Tornehave, J.: Equivariant maps of spheres with conjugate
orthogonal actions, Can. Math. Soc. Conf. Proc., Vol. 2
part 2, 275-301, (1982) .
[22] Traczyk, P.: On the G-homotopy equivalences of spheres of
representations, Math. Zeitschrift 161, 257 - 261 (1978).
Wolfgang Hick
Mathematisches Institut
der Georg-August-Universitat
BunsenstraBe 3-5
3400 Gottingen
Bundesrepublik Deutschland

SURGERY TRANSFER
by W.Lii'ck and A.Ranicki
Int roduc t i on
Given a Hurewicz fibration F »E " »B with fibre an
n-dimensiona1 geometric Poincare complex F we construct
algebraic transfer maps in the Wall surgery obstruction
group s
P! : Lj/ZttfjdO ]) ► Lm+n(Z[*l(E) ] } (m^°>
and prove that they agree with the geometrically
defined transfer maps. In subsequent work we shall
obtain specific computations of the composites p'p, ,
p,p" with p , :Lm(Z[*j(E) ] ) >L„/Z [ K j ( B ) ] ) the change of
rings maps, and some vanishing results.
The construction of p" is most straightforward in
the case when F is finite, with L* the free L-groups
h '
L*. In §9 we shall extend the definition of p' to
finitely dominated F and the projective L-groups L§, as
well as to simple F and the simple L-groups L* , and
also to the intermediate cases.
There are two main sources of applications of the
surgery transfer. The equivariant surgery obstruction
groups of Browder and Quinn [ 1 ] were defined in terms
of the geometric surgery transfer maps of the normal
sphere bundles of the fixed point sets. An algebraic
version will necessarily involve the algebraic surgery
transfer maps. (In this connection see Luck and Madsen
[8].) The recent work of Hambleton, Milgram, Taylor and
Williams [3] on the evaluation of the surgery
obstructions of normal maps of closed manifolds with
finite fundamental group depends on the factorization
of the assembly map by twisted product formulae which
are closely related to the algebraic surgery transfer.
Our construction of the quadratic L-theory
transfer maps is by a combination of the algebraic

168
surgery theory of Ranicki [14], [19] and the method used
by Luck [7] to define the algebraic K-theory transfer
maps p ' :Km(Z[*j(B) ] ) >Km(Zltfj(E ) ] ) (m=0,l) for a
fibration with finitely dominated fibre F.
The algebraic surgery transfer maps p" for a
are a special case of transfer maps
»L (B) (m^O) defined in abstract
algebra. Here, A and B are rings with involution, C is
an n-dimension a 1 f.g. free B-module chain complex with
Poincare duality chain equivalence
and U:A »R=H0(HomB(C,C))op is a
morphism of rings with involution from A to the
opposite of the ring of chain homotopy classes of
B-module chain maps f :C >C , with the involution on R
1 *
f i bra t i on
(C.a.U) ! :Lm(A)-
a symme trie
* n-*
a~a -c >C
de fined by
represented
( M , y/, : M-
M )
T(f) = a-if a. An element of L- (.a; is
by a nonsingular (-) -quadratic form
on a f-g. free A-module M = ©A. We define
(C,a,U)'(M.yfr)-(D»0)£L +2i<B) to be the cobordism class
of the (n+2i)-dimensiona1 quadratic Poincare complex
(D,0) given by
9 .
U(^)(ea * ) if s=0
k
0 i f s*0
Dn+2i-r-s = ecn+i-r-s
-» D
ec
There is a similar formula in the case m=2i+l, for
which we refer to §4.
The algebraic transfer maps of fibration F >E—E-^B
with fibre an n-dimensiona1 geometric Poincare complex
F are given by
p ■ = (C(F) ,a,U)
Lm(Z[*l(B)1}
Lm+n<Zf*l<E>l>
with C(F) the cellular Zl*,(E)]-module chain complex of
the cover F of F induced from the universal cover E of
E, a=( [F]A-)_I :0(9-) >C(F)n~* the Poincars duality

169
chain equivalence, and
transport .
determined
the fibre
Here is the main idea in the identification of the
algebraic and geometric surgery transfer. We know from
the identification of the corresponding K-theory
transfers in Luck [7] how to handle in algebra the lift
of CW structures from the base to the total space of a
fibration. We use the ultraquadratic L-theory of
Ranicki [16,^7.8] both to encode the algebraic surgery
data in the base spaces as CW structures, and to decode
the algebraic surgery data from the lifted CW
structures in the total spaces.
The paper was written during the second named
author's visit in the academic year 1987/1988 to the
Sonderforschungsbereich SFB170 in Gottingen, whose
support is gratefully acknowledged.
The titles of the sections are:
Introduction
§1. The algebraic K-theory transfer
%2. Maps of L-groups
§3. The generalized Morita maps in L-theory
§ 4 . The quadratic L-theory transfer
§5. The algebraic surgery transfer
§6. The geometric surgery transfer
§7. Ultraquadratic L-theory
§8. The connection
§9. Change of K-theory
Appendix 1. Fibred intersections
Appendix 2. A counterexample in symmetric L-theory
References

170
§ 1 . The algebraic K-theory t ran s f er
We recall from Luck [7] the construction of the
algebraic K-theory transfer maps, and the connection
with topology.
Given a ring R let Rop denote the opposite ring,
with the same elements and additive structure but with
the opposite multiplication.
Definition 1.1 A representation (A ,U) of a ring R in an
additive category /k is an object A in /k together with a
morphism of rings U:R iHom, (A,A) .
Given an associative ring R with 1 let B(R) be the
additive category of based f.g. free R-modules R
(n^O). A morphism f :R
is an R-module morphism,
corresponding to the mXn matrix (a, ),^,^ ,^--^- with
* 6 i3 l^i^m,1<J^n
entries a,.PR, such that
f = (,
i 3
)
(x,)
( £ x .a . . )
j-1 J U
Examp1e 1.2 The universal representation (R,U) of
B(R) is defined by the ring isomorphism
-» HomR(R,R)°P ; r
( s
sr )
which we shall use to identify R=Hom_(R,R)
op
A functor of additive categories F:A-
required to preserve the additive structures.
Proposition 1.3 Given a ring R and an additive category
/k there is a natural one-one correspondence between
functors F:B(R) >A and representations (A , U) of R in
A.
Proo f : Given a functor F define a representation (A,U)
by

171
A = F(R) ,
U : R = HomR(R,R)°P ► Hom„(A,A)°p ;
(p:R >R) ► (F(p):A >A) .
Conversely, given a representation (A,U) define a
functor F = -®(A,U) :B(R) >A by
F(Rn) = An ,
F((a.j),Rn >Rm) = (U(a )) : An
-» A
Examp 1 e 1 . 4 A morphism of rings f :R >S determines
representation (S,U) of R in B(S) with
U
f : R
-» Homs(S,S)op = S ,
such that -g(S,U) = f, :B(R) ►B(S) is the usual change of
rings functor.
For any object A in an additive category A there
is defined a representation (A,l) of the ring
Hom.(A,A) p in /k. The corresponding functor is the full
embedd i ng
)(A,1) : B(HomA(A,A)°p)
A ;
Hom,(A,A)op
-» A
The functor associated to a representation (A ,U) of R
in A is the composite
F = -0(A,U) : B(R)
«,
- B(HomA(A,A)op)

172
-®(A,1)
- tk -
Given chain complexes C,D in A let Hom„(C,D) be
the abelian group chain complex defined by
dHomA(C,D) : HomA(C,D)
E Horn (CD )
q-p=r *
-» HomA(C,D)r_1 ; f
dDf + (-)qfdc
There is a natural one-one correspondence between chain
maps f:C »D and O-cycles f '£Hom.(C,D)_, with
<-)"f = C.
- Dn (n£Z)
Similarly for chain homotopies and 1-chains. Thus
H„(Horn, (C,D)) is isomorphic to the additive group of
chain homotopy classes of chain maps C > D.
A chain complex C is finite if C =0 fo
there exists niO such that C =0 for r>n.
' r
r r < 0 and
Definition 1■5 Given an additive category A let D(A) be
the homotopy category of /k, the additive category of
finite chain complexes in /k and chain homotopy classes
of chain maps with
HomD(A)(C,D) = H0(HomA(C,D)) .
For a ring R we write D(B(R)) as J3(R).
We refer to Ranicki [17], [18] for an account of
the algebraic K-groups K (A) (m=0,l) of an additive
category A with the split exact structure, and the
application to chain complexes. In particular, the
class of a finite chain complex C in A is defined by
[c] = e (-)r[cr] e k0(A) ,
r = 0
and the torsion of a self chain equivalence f :C »C is

173
de fined by
7(f) = 7(d+r:C(f)
odd
>C<f>even> G K1(A)
for any chain contraction I":Os:l:C(f) >C(f) of the
algebraic mapping cone C(f).
De f i ni tion 1 ■ 6 The generalized
Mor i t a maps
JU:K (D(A)) >K (A) (m = 0,l) are defined for any additive
mm' J
category A by:
for m = 0 ju sends the class [ C ] £K Q(1D( A ) ) of an
object C in £)(A) to the class [C]€Kq(A),
for m=l ju sends the torsion 7(f)€K, (D(A )) of an
automorphism f:C >C in D(A) to the torsion 7(f)€K,(A)
of any representative self chain equivalence.
A morphism in D(A) is a chain homotopy class and
the definition of ju involves a choice of representative
chain map. The generalized Morita maps ju are therefore
not induced by a functor D(A) >A-
Examp 1 e 1 . 7 (Luck [7]) A Hurewicz fibration F >E ° iB
with the fibre F a CW complex determines a ring
morph i sm
U : Zttfj (B) ]
- H0^OmZlKl(E))^C(F^C(F)^
op
with C(F) the cellular based free Z[K■(E)]-modu1e chain
complex of the pullback F to F of the universal cover
E of E, and U the chain homotopy action of
HQ(OB)=Z[K, (B) ] on C(F) determined by the homotopy
action of the loop space QB on F. For finite F this
defines a representation (C(F),U) of Z[k,(B)] in
D(Z[tf,(E)]). For the identity map p=l:E >B = E with
F=<*> this is the universal representation (R,U) of 1.2
for R = Z[*! (B) )'Z[rtl (E) ] .

174
The transfer map in the torsion groups associated
to a representation (C , U) of a ring R in D(A) is the
compo site
U
M
(C,U)- : Kj(R) = K,(B(R)) =—► Kj(D(A)) ► Kj(A)
of the map U, induced by the functor
(C , U)g|-:B(R) >ID(A) and the generalized Morita map ju.
The torsion 7(f)€K.(R) of an automorphism f:R
i L
• Rk is
sent by (C,U)- to the torsion 7(U(f))€K, (A) of the self
chain equivalence U(f):©C >©C .
k k
The idempotent completion of an additive category
A is the additive category A with objects pairs
( A = object of A , p = p : A
A )
and morphisms f :(A,p) >(A',p') defined by morphisms
f:A >A' in A such that p'fp = f:A >A ' . The evident
functor ID(A) >D(A) is an equivalence of additive
categories, since every chain homotopy projection in A
splits (Luck and Ranicki [9]).
For any ring R the additive category P(R) of f.g.
projective R-modules is equivalent to the idempotent
completion B(R) of the additive category B(R) of based
f.g. free R-modules, with an equivalence
S(R)
'(R)
(RR,P)
-► im( p )
For any representation (C,U) of a ring R in D(A)
the functor (C,U)g)-:B(R) >& (A) extends to a functor
P(R) ►ID(A) (cf. Lemma 9.3), and so determines a
transfer map in the class groups
(C,U) • : KQ(R) = KQ(P(R))
- K0(D(A))

175
M
- KQ(A)
2 k k
The class [ im(p) ]£Kn(R) of a projection p = p :R >R is
i *■
sent by (C,U)' to the projective class [©C,U(p) ]£Kn(A )
k U
2
of the chain homotopy projection U(p)^U(p) :©C >©C .
k k
Example 1.8 (Luck [7]) A representation (C,U) of a ring
R in D(P(S)) induces algebraic K-theory transfer maps
(C,U)' : K (R) = K (P(R))
m m
-> K (S) - K (P(S))
m m
for m = 0, 1 .
The algebraic K-theory transfer maps of a
tibration F ► B with finite (or finitely dominated)
fibre F defined for m=0,l by
p- = <C(F),U>- : Km(Z[ff1<B)])
- Km<Zt,rl<E) ] }
were shown in [7] to coincide with the geometric
transfer maps using the following property of the
f unc tor
if
pw = -®(C(F),U) : BCZIff^B)]) ► DCZIff^E)])
Pr opo s i t i on 1 ■ 9 Let (X' ,X) be a relative CW pair such
that X' is obtained from X by adjoining cells in
dimensions r,r+l
X' = X V Uer V Uer+1
Given a map X' >B to a connected space B let (X' ,X) be
the pullback to (X',X) of the universal cover B of B,
and let
d : C(X',X)
r+1
![*! <B>]
-* C(X' ,X)r = ©ZfTTj (B)]
J I
be the differential in the cellular based free

176
B be a
is a CW
fibration
—»B. with
Z(fi (B) ]-module chain complex. Let F >
Hurewicz fibration such that the fibre F
complex. Let F '(Y'.Y) »(X',X) be the
obtained from p by pull back along the map X'
(Y'.Y) the pullback to (Y*,Y) of the universal cover E
of E. Then (Y',Y) is homotopy equivalent to a relative
CW pair (also denoted by (Y',Y)) with cellular base, d
free Z[^.(E)[-module chain complex
C(Y',Y) = SrC(p*(d) :eC(F)-
30(F))
the r-fold suspension of the algebraic mapping cone of
a chain map in the chain homotopy class
p (d )
P#«
:[*,(B) ]) = ec(F)
1 j
P*0
*, (B) ] ) = ec(F)
I
Proof
See L ii c. k [ 7 ] .
§2. Ma p s o f L-groups
We refer to Ranicki [14], [19] for the definition
of the quadratic L-groups L (A) (n^O) of an additive
category A with involution *:A >A» as the cobordism
groups of n-dimensiona1 quadratic Poincare complexes
(C,i/r£Q (C)) in A, and for the proof that these groups
are 4-periodic, with L„,(A) (resp,
L2i+1(A)) the Witt
group of nonsingular (-) -quadratic forms (resp,
formations) in A.
We now put an involution on the notions of §1.
Definition 2 . 1 An i nvo1u t i on on an additive category A
is a contravariant functor
: A
A ; M
-» M

177
(f:M >N) ► (f :N >M )
together with a natural equivalence
: A ► A ;
M ► (e(M):M »M )
such t ha t
* _ i * * * * *
e(M ) = (e(M) L) : M ► M
□
We shall use the natural isomorphisms
* * * *
e(M):M >M to identify M =M.
Examp1e 2.2 Given a ring R with involution
: R ► R ; r ► r
let the additive category
B(R) = {based f.g. free R-modules)
have the duality involution
(Rn)* = Rn , (aij>*
s uch t ha t
Ln(B(R)) = Ln<R) (n£0) •
By definition, a quadratic Poincare complex over R is
the same as a quadratic Poincare complex in B(R)-
□
No tat ion 2.3 Let /k be an additive category with
i nvo1u t i on .
id
tk
J1

178
i) A chain complex C in A is n-dimensional if C =0 for
r < 0 and r >n.
ii) The n-dua1 of an n-dimensional chain complex C is
the n-dimensional chain complex C
in A with
dcn-* = (-)r(dc)
<cn" )r = c"~r = (cn_r)
r» — *
(cn )
r- 1
iii) For n^O let D (A) be the additive category of
n-dimensional chain complexes in A and chain homotopy
classes of chain maps, with the n-duality involution
T = n-* :Dn(A)-
»Dn(A) ;C-
A functor of additive categories with involution
F:A >B is a functor of the underlying additive
categories together with a natural equivalence
G : F* i * F;A >B , such that for any object M in A there
is defined a commutative diagram in B
e,R<F(M)) **
F(M) B , F(M)
F(eA(M))
G(M)
** G(M*) * *
F(M ) ► F(M )
Notation 2.4 A functor F:A-
of additive categories
with involution induces morphisms of the quadratic
L-groups which we write as
F, : Ln(A)
- Ln(B) ;
(C,,/,)
-> <F(C),F(vO) (n^O) .
Examp1e
_5_ A morphism of rings with involution f : R-

179
determines functors of additive categories with
involution f,:B(R) >B(S) which induces change of rings
morphisms in the. quadratic I.-groups f
(n£0) .
Ln(R)-
.Ln(S)
Definition 2,6 Given a nonsingular symmetric form
5 —*—
(A,a=a :A >A ) in an additive category with involution
A let the ring Hom«(A,A) " have the involution
: H o m « ( A , A )
op
(f :A >A)
-> Hoi. ( A , A )
- 1 *
( a f a :A-
op
•A) .
By analogy with Definition 1.1:
De f i n i t i on 2.7 A s ymme trie representation ( A , a , U ) of a
ring with involution R in an additive category with
involution A is a nonsingular symmetric form (A,a) in A
together with a morphism of rings with involution
U:R iHom, ( A,A)op .
In particular, (A , U) is a representation of R in
the additive category /k in the sense of 1.1.
By analogy with Example 1.2:
Examp1e 2.8 The un i ve r s a 1 symmetric representation
(R, a,U) of a ring with involution R in B(R) is defined
by
-» R
(
■• sr )
with U the isomorphism of rings with involution

180
U : R ► HomR(R,R)°P ; r ► ( s ► sr ) .
We shall use U as an identification of rings with
involution R=HomR(R,R) p .
□
By analogy with Proposition 1.3:
Proposition 2 . 9 Given a ring with involution R and an
additive category with involution A there is a natural
one-one correspondence between functors of pairs of
additive categories with involution F:B(R) >A and
symmetric representations ( A , a , U ) of R in A.
Proof : Given a functor F define a symmetric
representation ( A , a ,U ) by
A = F(R) ,
* * *
a » G(R) : F(R ) = A ► F(R) = A ,
U : R = HomR(R,R)°P ► Hom„(A,A)°P ;
(p:R >R) ► (F(p):A >A) .
Conversely, given a symmetric representation (A , a,U)
define a functor F = -®( A , a ,U) :B(R) >A by
F(R) = A ,
* * *
G(R) = a : F(R ) = A ► F(R) = A ,
F((ai:j):Rn >Rm) = (U(a,.,)) : An ► Am .
□
By definition, a nonsingular symmetric form (C,a)
in D (A) is an n-dimensiona1 symmetric complex C in A
together with a self dual chain homotopy class of chain
equivalences a^Tct:C >Cn~ .

181
Propo s i t i on 2.10 A symmetric representation ( C , a , U ) of
a ring with involution R in D ( A ) determines a functor
F = -g)(C, a, U) :B(R ) >ID (A) inducing morphisms in the
quadratic L-groups
F, = -®(C,a,U) : Lm(R) ► Lm(Dn(A)) <m£0> •
Proof : Immediate from 2.4 and 2.9.
□
Given a ring with involution S let D (S)=D (B(S)),
the additive category of n-dimensiona1 chain complexes
of based f.g. free S-modules and chain homotopy classes
of chain maps with the n-duality involution C >C . A
symmetric representation (C,a,U) of a ring with
involution R in D (S) determines by 2.10 a functor
F = -g)(C , a, U) :B(R ) >D (S) inducing morphisms in the
quadratic L-groups
F, = -0(C,a,U) : Lm(R) ► Lm(Dn(S)) <m£°> •
§3. The generali zed Mor i t a maps in L-theory
By analogy with the algebraic K-theory generalized
Morita maps ju : K (D(A)) >K (A) (m=0,l) of §1 we define
generalized Morita maps in the quadratic L-groups
U:Lm(IDn(A) ) >Lm+n<A> (m,n£0) by passing from
nonsingular quadratic forms and formations in D (A) to
quadratic Poincare complexes in A. The L-theory \x is
the identity for n = 0 , since Dn(A)=A. For n^ 1 the maps ju
are not isomorphisms and are not induced by functors of
additive categories with involution; a morphism in
D (A) is a chain homotopy class and as in K-theory the
definition of ju involves a choice of representative
cha i n map .
Propo s i t i on 3.1 i) A nonsingular (-) -quadratic form in

182
D (A)
•i * *
(M,©ecoker(1 -( - )xTiHom^ (..(M,M ) ► Hom^ (A)(M,M )))
is represented by an n-dimensional chain complex M in A
together with
( 1 + (- )iT) 9=9+(.- )n + i 9* :M
chain map © : M-
such that
■M is a chain equivalence,
ii) The cobordism class (C,^)£L +,-(A) of the
(n+2i)-dimensiona1 quadratic Poincare complex in A
(C,i/0 defined by C=Mn+1~ and
9 if s=0
0 if sil
,n + 2 i -r -s
M
-» C
M
depends only on the class
9 £ coker( l-(-)1T:HQ(Hom„ (M,Mn~ )) >HQ(Hom^(M,Mn~ )))
= coker ( 1-(-) 1T iHom^ ,„..(M,Mn~ )-
.HomDn(A)(M,Mn- )) .
iii) Suppose given (C , yfr) as in ii), an n-d imens i ona 1
chain complex L, a chain map j : L >M and
<x€Hom,(L ,L )|r^0> defining a chain homotopy
., .n+i + 1 * ,* , ,
-» L
L l s a
such that the chain map (j (l+T)i/rQ 0):C(j)-
chain equivalence, with C(j) the algebraic mapping cone
of j. Then (C,^)=OeLn+2,(A)•
Proof : i) Trivial.
ii) The isomorphism of abelian groups
Q()i(M) = coker( 1-(-)iT:HomD (A)(M,M )
,HomDn(A)(M'M )}
Qn+21(C) ;

183
[6>:M >Mn *] ► <^seHomA(Cn + 2i_r S , Cr ) | r , s£0 >
defined by
^0 = G ' ^s ~ ° for S^[
sends the class of 9 to the quadratic structure
*€Qn+2i(c).
iii) Define an ( n + 2i + 1 )-dimensiona1 quadratic Poincare
pair in A (f:C >D,(8^,^)) by
f - j* s C - Mn+i~*
-» D = L
n+i
s , _ nn + 2 i +1-r _ .
_i_l
-» D = L
r
5^ , = 0 for s£1 .
We refer to §2 of Ranicki [19] for the definition
of a nonsingular (-) -quadratic formation
(F ,G) = (F, I YJG) in an additive category with involution
A, and fof ■dhe result that (F , G) =0£L 2 j (A) if and only
if there there exist a (-)1 -quadratic form in A (H,f)
and a morphism j:F >H such that the morphism defined
in A by
M
j
Y j
r+(-)i + V
j
; F©H
-» G ©H
is an isomorphism.
Propos i t ion 3.2 i) A nonsingular (-) -quadratic
formation (F.I |G) in D (A) is represented by

184
n-d i mens i ona 1 chain complexes F,G in /k together with
chain maps Y:G-
homo t o py
IX-
, 9:G-
and a chain
X = Y*M * 9 + (-)n+i + V : G
-> G'
such that the chain map
X+(~)n+iX* Y*
c(MV+1~*
C(m :F-
n-*
>Gn )
is a chain equivalence.
ii) The cobordism class ( C , y/, )€Ln+2 ± + j ( A)
( n+2i + 1)-dimensiona1 quadratic Poincare
(C = S1C(ju ) tVJr) in A with
of the
complex
d* (-)r_3TW
0 d„
Cr = Gn"r+1eF
r-i-1
- c . = Gn~r~1 + V . ,
r - 1 r-i-2
(.)(n+l)(r-l)x 0
(-)n<r_1>Y Q
cn+2i+l-r = ,
r- l - 1
- c = Gn~r+1eF , .
r r- l - 1
f (_)(n+l)r+le Q 1
0 0
cn+2i~r = g ,eFn-r+i~1
r- l
-» c
Gn~r+ieF
r-i-1

185
0 : C
n+2 i + 1 -r-s
-► C for s£2 ,
r
depends only on the, chain homotopy classes
Y € Hn(HomA(G,F) ) = Hom^ (^)<G>F) »
M € H0(HomA(G,Fn-*) ) = Hom^, (£)<G>Fn~*> •
iii) Suppose given ( C , i/r ) as in i), an n-dimensional
chain complex H in A and chain maps j : F i H ,
n- *
f:H >H such that the chain map
M
Y j
, r+(_)n+i+lj*
F0H
|-» — * |-» — *
-» G 0H
is a chain equivalence. Then (C,i/O=0£L +2-+i(A).
Proof : i) The inclusion of the lagrangian
(G,0) >H,_..i(F) extends to an isomorphism of
(-) -quadratic forms in D (A)
Y Y
M M
H()i(G)
-► H()i(F)
which is represented by a chain equivalence in A
Y Y
M M
n-* n-*
: G0G ► F0F
with chain homotopy inverse
Y Y
M M
M
(-)XY
i *
FffiF
-» GffiG

186
*J n — * n — *
For any representative chain map ju; G >F there
exist a chain map 1>:F >F and a chain homotopy
'"* ^ , .n+i+1 *
J] ■ MM * v + (.-) V
-» F
The chain maps in IK defined by
f =
X+(->n+1X* Y*
: C(M*)n+1-*
-► c(m )
g =
0 M
M *+(-)n+V
c(m*)
-► C(ju )
ire such that there are defined chain homotopies
C Y
0 5
: gf *
1 0
& 1
automorphi sm
c(mV+1~*
-► C(ju )
o 1
fg a
1 6 1
0 1
automorphism
C(m*)
■* C(u )
n+i
n+i.
with £«m(x+(-) X ) + (77+(-) 7? )Y, and S,e chain
homo t op i e s
5 : MY +(-)XMY a 1 : Fn
-» F
~ * i "•■ *
C : MY+(-) YM- 1 : G
-» G .
Thus both fg and gf are chain equivalences, and f is a
chain equivalence with chain homotopy inverse
(gf)"1g^g(fg)"1.

187
ii) With | M as in i) there exist chain maps in A
IM mJ
Y : G'
-> F , u : G
-> F'
9 i G'
-» G
and a chain homotopy
X : Y M - © + (-)
© : G'
-» G
such that the chain map
X+(-)n+1X Y
M * n + 1 - *
C(W )n+1
-► c(u )
chain equivalence in th~ Let (C,i/r)
be
the
(n+2i+1)-dimensiona1 quadratic Poincare complex derived
r» — * ^ ^ ^ ^
from (F,G >Y,U,0,x) ln tne waY (C.yO is derived from
(F,G,Y,M,9,x)• Define an (n+2i+2)-dimensiona1 quadratic
Poincare cobordism ((f f ) : C©C >D , ( Si^ , \pQ-\p ) ) by
D = S1+1F
Sl/r
0
(0 1 )
c = Gn~r+i©F . .
r r - l - 1
= F
(0 1)
G .©F .
r-l r-l- 1
= F
r-i-1
Thus (C,yt) »(C ,yt)€L +2i+i (A) • Since 9 and x can be
chosen independently of 9 and x it follows that the
cobordism is independent of these choices also. Given
(F,G,Y,U>#>x) and chain equivalences h:F >F ' , k:G >G*
it is possible to define (F',G',Y,,Ll',©,,x') such that
the corresponding quadratic Poincare complex (C ' , \p ' ) is
homotopy equivalent to (C, <p) , and so
(C tf ) = (C,^)eLn+2 i+1 (A) •
iii) Define an ( n + 2i + 2)-dimensiona1 quadratic Poincare
pair (f:C ^.(Syfr.yfr)) by

188
D = H
n+i+ 1
f = (0 j)
-» D = H
r
n+i + 1-r
S^n " f
Dn+2i + 2-r „ > „
r - l - 1 r
n+i+1-r
8^ = 0 for s£l .
This is a quadratic Poincare nu11 -cobordism of (C.yO,
so that (C,^)=0eLn+2,+1(A)-
Definiti on
3. 3
For
add i t i ve
ca te gory
involution A define the generalized Mor i t a ma ps
with
M
Lm(Dn<A>>
Lm+n<A>
(m,n£0)
for m=2i (resp. 2i+l) by sending a nonsingular
(-) -quadratic form (M,©) (resp. formation (F,G)) in
D (A) to the cobordism class of the (m+n)-dimensiona1
n /
quadratic Poincare complex (C,i/0 in A defined in
Proposition 3.1 ii) (resp. 3.2 ii) ) . The verification
that the maps jn are well-defined is contained in
Propositions 3.1 iii) (resp. 3.2 iii)).
For a ring with involution R apply 3.3 to A=B(R)
tai
(m,n^O).
to obtain generalized Morita maps ju: L (D (R))
L _,_ (R)
m + n
§4. The quadra tic L-theory t rans f e r
As before, let A be an additive category with

189
involution, and let D (A) be the chain homotopy
category of n-dimensiona1 chain complexes in A with the
n-duality involution.
Defini ti on 4. 1 The quadra tic L~theory t ran s f er maps of
a symmetric representation (C, a,U) of a ring with
involution R in D (A)
(C.ct.U)' : Lm(R)
Lm+n(A) <m*0)
are the composites
(c.a.U)• : Lm(R) = Lm(B(R))
)(C,a,U)
M
Ln,<<D„(A))
m n
- *•«+„<*>
of the maps -g)(C,a,U) of 2.10 and the generalized
Morita maps jn of 3.3.
Examp 1 e 4 . 2 Let A be the additive category B(S) of
based f.g. free S-modules with the duality involution,
for a ring with involution S. The transfer maps
determined by an n-dimensiona1 symmetric representation
(C, a, U) of a ring with involution R i n (D (A)=(D (S) are
morphisms of quadratic L-groups
(C.a.U)
!
Lm<R>
m+n
Lm + n(S) (m.nJsO)
Examp1e 4 . 3 Given a Hurewicz fibration F ► B with
the fibre F a finite n-dimensiona1 geometric Poincare
complex we shall define in §5 below a symmetric
representation (C(F),a,U) of Z [ * , ( B ) ] in D (Z[K, (E) ]) ,
with F the pullback to F of the universal cover E of E
and a=( [F]A-)~* :C(F) >C(F)n~ the Poincare duality
chain equivalence. The algebraic surgery transfer maps
will be defined in §5 to be

190
Palg = <C<F>.<*.">
Lm(Ztfr1<B) ])
Lm + n(Ztff1<E) ]) (mJtO)
In §6 we shall recall the definition via the lifting of
normal maps of the geometric surgery transfer maps
I t
p , which will be identified with p'. in §8.
*geo * *a1g
Examp1e 4.4 Given a morphism of rings with involution
f :R »S define a symmetric representation ( C , a , U ) of R
in D0(S)=B(S) by
-► C
0
( t
-» ts ) ,
Cr=0 for r#0 ,
-> H0(Homs(C,C))op = S
In this case the transfer maps are just the change of
t
rings morphisms (C, a ,U) " =f, :L (R) >L (S). For
f = l:R >S = R (C, a ,U) is the universal symmetric
representation (2.8) of R in B(R).
Examp 1 e 4.5 Given a ring with involution S and an
integer k^ 1 let R = M, (S) be the ring of kXk matrices
( s . . ) . . . ., with entries s..£S, with the involution
-» R
; (-^)
<-ji> •
Define a symmetric representation (C, a,U) of R in
D0(S)=B(S) by

191
'0
k
ES
1
k
ES
1
1 * S 2 S k
0 for r # 0 ,
0
-» C
(ES)
1
1 *2*****lc
■* tl8l+t282-
•+tRsk) ,
U = 1 : R = MR ( S)
-> H0(Homs(C,C))°P = MR(S) .
The generalized Morita maps ju:L*(R) >L*(S) in this
case are just the usual Morita maps, which are
isomorphisms for the projective and round L-groups. See
Hambleton, Taylor and Williams (5] and Hambleton,
Ranicki and Taylor (4] for Morita maps in quadratic
L-theor y.
Example 4 . 6 Let F = U<*> >E—K-» B be a k-sheeted finite
covering, so that k>(E) is a subgroup of k,(B) of index
k. There are evident identifications of spaces
F = tf, (B) = Utf,(E) C B = E ,
1 k L
and also of Z-module chain complexes
C(F) = ZUjCB) ]
:[*!<£) ]
The symmetric representation (C(F),a,U) of Z ( tf < ( B ) ] in
Dq(Z[k1 (E) ] )=B(Z[?r1 (E) ] ) associated to p:E >B (as in
4.3) is given by
U : Z[Kl(.B)] = H0(Homz, ( B) j (C(F ) ,C(F) ) )
op

192
restrict!on
H0(Homz[)t (E)](C(F),C(F)))°P = MR ( Z [ K l ( E ) ] )
a = ©1 : C(F) = ©ZU,(E)]
k k
Homz[)r (E) j (C(F) ,Z[Kl (E) ] ) = ©Z(*j(E)]
The algebraic transfer maps in this case are the
compo sites
U
Palg : ^(Zt^^B)!)
Lm(Mk(Z[frj <E) J))
M
Lm(Z[frj (E) ])
with U, induced by U as in 2.5 and jn the Morita maps of
4.5. In this case p", can be described more directly
alg J
by the restrictions of Z t ft i ( B ) ] -modu 1 e actions to
Z t ft i ( E ) ] -modu 1 e actions, and it is clear that
t t
*alg rgeo
Examp1e 4.7 The algebraic S -bundle transfer maps of
Munkholm and Pedersen [10] and Ranicki [16,^7.8]
p • : L (R)-
ralg m
■L ,,(S) are defined for any ring with
m+1 jo
involution S, with R=S/(t-l) for a central element t£S
such that t*=t . (We are only dealing with the
orientable case here). From our point of view these are
t j
the quadratic L-theory transfer maps p' . =(C, a,U) of
4.1 with (C, a, U) the symmetric representation of R in
Dj ( S) given by
1
C, = S
-> co = s

193
t : C,
a =
-» c
-» c
For an S -bundle S *K—E_B one takes R=Z(*1(B)J,
S = ZU1(E)], t = f ibreG^! (E) .
$ ^• The a 1ge bra i c surgery t rans f er
A map p:E >B of connected spaces with homotopy
fibre of the homotopy type of a finite (or finitely
dominated) CW complex F determines a representation of
Z[Kl (B) ] in £)(2(^ (E) ])
(C(F) ,V:Z[Kl(h) } »H0(Homz()t (E}j(C(F),C(F)))op)
as in 1.7. We shall now show that if F is a finite
n-dimensiona1 geometric Poincare complex then for any
choice of orientation map w(B): 7T>(B) >Z2 in the base
there is defined a symmetric representation (C(F) , a ,U )
of Z(*j(B)J in D (Zt*i(E)])> and hence obtain from §4
quadratic L-theory transfer maps
Palg = <C<F>.a.u>
^(ZU^B) ])
Lm+n(Zt*l(E)]) (m^0>
In §8 below we shall identify these algebraic, surgery
transfer maps with the geometric surgery transfer
maps .
There is no loss of generality in assuming that
F >E-iU B is a Hurewicz fibration with the fibre
F=p~ (*) a finite CW complex F. If F is disconnected
then p:E »B is the composite of a Hurewicz fibration

194
p':E >B ' with connected fibre p'~ (*) and a finite
covering B' ► B . Since transfer theory is well-known
for finite covers (cf• 4.6) there is no loss of
generality in taking F to be connected. In fact, the
algebraic transfer maps are defined in exactly the same
way for disconnected F, and only the geometric
treatment of the orientation maps has to be modified by
using groupoids instead of groups.
Transport of the fibre along paths in the base
p
space gives a map QB >F which on x-. induces a group
morphism U:x, (B) ► (F , F ] to the monoid of homotopy
classes of self-maps of F (Whitehead [24,p. 186]).
Analogously, one has the pointed transport of the
pointed fibre along paths in E, defining a morphism
U : K. (E) >(F,F] to the monoid of pointed homotopy
classes of pointed self-maps of F. Homotopy along a
path defines a morphism 7r,(F) >(F,F] (Whitehead
(24,p.98ff]).
Propo s i t i on 5.1 The transport maps define a morphism
from an exact sequence of groups to an exact sequence
of pointed sets
ffjCF)
Kl(E)
P*
tf^B)
<1>
ff^F)
■* (F,F] +
- [F.FJ
<1>
We shall now use 5.1 in the case when F is a
geometric Poincare complex to lift an orientation map
w(B) for 7f,(B) to an orientation map w(E) for tt,(E).

195
De f i n i t ion 5 . 2 An or i en t at ion map for a group k i s a
morphism w; k ► Z 2 = < ± 1 > • Let Z t * ] W denote the ring Z t * ]
with the w-twisted involution
:(*]
'.[n] ; E n g
g€* 8
-► E nw(g)g
-1
g€rf
g
Given a chain complex C in IB ( Z t f ] ) let Wc denote the
n- *
n-dual chain complex C in IB ( Z t f ] ) defined using the
w n — *
w-twisted involution on Ztf]* If w is trivial C is
' denote the right ZtK)-modu1e
wr i 11 en as
Let
with additive group Z and
Zw X ZtffJ
?w ; (m, E n g)
gs
m( E w(g)n )
g€rf 6 g€rf
Let Z denote the left ZtK]-module defined in the same
way .
When w is clear we abbreviate ZtT] to ZtK)•
An n-dimensiona1 geometric Poincare complex X is a
(connected) finite CW complex together with an
orientation map w(X):tt. (X)-
c 1 as s
and a fundamental
IX] e Hn(x;zw(x)) = Hn(Zw®zUi(x)]c(x))
such that the Z t *i (x) ) -modu1e chain map
tX]A-:C(X)n~ »C(X) is a chain equivalence, with
X the universal cover. See Wall t21) f°r tne general
the or y .
'he orientation map w = w(X):7r=7r,(X) ► Z 9
of

196
n-dimensiona1 geometric Poincare complex X is
determined by the topology of X, since the cap product
with a fundamental class tx)€H (X;ZW) defines an
isomorphism of 2[?r]-raodules
[X]A- : Hn(wC(X)n~ )
- HQ(X)
If Hn(X) is defined to be H0(C(X)n~ ) using the
untwisted involution (g = g~ ) °n Z t * ] then we get
Hn(X)SwZ.
De finition 5.3 Let X be an n-dimensional geometric
Poincare complex.
i) The degr ee of a pointed self-map f :X >X is the
number d(f)£Z such that
Hn(X)
Hn(X ) ; 1
- d(f) ,
with f :X >X a lift of f to a self map of the universal
cove r X.
ii) The homot opy orientat i on of X is the monoid
morph ism
^ ^
w = w(X) : ( X , X ]'
»X
-> d(f) ,
with Z the monoid defined by Z and multiplication,
Let f :X »X be a pointed self homotopy
equivalence, inducing an automorphism f* :k *K of the
fundamental group x=X. (X) . A lift f: X——* X of f to the
universal cover X induces a 2-module chain equivalence
f:C(X) >C(X) which is f*-equivariant

197
f (gx)
f*(g)(x) e c(x) (g€*,xec(x))
The induced isomorphism of additive groups
. U " f V ^ — ~ It ,. H"/'V'\m~-3> i r, ~ 1 „ „ t
'v x - "" ► H (X)= Z is also f^-equivariant. Hence we
f :H"(X)=
have
w = w f * : x
* w
» K
and f* defines an automorphism f * : Z t f ]
!(*]w of the
ring with involution Z t * ] • The Z-module automorphism
f*:H„(X;ZW)'
'Hn(X;ZW)^
such that
,--»X
f*( (X])=d(f ) [X] , with d(f )=w(f )€< + l>=Z2CZ'v. In
particular, it follows that the orientation map w and
the homotopy orientation w are related by a commutative
diagram of monoid morphisms
*! (X)
< + !>
(X,X]'
„X
Propos i t i on 5.4 For any pointed self homotopy
equivalence f :X >X there is defined a chain homotopy
commutative diagram of Z-module chain complexes and
chain equivalences
w „ n_* d(f)(f V „
wc(x)n ► wc(x)n
tx]A-
tx]A-
c(x)
•* C(X)
with the horizontal chain maps f*-equivariant , and the

198
vertical chain maps if. (X)-equivariant .
Definition 5.5 An n-dimensiona1 Poincare fibration
F >E-JU B is a Hurewicz fibration with the fibre F an
n-dimensional geometric Poincare complex, together with
an orientation map w(B):;r.(B) >l£n . The lift of w(B) is
the orientation map
p!w(B) = w(E) : 7rt(E) ► Z2 ;
g ► w(B)(p*(g)) .w(F)(U+(g))
with U as in 5.1 and w as in 5.3.
Propo s i t i on 5.6 An n-dimensiona1 Poincare fibration
B determines a symmetric representation
(C(F).a.U) of Ztff^B) JW<B> in (Dn(ZU,(E) ]W<E>) wit
)
h
a=((F]A-):C(F) >C(F)n~ the Poincare duality chain
equivalence and (C(F),U) the representation of Z t f i ( B ) ]
in ID n(Z t K i ( E ) ] ) associated to p.
Proo f: We have to show that
U : ZtiCj (B) ]W(B) ► H0(Homz[(t (E) j j < C( F) . C( F) ) )op
is a morphism of rings with involution, or equivalently
that for every g€7r.(B) there is defined a chain
homotopy commutative diagram of 2[ff.(E)]-module chain
comp1e xe s

199
w(E)c(?)n~* U(S) , w(E)c(?)n-
[F]A- [Fjfl-
w(B)(g)L'(g~' )
C(F) ► C(F) .
This follows from 5.4 and the if, (E)-equivariant
transport along the fibre U: *. (B) ► ( F , F ] used to
i ft . \ c- )
define the ring morphism ij in Luck (7] •
Definit ion 5 . 7 The alge bra i c surgery t rans f e r maps of
an n-d imens i ona 1 Poincare fibration F »E ° i B are the
quadratic L-theory transfer maps of 4.1 associated to
the symmetric representation ( C ( F ) , a , U ) of
Z(*! (B) ]w(B) in Dn(Z(*j (E) ]W<E>) given by 5.6
palg = <C<^> >a>u> ' ■
Lm(Ztfr1<B) J) ► Lm+n(Zt*l(E)]) <m£0> •
By definition, the algebraic surgery transfer maps
are the composites
Palg : ^(Ztf^B)])
M
(P ),
-—> Lm(Dn(Z(7r1(E) ]))
Lm+n(Zt*l(E)]) ^°)
of the maps induced as in %2 by the functor of additive
categories with involution

200
p" = -®(c(F),a,u) : BCZtff^B)]) ► Dn(Ztff1(E)J)
and the generalized Morita maps jn of %3 .
§6. The geometrie surgery t r an s f e r
Wall [22] defined the rel3 surgery obstruction
cr*( f ,b)eLm(Z[ ?r j (X) ] ) for a normal map
(f,b):(M,3M) ► (X,3X) from a compact m-manifold with
boundary (M,3M) to a finite m-dimensiona1 geometric
Poincare pair ( X , 3X ) with 3f=f|;3M >3X a homotopy
equivalence, and b : v., ►$•„ a map from the stable normal
bundle of M to a topological reduction of the Spivak
normal fibration Vy, of X, with the w(X)-twisted
involution on Z(7f,(X)]. The surgery obstruction has the
property that cr*(f,b) = 0 if (and for m> 5 only if) (f,b)
is normal bordant rel3 to a homotopy equivalence of
pairs. Given a connected space B with finitely
presented x> (B) , and given an orientation map
w(B):7f,(B) >22i it is possible to realize every
element x£L (Zt*i(B)]) (m^5) as the surgery obstruction
of an m-dimensional normal map (f,b):(M,3M) ► (X,3X)
with a if. -isomorphism reference map X >B and
orientation map w(X):7f, (X) ► K, ( B ) —SLi—i ►g'
x = or*(f ,b) € ^(ZUj (B) ] ) .
The total space E of an n-dimensiona1 Poincare
fibration F >E P ) B over an m-dimensional geometric
Poincare complex B is homotopy equivalent to an
( m+n)-dimensiona1 geometric Poincare complex, with the
t
orientation map the lift w(E)=p'w(B) :x. (E) >Z2 in tne
sense of 5.5 of the orientation map w(B):7r.(B) >Z2
(Quinn (12], Gottlieb (2]).
Quinn (11] used the realization theorem for

201
surgery obstructions to define geometric transfer maps
in the quadratic L-groups for a fibre bundle (or even a
block fibration) F >E P > B with the fibre F a compact
n-man i fo1d
Pgeo : ^(2(^(8)]) ► Lm + n(Zt^(E)]) ;
a*((f ,b): (M.9M) >(X,9X))
► CT*((g,c) : (N.9N) ►(Y,9Y)) .
Here, ( g , c ) : ( N , 9N ) ► (Y,9Y) the (m + n)-dimensiona1
normal map equipped with a reference map Y >E obtained
from the n-dimensional normal map (f,b): M——♦ X by the
pullback of p along a reference map X >B.
The surgery obstruction of Wall [22] was defined
using geometric intersection numbers on the homology
remaining after surgery below the middle dimension. The
theory of Ranicki (14),(15] associates an invariant in
Lm(Z[K j (X) ] ) to a normal map ( f , b ) : ( M , 9M) ► ( X , 9x ) of
m-d imens i ona 1 geometric Poincare pairs, with b : 1>M ► !>..
a map of the Spivak normal fibrations and 9f:9M '9X a
homotopy equivalence. The quadratic kernel of (f ,b) is
an m-d imens i ona 1 quadratic Poincare complex (C(f *),i/f)
over Z(7r.(X)]. Here, C(f") is the algebraic mapping
cone of the Umkehr 2(K.(X)]-modu1e chain map
■•*
~ ~ ((x]A-)_1 „ _* f
f : C(X,9X) ► C(X)m * C(M)
(M]A
m-
. *
■* c(m,9m)
with X the universal cover of X, f :M >X a
if. (X)-equivariant lift of f to the pullback cover
M=f X of M. The Poincare duality chain equivalence is
given up to chain homotopy by the composite

202
(1+T)vfr0 : C(f)'
[Mjfl-
C(M)
C(M,3M)'
-» c(m,3m)
c(f •)
with e:C(M,3M) >C(f ') the inclusion. The quadratic
signature of (f,b) is the cobordism class
flr*(f.b) = (C(f!),vO € L^ZI*! (X) ]) .
A normal map from a manifold to a geometric Poincare
complex determines a normal map of geometric Poincare
complexes with quadratic signature the surgery
obstruct i on .
De f i n i t i on 6 . 1 The geometrie surgery t rans f e r map s of
an n-dimensional Poincare fibration F >E " > B with
finitely presented k. (B)
>geo • 1..(2(^(8)1)
Lm+n<*t*l<E>l> *
or*( (f ,b) :M »X)
- ff*((g,c):N ►¥) ( mjs 5 )
are defined using the quadratic signature of normal
maps of geometric Poincare complexes. Here, (g,c) :N >Y
is the (m+n)-dimensiona1 normal map obtained from an
m-dimensional normal map (f,b):M >X by the pullback of
p along a reference map X >B.
The orem 6.2 The geometric surgery transfer maps of an
n-dimensional Poincare fibration F ► B coincide
with the algebraic surgery transfer maps

203
t t
geo ralg
Lm(Ztfr1<B) J) ► Lm + n(Zt?r1(E) ]) <m£5> .
□
The proof of 6.2 is deferred to $8. The ideal
proof would express the quadratic kernel of the
pullback normal map of the total (m+n)-dimensional
geometric Poincare complexes (g,c):N >Y as a twisted
tensor product of the quadratic kernel of the normal
map of the base m-dimensiona1 geometric Poincare
complexes (f,b):M >X and the symmetric Poincare
complex (C(F),0). This would generalize the chain level
proof of the surgery product formula in RanickJ [15] in
the untwisted case p = projection :E = BXF >B
or*((f ,b)Xl :MXF >XXF) = a* ( f , b ) ga* ( F )
€ Lm+n(Ztfr1(B)Xfr1(F) J)
which expressed the quadratic signature of a product
(f,b)Xl as the tensor product of the quadratic
signature of (f,b) and the symmetric signature
a*(F) = (C(F) ,0)€Ln(Z[n'1(F) ] ) . However, this would
require the development of a fair amount of new
technology, translating the homotopy action of QB on
the geometric Poincare complex F into a chain homotopy
action of C(QB) on the symmetric Poincare complex
(C(F),0) over Z[K.(E)]. For the purpose at hand we can
assume by the realization theorem that the
m-dimensional normal map (f,b):M ► X is
[(m-2)/2]-connected. In the highly-connected case we
can give a chain level geometric interpretation of both
the element U. a * (f,b)£L (D (Z[K, (E) ] )) and its image
: * m n 1

204
under the generalized Morita map
M:Lm(IDn(Z(K'1(E) ])) >Lm + n(2t?r1(E) ]) • For a fibre bundle
F >E—£_► B it is possible to dispense with some of the
algebra, using instead the fibred intersection theory
of Hatcher and Quinn (6] as outlined in Appendix 1
be 1ow•
%7 . Ultraquadrat i c I,-theory
Ultraquadratic L-theory was developed in §7.8 of
Ranicki [16] in connection with the algebraic theory of
codimension 2 surgery. We use it here to recognize
quadratic Poincare complexes in the image of the
generalized Morita maps ju : l*m (Dn ( A) ) >Lm+n^^ °f ^3,
providing a tool for the identification in §8 below of
the algebraic and geometric surgery transfer maps.
Let A be an additive category with involution. As
in Ranicki [15),(19] define for any finite chain
complex C in A and e=+l the Z~module chain complex
W„C = V®-.- ,Hom„(C ,C) ,
'X" " ""fI1Z2 } A
with the generator T(
cting on Horn.(C ,C) by the
e-transposition involution T =eT and W the standard
free 2[2,]-module resolution of Z
1-T 1+T
- ztz2j ► ztz2j ► Z[Z21
1-T
-> ztz2j
An m-chain ^r£(W„C) is a collection of morphisms
* = <^s€HomA(C .C>m_sls»0>
such that for a cycle there is defined a chain map

205
(l+Te)^0:C
An
m-dimensional e-quadratic
(Poincare) complex (C , yfr) in A is an m-dimensional chain
complex C in A together with an
e1ement
C is a
^€Qm(C,e)=Hin(W%C) (such that ( 1+Tc)yjr0 : C" -
chain equivalence). The skew-suspension isomorphisms
S : Qm(C,0
(w(sc*-£) * *
s./,
are defined by ( Si/O =+<p (s^O), for any finite chain
complex C in A. The skew-suspension maps
S:L (A,e) >l. +0(A,-e) (m^O) in the. +e-quadratic
L-groups are also isomorphisms, so that
Lm(A.€> = Lm+2^.-€) = Lm+4(A,£) <m£0) .
For e=l we write Q (C,1)=Q (C), L (A,1)=L (A), and
m mm m
1-quadratic = quadratic.
Ultraquadratic complexes are e-quadratic complexes
(C,,/,) with ^,=0 for s^l.
For any finite chain complex C in A define the
abelian group
Qm(C) = Hjn(HomA(C ,C)) = H0(HomA(Cm ,C))
of chain homotopy classes of chain maps ifi ;C
De f in i t i on 7.1 An m-dimensional e-ultraquadratic
(Po i neare) complex i n A (C, yfr ) is an m-dimensional chain
complex C in A together with an element d£Q (C) (such
T m
A m- *
that (1+T )i/r:C >C is a chain equivalence).
There is a corresponding notion of cobordism of

206
e-ultraquadratic Poincare complexes in /k, with the
m-dimensional cobordism group denoted by L (A,e), and
by L (A) for e = +l. The c-u 11 r aquad r a t i c L-groups are
4-periodic, with
Lm(A'e) = Lm+2(A>-£) = Lm+4(A'e) (n^0)
by skew-suspension isomorphisms, just like for the
C-quadratic L-groups
If A=B(R) for a ring with
involution R we write L (A) as L (R).
m m
^ *>. **
Define a map Q ( C ) >Q (C,e);^ >*/, by ¢^¢, ^=°
III III y
(s^l). An m-dimensional e-ultraquadratic (Poincare)
complex (C,<^) determines an m-dimensional quadratic
(Poincare) complex ( C , yfr ) • The forgetful maps in the
cobordism groups
Lm(A,0
- Lm(A,€) ; (CyO
-► (C,^) (m£0)
are surjective for even m and infective for odd m.
The u1traquadratic L-group L (Z) was identified in
§7.8 of [16] with the cobordism group C _. of knots
k:Sm"1CSm+1 (m£4). A Seifert surface for a knot
k : Sm" CS111 is a codimension 1 framed submanifold
M™CSm+1 with boundary dM=k(Sm~*) . Inclusion defines an
m-dimensional normal map (f,b):(M,3M) >(D ,S )
with quadratic kernel a* ( f , b ) = ( C , <j/ ) such that
H*(C)-H* , (Dm+2,M)=H*(M) . The framing determines a map
M-
•Sm -M which induces a chain
map
yfr : C"
defining an m-dimensional u1traquadratic Poincare
complex ( C , ./0 over 2- The knot complement U = S -(open
nbhd. of k(Sm-1)) has boundary 9U=Sm-1XS1, and there is
de fined an
(U.8U) >(Dm+2,s,n-1)xs1
(m+1)-dimensiona1 normal map
which is a Z-homology

207
equivalence. Let (Lm +*;Mm,zMm) be the fundamental
domain for the infinite cyclic cover U of U obtained by
cutting U along M, and let
((e;f ,zf ) ,(a;b,zb)) :
, . m+ 1 wm wm ■.
(L ;M ,zM )
-> Dm+2X( [0,11; <0> , < 1 >>
be the corresponding (m+1)-dimensiona1 normal map of
triads. The inclusions j:M ► L, k:z M ► L induce
Z-module chain maps j,k:C=C(f') >D=C(g') such that
j - k : C-
•D
is a chain equivalence.
structure i^GQ (C) is determined
structure (1+T)./,:C
homo t o p y
and
The u1traquadratic
by the symmetric
since up to chain
(j-k) *j = £<<1+T>£> *
( j-k)'
k =
-TyK ( l+T)yO'
More generally:
Pr opos i t ion 7.2 Let ( C , \p ) be an m-d imens i ona 1
e-quadratic Poincare complex in IK~ A cobordism
((j k).C©C >D , ( 8^ ,^9-^) ) with j-k:C >D a chain
equivalence determines an e-u1traquadratic structure
y^GQ (C) with image i^£Q (C,e), such that
(C.yO = u(Cm~* ,£) € im(u:L0(Dm<A> .€) ►Lm(A,€))
m— * A
with (C , \p) a nonsingular e-quadratic form in ID (A).
Proof: Define a morphism in ID (A)
m
h = (j-k)
1 .
-» D
(j-k)"
-> C
By the chain homotopy invariance of the Q-groups we can
replace ((j k ) , ( Sy^ , *p®-\//) ) by a homotopy equivalent

208
cobordism ((h h-l):C©C >C, (5^,yJ-©-^)€Qm +r( (h h-l),e))
On the chain level
h2(yjr) - (h-l)z(yjr) = d(5^) € <w%c>m >
so that there Is defined a chain homotopy
(1+T£)5^0 : h(l+T£)^0 ~ ( 1+Tc)yjr0( 1-h*) :
-» C
The m-dimensiona1 e-u1traquadratic Poincare complex
(C,i/0 in A defined by the chain map
yj- = h( 1+T£)^0 -. C
.* n+Te)yfr0
-» c
-» c
A ^ *
is such that \p+Z\p a((l+T )yt0:C
x€(W%C)in+1 such that v?-^ = d(x+5^)€(W%C)in by
0 if s=0
C. Define a chain
hT 0, , if sjl
e's -1 '
Thus ^=^6Q (C,e) and
: C
m+1 -r-s
-» C
(c.yjr) = (c.yj-) = M(c .yO € Lm(A,c) .
Coro11ary 7 . 3 Let (f,b ) :M ► X be an (i-l)-connected
normal map of ( n + 2i)-dimensiona1 geometric Poincare
complexes, and let
((e;f ,zf) ,(a;b,zb)) : (L;M,zM) ► XX( (0, 1 ] ; <0> , < 1>)
be an (i- 1 )-connected normal bordism between (f ,b) and
a disjoint copy (zf,zb). If the ( i-1)-connected normal

209
map of (n+2i+1)-dimensiona1 geometric Poincare
comp 1 e >: e s
(e/(f=zf),a/(b=zb)) :
L/(M=zM)
XX( [0,1 ]/0=l ) = XXSJ
is a Z [ K> ( X) ]-homo]ogy equivalence then the bordism
determines an (-) -ultraquadratic structure
i^GQ (S C(f ')) with image the quadratic kernel
structure ^€Qn(S"iC(f!),(-)i)=Qn+2.(C(f!)). The
i —i ' n + 2i-* A
no nsingular (-) -quadratic form (S C(f") , yfr ) in
Dn(Z(*1(X)J) is such that
a*(f,b) = (C(fl)iVfr> = M(S_iC(f! )n + 2i"* ,^)
€ nn<u:L0(Dn(Ztfr1 (X) J) ,(-)1) ►Ln(Ztff1 (X) ] , (-)1))
= im(M:L21(Dn(Ztfr1(X) J)) ►Ln+2i(Ztff1(X) J)) .
Prop f : The kernel Zttfi (X) j-module chain complexes
C = C(f!:C(X) ► C(M)) , D = C(g! :C(XX(0 , 1 ] ) >C(L))
are i-fold suspensions of n-dimensional chain complexes
(up to chain equivalence). The inclusions M ► L, z M ► L
induce Z t it i ( x ) ] -modu 1 e chain maps j:C >D, k:C >D such
that j-k:C ► D is a chain equivalence. Let
h=(j-k) j:C >C for any chain homotopy inverse
(j-k)~ :D >C. The quadratic kernel
or*((e; f,zf),(a;b,zb)) = ((j k):C©C >D , ( Syfr , yJ-©-yO )
is the i-fold skew-suspension of a cobordism of
n-dimensional (-) -quadratic Poincare complexes over

210
Z t * I ( X ) ] satisfying the hypothesis of 7.2. It follows
that i^€Q . 2 -(C) is the image of the element
"" n + 2 i - *
^£Hn(Hom.(C ,C)) defined by the composite chain
map
* : C
n + 2 i-*
-» C
j (J-k)
► D
-1
-► C
with 0O= ( M] f\- : cn+ >C the Poincare duality chain
equivalence. The nonsingular (-) -quadratic form
(S-iCn+2l~* ,£) in Dn(Z(^1(X)]) is such that
flr*(f.b) = (C.^> = M(S_iCn+2i"*,^)
€ im(M:L0(Dn<Z[fr, (X) ]) . (-)1 ) ►Ln(Z[*1 (X) ] ,(-)1))
= im(M:L2l(Dn(Z[fr1<X) ])) >Ln+2 . (Z[*j (X) ] )) .
Proposi t ion 7.4 Let ((j j ' ) :C©C' ^0,(5^,^8-^1)) be a
cobordism of m-dimensional e-quadratic Poincare
complexes in /k, such that D, C(j) and C(j') are the
suspensions of (m-1)-dimensiona1 chain complexes (up to
chain equivalence), with m^1. The chain homotopy
classes of the chain maps
inclusion : G = S" D
S [C(j ')
jH = inclusi on
S * D
S_1C(j) * C(j')m
* = pm-l-*
are the components of a morphism of e-symmetric forms
in £) ,(A)
m— 1

211
lWJ
: <G,0)
-» H£(F) = (FSF111 *
0 1
e 0
such that Y ju= ( 1 +T_ c) &q ■ G *Gm l for a certain
element 0€Qm_l(Gm~l~ ,-e) determined by (5^,¢$-^)•
f vl m_1 _ *
If the morphism | 't | :G >F©F is a split injection
PI-
in (D^^A) and if 6>€im ( Qm_ { (Gm~ [ ~ * ) ^m-i^^"1 *.-£>>
then G is a lagrangian of the hyperbolic e-quadratic
form
H(r(F) = (F©Fm-1-*, f° H)
lo oj
and (F , G) is a nonsingular e-quadratic formation in
Dm_j(A) such that
(C.yjr) = u(F,G) € im(M:L1(Dm_1(A) ,C) ►Lm(A,c)) .
Proof ; Let (Dm+1~ ,0) be the (m+1)-dimens iona1
e-quadratic complex in A (not in general Poincare)
defined by the algebraic Thorn construction, the image
of ( Si/'/i/'©-!/' ' )€Qm+1 ( C( j j*),e) under the isomorphism
<<i+t )<5^0.^0e-^)z) l
Qm+1(C(j j'),0
Q„.,(Dm+1-*,e) = <>„, (^-^.-0
<m+l
'm-1
Up to chain homotopy
* - 1
Y M : G = S D
i nc1u s i on
S"'C(] j') * D1 = G
m-1

212
so that there exists a chain homotopy
Y M - (1+T_£)0O
-» G
m-1-*
and
"*"*'(: m
* *
Y M + £JU Y * 0
,m-l
as required for
,m- 1 -*
to be a lagrangian in H (F)
If
^m-l<G
*• A m-1 - *
e) is the image of eGQ^^G ) then
(G,0) is the hessian (-e)-quadratic form in ID _i(A)
required for G to be a lagrangian in H (F). The
algebraic Thorn construction defines a one-one
correspondence between the homotopy equivalence classes
of ( m+1 )-dimensiona1 e-quadratic Poincare pairs in A
and (m+1)-dimensiona1 e-quadratic complexes in A
(Proposition 3.4 of Ranicki [14]). Thus
((j j ' ) : C©C ' >D > ( Sy^ ,y^®y^ ' ) ) is homotopy equivalent to
the (m+1)-dimensional e-quadratic Poincare pair
((0 +1):3D >D,(0,30)) defined by
l3D
(-)1 (1+T_e)6>
3d
Dm_r©D
3D
r-1
Dm-r+leD
36>,
0
1
0 "J
0
3D1" * = Dr©D"
3D = Dm"r©D
r r *

213
Q9,
(-)B"r+Be o
3Dm-r-1 = Dr+1©Dm-r-1
3d = Dm"reD
r r *
Q9s = 0 : 3D
m-r - s
- 3Dr (s£2) .
Up to chain homotopy
M* : F = S_1C(j«)
inclusion
s V.( j j' ) * d'
,m-l-
so that there is defined a chain equivalence
f :C >C(jn ). Choosing a representative chain map
,m+l -*
6>:D-
D
X:Y M^( 1+T )0:D-
g:3D >C(ju ) by
•D
and
m+1-*
a chain homotopy
define a chain map
1 X
0 Y
3D = Dm"r©D
r r
-> C(W*)r = Gm r [©F
r-1
such t hat
f-(vfr) = gr(36>) 6 Qm(c(w ),£) .
%
•7.
Now (C(ju ),g,Od)) is the m-d imen s i ona 1 e-quadratic
Poincare complex in A constructed in 3.2 from the
nonsingular e-quadratic formation (F,G) in ID _i(A)» so
that
(C.vO "= (C(W ).f%(vO> = (C(W ),g%O0))

214
U<F.G> € im(u:Ll(®m_l(A) >0 ►T^CA.c))
§8. The connec t i on
We now connect the algebra and the geometry,
verifying the claim of Theorem 6.2 that the geometric
surgery transfer maps for an n-dimensiona1 Poincare
tibration F— B coincide with the algebraic surgery
t ran s f e r map s
I ;
Pgeo ~ *a 1 g
Lm(Ztfr1(B) J) ► Lm+n(Z[ff1<E) J) (m£0) .
We know from 1.9 how a CW complex structure behaves
under transfer on the cellular chain level. The
strategy is to encode the L-theory data in CW complex
structures, and to decode the lifted L-theory data from
the CW lifts using the u1traquadratic L-theory of %7.
We consider first the case ra*2i . By Chapter 5 of
Wall [22] every element x£L 2 ± ( Z I K j ( B ) ] ) (i£3) is the
Witt class of the nonsingular (-) -quadratic form in
BCZlffjCB) ])
( K.(M) , X:K,(M)XK. (M) ► ZU^B)] ,
M : Ki(M) ► ZtiTj (B) ]/<a-(-)iaIa€Zt7T1(B) ]> )
on the kernel 2[iti (B) ]-module
K.(M) = Xi + l(.f) - H.(f!) = ker(f„:H.(M) ^(X))
of an ( i - 1 )-connected normal map (f,b):(M,3M) ► (X,3X)

215
from a 2i-dimensional manifold with boundary (M,3M) to
a 2i-dimensional geometric Poincare pair (X,3X) , with
3f : 3M >3X a homotopy equivalence, and with a
7f, -isomorphism reference map X >B such that
w(X):?r1(X) ^ ( B) w^ B^ »Z2 • Th« adjoint of X defines an
isomorphism in IB ( Z t tfi ( B ) ] )
X : Ki(M)
- Ki(M) ;
( v
X(u.v) )
( K . ( M) , X , JLl) can be viewed as a nonsingular
i *
(-) -quadratic form (K.(M) , \j/) over Zttfi(B)], with \j/ an
equivalence class of 2[ffi (B) ]-module morphisms
yfr:K.(M) »K.(M) such that
V^+(-)V = X •• K.(M)
-► K. (M)
vKX(v))(X(v)) = u(v) (v€K1(M)) ,
*• ^ i +1 *
with \p equivalent to i/'+x+("") X for any
*
Z t K. (B) ] -modu1e morphism x=K-(M) ►K.(M). The surgery
obstruction is thus given by
x - or*(f,b) = (K.(M),X,M) = (Kt(M) >vO
€ L2i(Zlff1(B) ])
We shall be regarding modules as 0-dimensiona1
chain complexes, and for any q£Z we write S C for the
q-fold suspension of a chain complex C, with
dSqC
(S"C)
r-q
(SqC)
r-1
= C
r -q- 1
The quadratic kernel or* ( f , b ) = ( C ( f " ) , <p ) of the

216
(i-l)-connected 2i-dimensional normal map (f,b ) : M ► X
is an (i- 1 )-connected 2i-dimensiona 1 quadratic
Poincare complex over Z ( »f i ( B ) ] which is homotopy
i A ^
equivalent to (S K.(M),yfr)- Thus we can identify yfr/^yfr.
and up to chain homotopy
(l+T)yJ-0 = ^+(-)V
1
I 9 -i - *
c(f • )Zl
SXK.(M)
C(f!) = SXK.(M)
1
The quadratic structure i/f£Q2 • (c ( f ' ) ) is the equivalence
class of Zttfi (B) ]-module morphisms yfr:K. (M) ► K.(M)
described above. A choice of representative \p is a
choice of u1traquadratic structure ^£Q„.(C(f')) for the
quadratic structure ^6Q7 .(C(f ' )) . We now fix a choice
of \(,.
Let < v j » v2 vk> be
base for the f .g. free
Z[iti (B)]-iodule K.(M), and use the dual to define a
i *
base for K (M)=K.(M) . The functor of additive
categories with involution
it
)(c(F),a,U) :
KZtff^B) ])
-> Dn(Ztfr1(E) ])
sends the morphisms in B(Z(»fi(B)])
K,(M) = eZtff^B) J
-» K . (M)
k k
tff^B) ]
to chain homotopy classes of Zttfi(E)]-module chain
map s

217
P (yfr) » P it ) :
p*(K.(M)*) = eC(F)
1 k
- p"(K.(M)) = ©C(F)
1 k
such that there is defined a chain homotopy commutative
d i agram
©C(F)'
k
>(F]A-
©C(F)
k
//■*■*
// ^*
P <* >
©C(F)'
k
>(F]A-
©C(F)
k
and such t ha t
//^ !//■*■* // - 1 ~ ~
P <*> + (-) P (^ ) = P (X *) : eC(F) ► eC(F)
k k
is a chain equivalence.
In Lemmas 8.1,8.3 below we shall show that the
quadratic kernel or* ( g , c ) = ( C ( g ' ) , j\ ) of the pullback
(n+i- 1)-connected (n + 2i)-dimensiona1 normal map
(g,c):(N,3N) >(Y,3Y) is homotopy equivalent to the
( n + 2 i )-d imens i ona 1 quadratic Poincare complex ( D » 7? )
defined by the Z( * .(E) ]-module chain complex
D = Sip//(K, (M)) = ©SiC(F)
1 k
with the (u11ra ) quadratic structure
„„ : Dn+2i"* = eS^FV"*
u k
>[FJA-
// ^
P <*>
-» ©S1C(F)
k
-» D = ©S1C(F) ,
k

218
>?s = 0 : D
n+2i-r-s
Dr (s=5l) .
It will follow that the nonsingular (-) -quadratic form
(p//(Kj (M)*) .(©[FlA-rV'^)) in Dn(Z(tf, (E) ]) defined
(©(Fin-TV (^) :
// ^
P (K (M) ) = ©C(F) ► p"(K.(M)) = ®C(F)
k x k
x mn-)'
~ n - * It * n - *
©C(F)n = (p"(K.(M) ))n
k l
is such t hat
Pgeoa*(f>b) = OaU.c)
M(p//(Ki(M)*) ,(©tF)A-) [p*(^))
= Paig<Ki<M) •'A) = Palga*(f'b)
€ im<M:L0(IDn<Ztfr1<E) J) . (-)1) ►Ln(Z[fr1<E) 1,(-)1))
= im(M:L2l(Dn(Ztfr1<E) ])) >^n+2 i < * I * 1 < E > 1 > > •
verifying that p* =p", in the case m=2i.
' geo *alg
f
For the symmetric structure of or* ( g , c ) ■ ( C ( g ' ) , j\ )
we have :
Lemma 8 . 1 The symmetric kernel a ( g , c ) = ( C( g ' ) , ( 1 +T ) j\ )
is such that up to chain homotopy
O+T)7?0

219
C(g!)n + 2i~* = Sip*(K .(M)*)n_* = ©SxC(F)n
• [Fin- 4 „ t „
► eSxC(F) = sV(K. (M) )
k
p' ((1+T)vt0) . i // i -
Si ► C(g-) = sV(K.(M)) = eSXC(F) .
Proof : Represent the base elements v.£K. (M) (l^j^k) by
framed immersions v.:Sx >int(M x) with nu11 homotopies
in X, and with it. (B)-equi variant lifts
v .:S1»ff.(B)XS1 >M. Replace f :M »X by the inclusion of
M in the CW complex M\J\Je homotopy equivalent to X
k
(which is also denoted by X), so that
C(f!) ~ S"1C(X,M) = SXK.(M) = ©S1Z(*,(B)] .
3 k L
In the total spaces of the pullbacks g:N >Y is
replaced by the inclusion of N in the CW complex
NWFXe 1 + 1 , so that
k
C(g!) ss S_1C(V,N) = Sxp*(K<(M)) = eSxC(F) .
1 k
The Poincare duality chain equivalence is given up to
chain homotopy by the composite
*
<1+T>„0 : C(g!)n+2i-* — ► c(3,9S)n+2i-*
[N]n- ,.„ ~ „ e ,
► C ( N ) ► C ( N , 9N ) ► C ( g ' )
with e the inclusion.
For a sufficiently large number q^O the framed
immersions can be approximated by framed embeddings

220
v.sS1 >int(M 1XDq) with nu11 homotopies in X. Let V. be
a regular neighbourhood of v . ( S x ) in MX D q , and let
P .=closure(MXDq-V .), so that
MXDq = V V» P , 3P, = 3V A/3(MXDq) ,
J j J J J
<VJ,SVj> = vj(si)X(Di+q,Si+q_1) .
The intersection number
X, ,-, = X(v.,v.,> G Z(*,(B)] (Uj.j'^k)
j > j j j l
is the image of lEZttfi (B)] under the composite
Z[ifi (B) ]-module morphism
■ «?i
Hi(Si) = ZtffjCB) ]
V
- H.. (M) 9- H (MXDq,MXSq_1)
H,-u.„(MXI
. . X..^DM,P.) a H.^ (V..9V.
,)
HqCS1) = z[ ttl (b)] ,
which can also be expressed as
H.CS1) = ZtffjCB) ]
Vj'
-» H . ( M , 3M )
(tMin-)'
-» Hi(M)
v .
J
H1(S1) = ZtffjCB) ]
The pullbacks from the n-dimensiona1 Poincare fibration
imme r slons
F >E v » B define framed Poincare
w.:F XS >N with nu 1 lhomo t op i e s in Y, and with
tfj (E) -equivar iant lifts w.-.FXS1 >N (l^j^k). Let

221
W..Q.CNXD be the total spaces of the fibrations over
Vj.PjCMXD*1, so that
NXDq = WjV9w Q. , QQ. = 9W..V9(NXDq) ,
(Wj.aWj) = w.(FXSi)X(Di+q,Si+q-1) .
For any embedding D2x+qCint(V . )CM21XDq the pair
( (MXDq-int(D2i + q))U v.Di+1XDq , P .V/D1 + 1XS**" X )
j,X J
has a relative CW structure with one (i+q)-cell and one
(i+q+1 )-ce11 , such that the cellular chain complex in
BCZU^B)]) is \ , rZttf^B) ] ►ZUjU)]. By 1.9 the
chain homotopy class of the Z t it i ( E) ] -modu le chain map
p (X. .,):C(F) >C(F) coincides with the composite
J > J
C(F) ► S~1C(FXS1)
S iC(N,3N) ^ S_i qC(NXDq,3(NXDq))
■* S 1 qC(NXDq,Q.) s; S~1-qC(W . ,3W .) * C(FXRl)
-* C(F) ,
and hence also with the composite
C(F)
-► S 1C(FXS1)
wr
-» s 1c(n,3n)
([Nin->
-> SiC(N)'
« *
w .
■ * J i -*J ,i.n-*
-+ S1C(FXS1 )'
[F]n-
- C(F)1
- C(F)

222
The ( j , j * )-component of the 2[(T, (E) ]-raodule chain
equ i va1enc e
<<1+T)»I0>'
1
C(g-) * ©SXC(F)
k
- C(g" )
!,n+2i-*
* ©SXC(F)'
k
is thus the composite
P (x1 1,)
SXC(F) ► S1C(F)
((F]A-)
- S^F)"-
and up to chain homotopy
(1+T)^0 : C(g)
K n + 2i
)S1C(F)n
.* etFin-
,- ~ p ( a ; , ,-
©S1C(F) ► C(g-) = ©S1C(F)
k k
We extend the description of the symmetric
structure of a*(g,c) given by 8.1 to the quadratic
structure, using the u 11raquadratic L-theory of §7. A
choice of u11raquadratic structure yfr:K.(M) >K.(M) for
or* ( f »b) is used to construct a normal bordism between
(f,b):M >X and a copy (zf,zb):zM >zX which encodes
the quadratic se 1 f - i n t e r s ec t i on form jn in the CW
structure. The quadratic structure of or*(g,c) is then
decoded from the CW structure of the pullback normal
bordism between (g,c):N >Y and a copy (zg,zc):zN >zY,
using 1.9 and 7.3. The construction of the bordism is
motivated by the way in which the infinite cyclic cover

223
of a knot, complement can be obtained by cutting along a
Seifert surface.
Lemma 8 . 2 A choice of ultraquadratic structure \p for
(K.(M),X,JLl) can be realized by an (i-l)-connected
(2i+l)-dimensional normal bordism
((ejf,zf),(a;b,zb)) : (L;M,zM) ► XX( t0, 1 ] ; <0> , < 1 >)
between (f,b):M ► X and a disjoint copy
( z f , z b) : z M ► z X, such that the difference of the
Z [ * i ( B ) ]-module morphisms j ,k:K .(M) ► K.(L) induced by
the inclusions j :M >L, k: zM >L is an isomorphism
j-k-.K.(M) ► K.(L) with
(j~k)_1j = $\ : K.(M) ► K.(M) ,
(j-k)_1k = (-)i + 1^*X : Kj(M) ► KjCM) .
The (i-1 )-connected (2i+ 1 )-dimensiona1 normal map
(e/(f = zf ) ,a/(b = zb)) ; (L/(M=zM) .3MXS1 )
► (X,9X)X( (0, 1 )/0=1) = (X,3X)XS1
is a Z[K1 (B) )-homo logy equivalence, with the homotopy
equivalence 3fXl:3MXS >3XXS on the boundary.
Proo f : Every based f .g. free lagrangian of the
(-) ^quadratic form ( K ± ( M) , X , JLl) ©( K ± ( M) , -X , — jut > can be
realized by disjoint framed embeddings of S in
MVaMvrri 1 iZM with nu 1 1 homo t op i e s in X, such that the
trace of the surgeries on these framed embedded
i-spheres defines a normal bordism between (f,b) and
(zf ,zb). The realization of the lagrangian

224
i m(
L(-) V xj
: Kt(M)
-► K1(M)©K,(M))
has the required properties. (This lagrangian is a
direct complement of the diagonal lagrangian
im( J |j :K±(M) >K (M)©K±(M)) . The realization of the
diagonal lagrangian is the product (2i+1)-dimensiona1
normal map
(f,b)Xl : MX([0, 1 ] ;<0> ,<1>) ► XX([0, 1 ] ;<0> ,<1>) -
The required normal map (e,a) can also be obtained from
( f , b) X1 by surgeries on i-spheres in the interior of
MXt^.l] representing a base of K.(MX(0,1])=K.(M).)
We can now extend l.emma 8.1 to the quadratic
structure :
Lemma 8 . 3 The quadratic kernel a* ( g , c ) = ( C ( g " ) , j] ) is
such that up to chain homotopy
„, !,n+2i-* _ i #,„ , w s * s n
7?0 : c( g ) =Sp(Ki(M))
* r» — * i ^ n — *
- ©S1C(F)n
itFin-
)S*C(F> = S1p*(K1(M)*)
P (yfr)
-* C(g!) = sV'(K.(M)) = eSiC(F) ,
1 k
n s = 0 : C ( g ' )
n + 2 i-r-s
- C(g) (s^l) .
Proof : Let yfr , ( e , a ) , ( f , b ) , j , k be as in 8.2, and let

225
((h;g , zg) , (d ; c , zc) )
(P;N,zN)
YX([0,1];<0>,<1>)
be the (n+i-])-connected (n+2i+1)-dimensiona1 normal
bordism between (g,c):N >Y and a disjoint copy
(zg,zc):zN >zY obtained from ((e;f,zf),(a;b,zb)) by
pullback from F >E—^-*B along the reference map X >B.
The (n+i- 1 )-connected (n+2i+1)-dimensiona1 normal map
(h/(g=zg),d/(c=zc)) :
P/(N«zN)
VX((0,1)/0=1) = YXS
1
is a Z(X, (E) ]-homo 1ogy equivalence. By 7.3 the
quadratic kernel O * ( g > O i s determined by the chain
homotopy classes of the Z[ ff. (K) ]-module chain maps
i i t i 1 '
C(g") ►C(h'), C(zg') >C(h) and the Poincare duality
' n + 2 i — * '
chain equivalence C(g') »C(g"). We shall now
arrange CW structures for (e,a) in such a way that only
cells in dimensions i,i+l occur in the relevant pairs,
and 1.9 applies to obtain the Z[ff, (E) ]-module chain
homotopy data in the total spaces of the pullbacks from
F >E—E-»B as the algebraic transfers of Z[ff, (B) ]-module
da t a .
L is the trace of surgeries on (i-1)- and
i-spheres in M, so that (L,M) has a relative CW
structure with i- and (i+l)-cells, with the cellular
chain complex in B(Z(7r. (B)]) given by
d = j : C(L,M)i+1 = K.(M)
-♦ C(L,M) . = K . (L)
Replacing e:L >XX(0,1] by the inclusion of L in the
mapping cylinder it may be assumed that Lisa
subcomplex of X, such that (X,L) and (X,M) have
cellular chain complexes in B(Z(7r. (B)])
C(X,L)
Si+1Ki(L) ,
(j 1) : C(X,M).+1 = Ki(M)©Ki(L)

226
■* C(X,M).
Kt(L) .
The kernel chain complexes C(f'), C(e') are chain
equivalent to S~ C(X,M) , S C ( X , L ) respectively.
Replacing the inclusion C(f") >C ( e " ) by the inclusion
S C(X,M) ► S C(X,L) of the chain equivalent complexes
i i
corresponds in the pullbacks to replacing C(g') ► C ( h " )
by S-1C(Y,N) ►S~1C(Y,P), and by 1.9
C(Y,N) = p^CCX.M)
C(Y,P) = p*C(X,L)
Thus up to chain homotopy the inclusion C(g") >C(h')
may be identified with the ZtK■(E)]-modu1e chain map
P*(j) = C(g!) = Sip*(Ki(M))
C(h' ) = S1p*(Ki(L)) .
t i
Similarly, up to chain homotopy C(zg') >C(h') may be
identified with
p"(k) : C(zg!) = S1p#(Ki(M))
C(h!) = S1p*(Ki(L)) .
Th
e Zttfi(B)]-,nodule isomorphism j-k:K1( M) >K . ( L ) such
/^ .. x - 1 .,_ ,% . ., /wx »k.(M) lifts to (the chain
that (j-k) ij=^X:Ki(M)
lifts to
homotopy class of) a Ztfi(E)]-modu1e chain equivalence
p"(j-k)
P ( J )
P (k)
c(g- )
■* c(h- )
such that up to chain homotopy
P(j-k) p(j) = p (^X)
C(g') = ©SiC(F)
k
-► ©S1C(F) .
k
Applying 8.1 and 7.3 we have that the quadratic kernel
a*(g,c) is homotopy equivalent to the

227
(n+2i)-dimensiona1 quadratic Poincare complex
(®S1C(F),^) over Z(tf,(E)] with
k l
<1+T),0 : C(g!)n+2i-*
)SlC(F)n
®(F]r»-
i -- p (X ) , .
©S1C(F) ► C(g) = ©S1C(F) ,
k k
10 = p'(J\)(l+T)(l
0
c(g!)n+2i-* = ©S^fV
itFlA-
JSV.(F)
It ^
P <*>
- C(g!) = ©S1C(F) ,
k
Us = 0 : C(g!)n + 2i-r-s ► C(g!)r (s£l> .
This completes the proof of Theorem 6.2 in
case m=2i, and we proceed to the case m=2i+l.
the
By Chapter 6 of Wall [22] every element
x£L2i+1(ZtKj(B)]) (i^2) is the Witt class of the kernel
nonsingular (-) -quadratic formation over ZtTi(B)]
(F,G) = (K.+1(U,3U) ,K. + 1 (MQ ,3U))
of an (i-1)-connected (2i+1)-dimensiona1 normal map
(f,b):(M,3M) >(X,3X) with 3fr3M >QX a homotopy
equivalence, and with a if. -isomorphism reference map
X >B such that w(X):^j(X) ^ (B) M<> B^ >Z2 . Here, U is
the connected sum of a sufficiently large number k^O of
framed embeddings S Cint(M) with nu1lhomotopies in X to
generate the f-g- ZtK. (B) ]-modu1e K.(M), and
MQ=closure(M-U). Thus F=K.+1(U,3U) is a based f.g. free
Zlff^B) 1-raodule, and G = Ki + 1 (MQ , 3U) is a based f.g. free

228
lagrangian of the hyperbolic (-) -quadratic form
H(_ji(F)=(F©F*,f° H), with
F = G =
!(*! <B> ]
The i
inclusion
:G >F©F extends to an isomorphism of
hyperbolic (-) -quadratic forms
Y Y 1
M M
H ( _ j i ( G )
-► H(_)i(F)
Surgery on the framed embedded i-spheres in U defines
an (i- 1) -connected (2i+ 2)-dimensiona1 normal map of
triads
((Cif,f' ),(a;b,b')) : (L2i + 2;M2i + 1,M'2i+1)
■* XX (. (0, 1 ] ;<0> ,< 1>)
i
with (F ,G) the kernel nonsingular (-) -quadratic
formation of (f',b'), and
G = K.+1(L) , F = Ki+1(L,M') , F* - K.+1(L,M) ,
C(e') = S1 + 1G , C(e-,f) = S1 LF ,
C(e-,f-) = S1+1F
The quadratic kernel a*((e;f,f ' ) , ( a ; b , b' )) is an
(i-l)-connected (2i+2)~dimensiona1 quadratic Poincare
cobordism over Z ( ;f i ( B) ] with algebraic Thorn complex
homot opy
equ i valent
to
the
i-conne c t e d
i + 1 *
(2i+2)~dimensiona1 quadratic complex (S G ,0)
corresponding to the (-) -quadratic hessian form
(G,0) such that Y U=0+(-)1+1 9 :G >G . The base
elements of the f.g. free Z(Ki(B) ]-module G = 7r...(e) can
be represented by immersed ( i+1)~spheres in int(L )
with nu11 homotopies in X, so that the form (G,0) can be

229
expressed in terms of geometric intersection and
self-intersection numbers exactly as in Chapter 5 of
Wall [22].
The pullback of ( ( e ; f , f ' ) , ( a ; b , b ' ) ) from F >E—E-»B
along the reference map X >B is an ( n+ i - 1 ) -connec t ed
normal map of (n+2i + 2)~dimensiona 1 geometric Poincare
triads
<<h;g,g'>,<d;c,c'>> : (Pn + 24 + 2 ; Nn + 24 + l ,N •n + 2i+l )
- XX<[0,1];<0>,<1>)
The Z( k, ( E) ] -module chain maps C(h') ►C(h',g,')>
> ii
C(h') 'C(h'ig') defined by projections are given up to
chain homotopy by
It ' i 4- 1 # 1+1
p"(Y) : C(h-) = S1+1pffG = ©S1+1C(F)
- C(h!,g' !) = s1 + Vf = es1 + 1c(K) ,
k
(®(F]n-)~ V'(|Ll) : C(h!) = Si+V'G = 0Si + 1C(F)
k
P (m)
- Si + 1p"(F*) = ©Si + 1C(F)
(etFin-)
-1
©S C(F) =C(h,g') ^C(h.g).
k
The quadratic kernel or*((h;g,g,),(d;c,c')) is the
i-fold skew-suspension of an (n+1)-dimensiona1
(-) -quadratic Poincare cobordism over ZtKi(E)]
satisfying the hypotheses of Proposition 7.4, with
P*(Y)
(etFin-)"1 p*<w)
<P*G,0)
-► H . (p F)
(-)1

230
the inclusion of a lagrangian of the (-) -quadratic
hyperbolic form H .(p F) in ID (Z t it, ( E ) ] ) because it
(-)1 n l
extends to an isomorphism of ( -)x-quadratic forms
P"(r)
p//(Y)(©tF]A-)
(©(Fin-)"V(u) (e(F)n-)~V'(u)(©(F]n-) J
H(_)i(p"G)
-► H(_)i(p'rF)
Working as in the proof of Lemma 8.1 the hessian
(-) -quadratic form in ID (21 K, (E) ] ) may be expressed
as (p*G, (©[Fl/V)""1 p*(0) ) , with 9:G >G* an
(-) -ultraquadratic structure for (G,0). By
Proposition 7.4 the nonsingular (-) -quadratic
formation (p F,p G) in ID ( Z t * , ( E ) ] ) is such that
i // //
pgeo°*(f,b) = a*(-&»c'> = W(P F.P G>
Palg(F,G)
is
a*(f ,b)
€ im(M:L1 (Dn<Z[ff1 (E) ]) ,(-)1)
>Ln+l(Zt*l (E)1 .^-)1))
= im(W:L2,+1 (ID^Zt*! (E) ] ) )
>Ln + 2i+l<Zt*l<E)>>> •
i i
This verifies p" =p' also in the case m=2i+l,
geo ^alg '
completing the proof of Theorem 6.2.
We can now write the surgery transfer maps
unambiguously as
p- : Lm(Ztff1<B)]) ► Lm+n(Zt*l(E) ] } <m^0>

231
= 9. Change o f K-t he ory
We now extend the definition of the algebraic
surgery transfer maps (C , a, U) ' : L (R) >L (S) to the
intermediate L-groups, and show that they are
compatible with the Rothenberg exact sequences.
An involution R >R ; r >r on a ring R determines a
duality involution *:P(R) >P(R);P >P =HomR(P,R) on
the additive category P(R) of f.g. projective R-modules
by
R X P
- P ; (r,f)
- (x
- f (x).r) ,
e(P) : P
-► P ; x
- (f
- f(x))
The duality involution on P(R) determines involutions
on the algebraic K-groups
* : KQ(R)
- KQ(R) ; (P)
(P I ,
Kj (R)
-► Kj (R) ;
T(f :P >Q)
* *
-> 7(f :Q -
•P )
and also on the reduced K-groups
K.(R) = coker(K.(Z) >K.(R)) (1 = 0,1) .
The intermediate quadratic L-groups L*(R) of a
ring with involution R are defined for '-invariant
■j *
subgroups XCK.(R) (1=0,1), such that x £X for all x£X.
The intermediate L-groups for X=<0>,K.(R) are written
as
Kn(R) n <0>CK.(R)
L* (R) = l£(R) , L* (R) = L*(R)
<0>CKn(R) K.(R) .
L* U (R) = L* (R) = L*(R) = L*(R)

232
For '-invariant subgroups XCX'£K.(R) there is defined a
Rothenberg exact sequence
... ► Ln(R) ' L*'(R) ' Hn(Z2;X'/X)
► L*_j(R) ► ...
with
Hn(Z2;X'/X) =
<aex'/X!a*=(-)na>/<b+(-)nb*|b£X>/X> .
See Ranicki (13),(14] for further details.
We consider first the torsion case XCk.(R).
A representation (C,L') of R in ID(S) determines a
transfer map in the absolute torsion groups
( C , U) " : K . ( R ) ►K.(S) (Example 1.8), and also in the
reduced torsion groups (C,U) ' :K. (R) >K. (S). By
definition, ID ( S) is the homotopy category of finite
chain complexes of based f.g. free S-modules. We shall
now make use of the bases.
Proposit i on 9. 1 Let (C, a ,U) be a symmetric
representation of R in ID (S), for some rings with
involution R,S.
i) For any '-invariant subgroups XCKj(R), YCKj(S) such
that (C,U)"(X)CY and 7( a:C >Cn_ )£Y there are defined
transfer maps in the intermediate torsion L-groups
(C,a,U)! : Lm(R) ► Lm + n(S) (n£0) .
ii) For any '-invariant subgroups XCX'Ck.(R),
YCy'Ck^S) such that (C,U)!(X)£Y, ( C , U ) ! ( X * ) CY ' , 7(a)€¥
there is defined a morphism of Rothenberg exact
sequences

233
Lm(R) ' Lm'(R) ' Hm(Z2;X'/X)
(C.a.U)
(c , a, u)
(c, V )
Lm+n(S) -» LmIn(S) ^ Hm+n(Z,;Y'/Y)
-► ...
Prop f : The transfer map in the reduced torsion groups
(C,U) " :Kj (R) ►Kj(S) is such that
(C,U)' = (-)n(C,U)-* : K.(R)
■* Kj (S) .
Let m = 2i. For any nonsingular (-) -quadratic form (M,i/0
on a based f ,g. free R-module M = R the n-d i mens i ona 1
(-) -quadratic Poincare complex (©C,0) representing
k
(C,U)'(M,yfr) has reduced torsion
T((1+T)0n:©c'
k
► ®C)
k
= (C,U) ! 7(^+(-) V* :M >M*> € K,(S)
,- *
the image of 7(^+(-)^ )€K.(R). Similarly for m=2i+l
and formations.
Next, we consider the projective case XCK„(R). It
is more convenient to work with the preimage of X in
K~(R), so we regard X as a '-invariant subgroup of
KQ(R) such that [R1€*.
Given a ring S let E( S ) =£) (P( S ) ) , the homotopy
category of finite-dimensional f-g- projective S-module
chain complexes. A representation (C,U) of a ring R in
E(S) determines transfer maps in the algebraic
K-groups
(C,U)' : K.(R)
K±(P(R))

234
K .(S) = K.(P(S)) (i=0, 1 )
(Example 1.8). For n^O let E ( S) =D (P(S)), the full
subcategory of E(S) with objects n-dimensiona 1 f.g.
projective S-module chain complexes. An involution on S
n- *
determines the n-duality involution C ► C on E (S) .
Propos i t i on 9.2 Let ( C , a , U ) be a symmetric
representation of R in E (S), for some rings with
i nvo1u t i on R,S .
i) For any '-invariant subgroups XCKn(R), YCKn(S) such
0'
'0
that [R]€X, tS]€Y, (C,U)'(X)CYCKQ(S) there are defined
transfer maps in the intermediate class L-groups
(C.a.U)
Lm<R>
L' (S) (n^O)
m+n v ' \ -« /
■invariant subgroups XQX'£Kn(R),
i i) For any
YCy'Ckq(S) such that [R]€X, tS]€Y, ( C , IJ ) ' ( X) Cy ,
(C,U)'(X')Cy• there is defined a morphism of Rothenberg
exact sequences
m
m
Hm(Z2;X'/X)
(C.a.U)
(c,a,u) •
(c , u)
L* (S)
m+nx
Y '
L * (S)
m+n
Hm+n(Z2;Y'/Y)
The proof of 9.2 is somewhat more involved than
that of 9 . 1 .
A splitting (B,r,i) in A of an object (A,p) in the
idempotent completion A is an object B in A together
with morphisms r:A >B , i:B >A in A such that
r i = 1 : B
-> B , i r = p
-► A .
Lemma 9 . 3 A functor of additive categories F : A >E

235
extends to a functor r; u\ < id 11 ana uniy n
object (A,p) in A the object (F(A),F(p)) in
splitting in B. Any two such extensions o
naturally equivalent.
ich
if and only if for e;
F are
A'
-» F(A* ) = B*
An additive category A is idempotent complete if
the functor A >A; A >(A,1) is an equivalence of
categories. Applying 9.3 to 1 -. A >A we have that A is
idempotent complete if and only if every object (A,p)
in A splits in A • If B is idempotent complete every
*■* *■*
functor F:A >B extends to a functor F: A >B , namely
the composite of F: A >B and an equivalence B 'B•
For any ring S the additive category P(S) of f.g.
projective S-modules is idempotent complete, with every
object (A,p) in P(S) split by the triple (B,r,i)
de f i ne d by
-► B = im(p)
-► p( x)
i = inclusi on
This is the special case n=0 of
Lemma 9.4 For any ring S and any n^O the homotopy
category E (S) of n-dimensional f • g • projective
S-module chain complexes is idempotent complete.
2
Proof : For every chain homotopy projection p^p :D >D
of an object D in E (S) there exists by Lemma 3.4 of
Luck (7] an (n+1)-dimensiona1 infinitely generated

236
projective S-module chain complex C with chain maps
r:D >C, i :C■ > D and chain homotopies
-► D
Since C is dominated by an
it is chain equivalent to an object
Proposition 3.1 of Ranicki (17],
object in E (S) (namely D)
in En(S),
by
The idempotent completion of an additive category
A with an involution * : A >A is an additive category A
with the involution
tk
■+ A
(A,p)
■* (A ,p )
For a ring with involution R the functor
-^ k
E(R) »P(R);(R ,p) >im(p) is an equivalence of
additive categories with involution. Both 9.3 and 9.4
have evident versions for additive categories with
i nvo1u t i on .
Def i n i t ion 9 . 5 Let R,S be rings with involution. The
su rgery t rans fe r maps of a symmetric
project ive
representation (C, a ,U) of R
in En(S)
( C , a , V ) '
m
L^ (S)
m + n
(m^O)
are the composites
(C.a.U) ' : Lm(R) = Lm(P(R))
Lm<En<S>>
M
-► L
m + n
(S)
with [x the generalized Morita maps of 3.3 for A=P(S)
and F induced by the functor of additive categories
with involution F : B ( R ) aP (. R )-
t o t he functor
E (S) associated by 9.4

237
)(c,a,u)
!(R)
- En(S) ; R
The proof of 9.2 is now completed by observing
that the transfer map in the projective class groups
(C,U) • :KQ(R)-
■Kq(S) is such that
* < C , U )
(-)n(C,U)!*
KQ(R)
-* KQ(S)
Remark 9 . 6 Our methods also apply to construct
algebraic surgery transfer maps in the round L-groups
r X
L* (R) of Hambleton, Ranicki and Taylor (4], which are
defined for '-invariant subgroups XCk.(R). For any
symmetric representation (C, a,U) of R in E (S) and any
'-invariant subgroup XCk.(R), YCk.(S) such that
(C, U) ' (X)Cy there are defined round L-theory transfer
map s
(C.a.U)
LmX(R)
LrI (S)
m + n v
(m^O)
which are compatible with the round L-theory Rothenberg
exact sequences.
Remark 9.7 The connection established in §8 between the
algebraic and geometric surgery transfer maps extends
to the intermediate cases, and also to round L-theory.
Remar k 9 . 8 Our algebraic constructions apply also to
the e —quadratic L-groups Lt(R,c), which are defined for
a ring with involution R and a central unit e£R such
that ee=l. L2i(R,e) (resp. L2.+,(R,e)) is the Witt
group of nonsingular (-) e-quadratic forms (resp.
formations) over R. A symmetric representation (C, a,U)
of R in ID (S) such that U ( e) = 7} : C ► C for a central
un i t

238
7}£S with 7?7?=1 induces transfer maps
(C,a,U)! : Ljn(R,e) ► Lm + n(S,'?) (m£°> •
Hitherto wo. considered the case e = 1 £ R for which
L* (R, 1 )=L* (R) , with 7j = ies.
D
Appendix 1 . F i bred intersect i ons
i i
The proof of p' = p ' in %8 makes heavy use of
geo*alg '
the algebraic properties of the L-groups. For a fibre
bundle F ► B with the fibre F a compact
n-dimensiona1 manifold it is possible to verify that
the algebraic and geometric surgery transfer maps
coincide more directly, using the bordism intersection
theory of Hatcher and Quinn (6] to obtain fibred
versions of the geometric intersection forms (resp.
formations) used by Wall (22] to define the surgery
obstruction of a high1y-connected even (resp. odd-)
dimensional normal map. The quadratic kernel of the
pullback normal map is the fibred intersection form
(resp. formation) both algebraically and geometrically.
We now sketch the argument for the intersection pairing
X in the even-dimensional case, leaving the
se 1 f - i n t e r se c t i on function ju and the odd-dimensional
case to the interested reader.
Given two maps v. :Q. >M (i = l,2) let E(v. , v2 ) be
the pointed space of triples (x. , x_,to) defined by
points x €Q. and a path w:(0,l] >M from w(0)=v (x.) to
w( 1 )=v2(x-) , so that there is defined a homotopy fibre
square
E(v j , v2) ► Qj
vl
■

239
Given a stable vector bundle r\ over a space M let
f r
Q (M, 7}) be the bordism group of n-manifolds N equipped
with a map N >M and a compatible stable bundle map
For trivial J] this is the usual framed
group 0nr(M)=*^(MV<*>). For
<*> >M the homotopy pullback is the loop
space , E( * , * )=QM.
V
N
>n
cobord i sm
vl"v
2 :(^=02
Now suppose that M is an m-manifold, and that
v. :Q. >M is an immersion of a q. -manifold Q. (1 = 1,2)
such that v.(Q.) intersects v„(Q,) in general position.
Let Q.nQ2 denote the corresponding
(q.+q0-m)-dimensiona1 submanifold of M. The bordism
invariant of the intersection ((6,2.1]) is the bordism
class
X(v v ) = tQ,AQ9]
1 ' v2
l"v2
€ Oq"+q2_m(E(v1,v2),^Qi©^Q2©rM) .
If Q. and Q„ are (q.+q„-m+1)-connected the map
E ( * , * )=QM >E(v. ,v- ) induces an isomorphism ((6,3.1])
)fr
ql+q2 ~m
(E(*,*)) = 0
,fr
f r
qj +q2-m
(OM)
o^+q2_m<E(v1,v2),yQieyQ2eTM)
which is used as an identification,
Let
( f , b ) : M-
be
(i-l)-connected
reference map X >B, with the surgery obstruction
a*(f ,b) = (Ki(M) ,\,y)€L2i(Z( tTj (B) ] ) defined as in Chapter
5 of Wall (22]. Let v.,v9,...,v, be a base of the
kernel f-g- free Z(Kj(B) ]-module K (M)=Ki + j(f) -
Represent each v .£K.(M) by a pointed framed immersion
v .:S >M with a nullhomotopy in X. The values taken by
the (-) -symmetric form (K.(M),\) on the base elements
are just the bordism intersections

240
X(v v ,) £ OQr(E(v v ,),J> 0V i®7"M)
J J J J 5 5
= OQr(OM) = H0(OM) = ZfTTjCB)]
Now
let
( g , c ) : N-
be
the
(l^j.j'Sk) •
(i-1)-connected
( n + 2 i ) - d imens i ona 1 normal map with a if .-isomorphism
reference map Y >E obtained from (f,b):M >X by the
pullback of the fibre bundle F >E—E-»B along X >B. The
pointed framed immersions v.:S >M (l^j^k) with
nu11 homotopies in X lift to pointed framed immersions
On the chain
the kerne 1
w .:S Xf >N with nullhomotopies in Y.
J
level this corresponds to lifting
!(*, (B) ]-module chain complex C ( f " ) = S x K . ( M) =©S 1Z[ K. ( B ) J
1 1 k
to
the
kernel
![?r, (E) ]-module chain complex
C(g')=©SXC(F). The bordism intersections
k
X(w.,w.,) € 0^r(E(w., w,,),V . ®V , ©TM)
J J J J SXXF S]XF W
= Onr(nMXF,vF) (UJJ^k)
are the images of the bordism intersections X ( v . , v , )
under the geometric bordism transfer map
■XF : 0Qr(0M)
-► nnr(0MXF,vF) ;
-► XXF .
The Poincare duality isomorphism of based f • g
Zttfi(B)]-modules
free
(X(v.,v.,))
c(f • )
2i_*
SXK.(M)
-► C(f ' ) = S1Ki (M)
is lifted to the Poincare duality chain equivalence of

241
chain complexes of based f.g. free ZtKi(E)1-modu1e!
/%/ •>•> ^/'\ n + 2 i - *
(X(w . ,w . ,)) : C(g )
)S1C(F)n"
-► C ( g ' ) = ©S 1 C ( F ) .
k
Using the Poincare duality Z t it i ( E ) 1 -modu 1 e chain
~ n _ * ~
equivalence (F]fi-:C(F) >C(F), the action of QM on
the if, (E)-equivariant homotopy type of F and Hurewicz
maps there is defined a commutative diagram
0jr(0M)
- HQ(OM)
-®[ F]
I"
H0<"°"'Z[»1(E)]<C<F>'C(F>>OP
Jmn-
Onr(OMXF, vF) —» Hn(OMXF) —» Hn(C(F)0zt7r (E) ]C(F)) .
The anticlockwise composition gives the geometric
surgery transfer p on the level of intersections,
6 7 Fgeo
while the clockwise composition gives the algebraic
i
surgery transfer p'
g
Append i x 2 _. A, counterexample i n s ymme trie L- t heor y
An n-dimensional Poincare fibration F >E r ,g does
induce transfer maps in the symmetric
e i t he r
not in general
t.-groups p ! :1^(2(^, (B) ] )
1
.Lm+n(Zt7r1(E)])
algebraically or geometrically. It is not possible to
i
define p' geometrically since the symmetric L-groups
are not geometrically realizable (Ranicki (16,7.6.8]).
There are two obstructions to an algebraic definition
of p", which requires the lifting of an m-dimensiona1
symmetric Poincare complex (C,0) over Ztfi(B)l
representing an element (C,0)£L (Ztrt\(B)]) to an
(m+n)-dimensional symmetric Poincare complex (C',0")
over Ztfi(E)l representing the putative transfer
p! (C,0) = (C! ,0! )€Lm + n(Zt?r1(E) ] ) . The symmetric L-groups
are not 4-periodic, so it cannot be assumed that (C,0)

242
is highly-connected as in the quadratic case. In the
following discussion we assume that the fibre F is
finite, and that the chain complex 0 consists of based
f.g. free Z(tf;(B)]-modules. The two obstructions to
lifting (C,0) to (0',0') are given by:
i) it may not be possible to lift C to a based
t
f.g. free Z[;r. (E)]-module chain complex C ' with a
filtration F„C"CF, C ' C. . .CF C'=C" such that the
0 1 m
connecting chain maps between successive filtration
quotients are given up to chain homotopy by
9 = p/'(dc) : FrC!/FrlC! = Srp#(Cr) ►
S(FrlC! /Fr_2C! ) = Srp/'t(Cr_1) (l^r^m)
r it
where S denotes the r-fold dimension shift and p is
the functor of 51
P# = -®(C(F),l!) : BCZIff^B) J) ► Bn<Z[Kl(E) ]) ,
i i ) even if C' exists, it may not be possible to
lift the m-dimensiona1 symmetric Poincare structure 0
on C to an (m+n)-dimensiona1 symmetric Poincare
t i
structure 0" on C".
If C can be assembled over B in the sense of Ranicki
i
and Weiss [2 0] then it can be lifted to C', but in
general it is not possible to assemble 2[ffi(B)]-module
chain complexes, so already i) presents a non-trivial
obstruction to the existence of transfer in symmetric
L-theory. Even if the obstruction of i) vanishes (e.g.
if B is an Ei1enberg-MacLane space K(X■ (B) , 1 )) then ii)
may present a non-trivial obstruction. This is
illustrated by the following example, which exhibits
the failure of a projection of rings with involution
p:S >R=S/(l-t) (t = central unit £ S, t=t~ £S) to
induce an S -bundle symmetric L-theory transfer map
i o 1 1
p':L (R) >L (S) analogous to the S -bundle quadratic
L-theory transfer map p':Ln(R) >L.(S) (cf. 4.7). The

243
transfer p"(C,0) = (C',0') of a O-dimensional symmet r i c
/
Poincare complex (= nonsingular symmetric form) (C,0)
over R with C^ = R is defined if the symmetric kXk
0
mat r i x
*0 = (<V € MR(R)
can be lifted to a kXk matrix 0A€M, (S) such that
p(0(l))=0o€Mk(R) and
t0Q - (0O)* = (l-t)0J € MR(S)
t t *
for some symmetric- kXk matrix 0 ." = (0 ". ) £M,(S), so that
,, ilk.,
(C',0') is a 1-dimensiona1 symmetric Poincare complex
over S with C'=C(]-t:S >S ). In particular, for
S =
_o [ OL-y J\wur\
[t,u]/(t2-l ,u2-l) ,
t = t , u = t+u+1 ,
P : S
♦ R = Z2[Z2] = Z2[u]/(u--l) ;
t ► 1 , u ► u
the transfer is not defined for the O-dimensional
symmetric Poincare complex (C,0)=(R,u) over R, for
although C can be lifted to C' and 0n can be lifted to
0^ there does not exist a symmetric 0j- Both the
obstructions to i) and ii) vanish for the visible
symmetric L-groups VL (Z [ ;f ] ) of Weiss [23] provided
that B is an Eilenberg-MacLane space K(?r. (B),l), in
which case there are defined transfer maps
P! :VLm(Z[tfj (B) ]) ►VLm + n(Z[7r1 (E) ]) .
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244
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(19] Addi t ive L-theory
Mathematica Gottingensis 12 (1988)
[20] A.Ranicki and M.Weiss
Chai n complexes and ass embly
Mathematica Gottingensis 28 (1987)
[21 ] C.T.C.Wall
Poincare complexes
Ann. of Maths. 86, 213-245 (1970)

246
[22] Surgery on c ompac t man i f oId s
Academic Press (1970)
[23] M.Weiss
On t he definition of t he s ymme trie L-groups
preprint
(24] G.W.Whi tehead
E1emen t s o f homot opy theory
Springer (1978)
W.Luck: Mathematisches Institut,
Georg-August Universitat,
Bunsenstr. 3-5,
34 Gottingen,
Bundesrepublik Deutschland
A.Ranicki: Mathematics Department,
Edinburgh University,
Edinburgh EH9 3JZ,
Scotland, UK.

SOME REMARKS ON THE KIRBY-SIEBENMANN CLASS
R. J. Milgram
In this note we study the relations that hold between the Kirby-Siebenmann class
{A'S} 6 Hi(BsTOPi Z/2) and the first Pontrajagin class.
The first result is that that the natural map po : Bstop —* Bsg does not detect
{A'S} no matter what coefficients might be used. However, the homology dual of {A'S}
is in the image of the Hurewicz map
■Ki{BsTOp) —> Hi(BsTOP\ Z/2).
In fact there is a unique non-zero element [A'S] 6 *i{BsTOp) °f order 2, and po([KS}) ^
0 6 tt^Bsg)- In particular this implies that ji>4 + {A'S} is a mod(24) fiber-homotopy
invariant of SPIN-TOP bundles. However, it is interesting to ask what happens when w2
is non-zero. To understand this we introduce an intermediate classifying space, Btsg for
which we have a factorization
/ P
Po=p-f, Bstop >Btsg—>Bsg-
Btsg is univeral for the vanishing of transversality obstructions through dimension 5.
Additionally, Btsg is built out of finite groups (Z/2-extensions of the symmetric groups
Sn) in the same way that Bsg >s constructed from the Sn. As a result, explicit construction
of homotopy classes of maps into Btsg >s often possible.
We show that H4{BTsg; Z/2) = Z/2 © Z/48 and that the homology dual of the
Kirby-Siebenmann class maps to 24 times the second generator. Thus, this transversality
theory does detect {KS}. But note also the Z/48. Our main question is the extent to
which it gives rise to a fiber homotopy invariant of topological Rn-bundles. The general
result is
Theorem I: Let f, tp be two stable Hn-bundles over X, and suppose they are fiber
homotopy equivalent. Then there is a 6 H2(X; Z/2) and
24a2 + Ptf) + 24{KSU)} = P^) + 24{KS{^)}
in H4(X\ Z/48) where Pi(() is the Z/48 reduction of the first Pontrajagin class.
In other words, there is an element A G H4{BTsg; Z/48) with f'{A) = Pi + 24{KS},
and (I) gives the effect of different liftings of a map po ■ 9 '■ X —> Bstop —* Bsg on A.
H2(Bstop; Z/2) = Z/2 with generator w2, so the possible factorizations of po
through Btsg differ in their effect on A only by 24w\. In particular this gives
Corollary: If M4 is a compact closed topological manifold with even index, and u is its
stable normal bundle, then w\ = 0 £ H2(M; Z/2) and
u'f'(A) = P1(u)+24{KS(u)}
is independent of the choice of f factoring po.
This note came about in answer to a question of Frank Quinn. He pointed out that
in [M-M] the exact structure of Bstop, and the various surgery maps in dimension 4 were
never worked out. But currently it appears very useful to understand them. Of course, we
do not attempt to work out explicit geometric methods for evaluating the new invariants.
But knowing what they are and how they fit together should make that fairly direct.

248
' z
Z/2
Z/2
Z/24
0
. Z/2
i =0
2 = 1, generator r\
2 = 2, generator ftj
2 = 3, generator v
i = 4, 5
2=6, generator k.2
The homotopy types of Bso■, Bsg in dimension < 7
A Postnikov system for Bso through dimension 7 is given by
(1) Bso >K(Z/2, 2) >K(Z, 5)
vrithK-inva.n&nt2{Sq2Sq1{i2) + i2-Sq1{i2)}. (Note that if 5(A'(Z/2, 2); Z) = Z/4 with
generator having mod( 1) reduction 7 and
(2) 7 = Sq'Sq1^) + 12-Sq1 (12).
Moreover,/^4(4) = 7,)
The stable homotopy of spheres is given in the first 6 dimensions by
(3) <(5°) =
and we will use the same names for the corresponding elements in t1+j(5sg) — ir*(S°).
One relation that should be kept in mind is tjki = \2v, since it also holds in tt,(Bsg),
though the relation r\2 — k\ which holds stably does not hold in ~k+{Bsg)-
Lemma (4): A Postnikov system for Bsg through 7 is given by
K(Z/2, 2) x K{Z/2, 3) x K{Z/2, 7) >K{Z/24, 5)
wiiere the A'-invariant is 2{Sq2Sql(i2) + t2 • Sql(t2)} + 4{Sg2(i3)}.
Proof: With Z/24-coefficients the AT-invariant for Bsg maps back to the image of the cor-
risponding AT-invariant for Bso- Hence, the class in (2) must appear in the AT-invariant.
Also, the kernel of the map if5(AT(Z/2, 2,3); Z/24) —► if5(AT(Z/2, 3); Z/24) is
generated by 4Sq2{i3), It follows that 4Sg2(t3) is the only term which can be added to the
A'-invariant. But, in fact, this term must be involved in the AT-invariant because there is
the homotopy relation which we have already noted tjki = \2v, since r\ is detected by Sq2.
In order to understand the integral homology of Bsg, Bstop, and the intermediate
space Btsg which we will introduce shortly, we need a method for obtaining Bochstein
information from AT-invariants. The following result will suffice.
Lemma (5): Let AT(Z/2\ j) x A'(Z/2, ; + 1)-AT(Z/2% j + 1) be given with
k - 2^-) + 2-^+1),
then the fiber B of the map k is K{Z/2t+,-w-1 x Z/2"1).
Proof: The homotopy exact sequence of the fibration in dimensions j, j + 1 is
(6) 0 >*j+i(E) >Z/2-^Z/2s >*j{E) »Z/2J >0

249
But the term 2'~1l]+i in k*(lj+i) implies that k, is injective in (6), Thus E is a K(it, j)
and -k is given as an extension in the sequence
0—>Zj2'~l >*j(E) »Z/2'—>0.
The type of this extension is determined by the term 2w{(3(lj)) in k'(i] + 1). From this (5)
follows.
(4) and (5) imply that there is a mod(8) Bochstein
M4) = {Sq'i*)} m H'(Bsg; 2./2).
Additionally, the Hurewicz image of v is {w\~) + 2{i^'"} since this is already true in Bso,
where it is well known. As a consequence Hi(BsG\ Z) = Z/2 © Z/24 with generators
{w^}, {uC*} respectively, and \1v is in the kernel of the Hurewicz map.
The structure of Bstop through dimension 7
From the fiberings
G/0 > Bso Bsa
(7) 111
G/TOP ► Bstop > Bsa
and the well known result of Kirby-Siebenmann that t4(G/TOP) = t4(G/0) = Z, but
that the map between them is multiplication by 2, we get the diagram of extensions in t4,
0 Z -^+ Z ► Z/24 ► 0
(8) -2
0 Z ► ^i(BsTOp) > Z/24 > 0
The only way this diagram can commute is if ^t(BsTOp) — Z/2 © Z with the element
of order 2 mapping to 12 ■ v, and the generator of the Z-summand mapping to v.
(Z/2 t = 2
Lemma (9): ~x%{Bstop) — { Z$ Z/2 i = 4 Moreover, a Postniicov system /or
U 4 < i < 8.
Bstop through this range is given by
(10) K(Z/2, 2) xK(Z/2, 4) >K(Z, 5)
with if-in variant 2{Sq2Sq1(i2) + 1-2 • Sql(i-2)}-
(This is clear.)
In particular, the class {KS*} € Hi(BsroP', Z) which is in the Hurewicz image of
the element of order 2, must go to zero in H^Bsg; Z), since, in homotopy, it goes to
12f. This shows that {KS'} has no homology (or cohomology) relations implied by the

250
map into Bsg- However, in homotopy, the fact that it maps to 12j/ should have some
consequeences.
The space Btsg
The failure to detect the Kirby-Siebenmann class in H,(B$q; Z) is the influence of
the first exotic class 13. In fact, the term 4Sq2(i3) in the 5-dimensional K-invariant (4)
is exactly the difficulty. (For example, if we kill u>2 but leave 13 in H'(Bsg; Z/2) the
resulting space has only Z/4-torsion in if4( ; Z).) Hence it is natural to consider the
classifying space Btsg obtained from Bsg by killing the exotic class 13. For definiteness,
recall that 13 is detected with O-indeterminacy in the Thom-complex MSG by applying
the twisted secondary operation associated to the relation (u>2 + Sq2){u<2 + Sq2) to the
Thorn class, and using the Thorn isomorphism to bring the class back to Bsg- For details
see [R].
We have the fibration sequence
(11). A'(Z/2, 2) yBTsG-^BSG-^K(Z/2, 3)
with .fiT-invariant 1.3. This is the universal space for fiber homotopy transversal!ty to
hold in the Thorn space, at least through dimension 5 (Compare [B-M]). Indeed, a fiber
homotopy sphere bundle £ —> X and reduction to Btsg is equivalent to the condition
Lz{V) — 0 £ H3(X; Z/2), together with a specific choice of 2-dimensional cochain c so
6c = f*(i3)
where / : X —► Bsg classifies f. This situation is very close, but certainly not the
same as the situation studied in [F-K]. Also, there is a factorization of the canonical map
Bstop-^Bsg as
Bstop +Btsg >Bsg-
Precisely, there are exactly two such factorizations differing by a map
Bstop — #(Z/2, 2),
Now, we look at the 6-skeleton of Btsg- This is the 6-skeleton of the 2-stage Postnikov
system
K{Z/2, 2) x K(Z/3, 4) >K(Z/8, 5)
with X-invariant 2{Sq2Sq*(12) +12 •Sq1(t2)}- From (5) the resulting space has Ath integral
homology group given as
Ha(BTsg; Z) = Z/2 0Z/48
with generators (w^)', (u^)* respectively. Here, W2 can be identified with 12- Note that
this implies that the Kirby-Siebenmann class maps non-trivially to 24((w2,)").
The proof of theorem (I)
Lemma (12): Let X —'Btsg be given ancf suppose /' is the composite
X »if(Z/2, 2) x Btsg >BTsg

251
where fj. is the principal bundle map K(Z/2, 2) x Btsg —* Btsg, then
/'•'(vl) = /*{^}+24a:eif4(X; Z/48).
Proof: #4(A'(Z, 2) x Btsg] Z/16) = (Z/2)2 © Z/4 © Z/16 with generators
8( i2 ® u)2), 8(1 ®u>4) of order 2, (4i2 ® 1) of order 4, and (1 ®u>?,) of order 16.
We will show that /i*(u>?) = 8(t5 ® 1) + 1 ® u>?. We first note, by naturality and the
primitivity of w\ in H*(Bso', Z) that 8( i2 ®u>2) is not in this image. Next, we look at
the cohomology Serre spectral sequence of the fibering
A'(Z/2, 2) >BTsg -Bsg
with Z/16-coefficients. E° 4 = #4(A'(Z/2,,; Z/16) = Z/4, with generator 4ti Also,
£| ° = H*(Bsg; Z/16)"= Z/2©Z/8 with generators 8u»4, 2(^), and
E* ° = Hs(Bsg; Z/16) = (Z/2)3 + Z/8.
Here, only the Z/8 is of interest. It has generator Sq2(i3), so ^5(4^) = 4Sq2(i3), and
at E'J, i + j = 4, only E° 4 = Z/2, E4 ° = Z/8 © Z/2 are non-zero. Thus there is a
non-trivial extension for Hi(BTSG', Z/16)
0 >Z/8 (generator 2w\) ^Z/16 >Z/2 (generator 8i|) »0.
But this forces the result.
Theorem (I) is direct from (12). The corollary follows, also, since the assumption of
even index implies that tu2(M4)2 = 0 (mod 2). Hence, either lifting gives the same map
in cohomology with Z/48-coefRcients.
Concluding remarks
From Quillen's work we know that Bsg®Z2 can be identified with B(B+(SO(F3))) in
dimensions < 6, and as B(B+(Soo)) in all dimensions. Here, Sx, is the infinite symmetric
group. Similarly we can describe Btsg as B(B+(SO(F3))) in this same range. Moreover,
Btsg can be given as B(B+(S<x,)) in all dimensions. Here, these new groups are described
by central extensions
Z/2 >SO(F3)—>SO(F3) »0
Z/2—xSoo—^00—>0
where, for S<x, the extension is the (unique) non-trivial one for which the transposition
(1, 2) continues to have order 2. This might be very useful in understanding Casson's
recent results on the Rochlin invariant.
It seems direct to use the description above of Btsg by finite models to calculate the
order of the classes which carry the remaining Pontrajagin classes. I hope to return to
this later.
Also, there is a second factorizing space for the map Bstop —* Bsg, namely the
space where we kill all the exotic classes ff(e2. _i,2'-i)- The precise structure of these
classes is not entirely known, but there is considerable information in [R]. So it should

252
be possible to understand the higher torsion in the cohomology and homology of this
intermediate classifying space. Moreover, it is likely that it is the universal space for the
vanishing of transversality obstructions.
Bibliography
[B-M] G. Brumfiel-J. Morgan, ffomotopy theoretic consequences ofN. Levitts obstruction theory to transver-
saJity for spherical fibrations, Pac. J. Math (1976) 1-100
[F-K] M. Freedman-R. Kirby, A geometric proof of Rochhn's theorem, Algebraic and Geometric Topology,
A.M.S. Proceedings of Symposia in Pure Mathematics, Vol. XXXII(l) (1978) 85-98
[M-M] lb Madsen-R.J.Milgram, Classifying Spaces for Surgery and Cobordism of Manifolds, Ann, of Math
Studies #92, Princeton U. Press (1979)
[R] Doug Ravenal, Thesis, Brandeis University (1970)
November, 1987
Sonderforschungsbereich 170
Gottingen Universitat

The Fixed-Point Conjecture for p-Toral Groups
by
Dietrich Notbohm
1. Introductlon
Suppose that X is a space with an action of the topological
group G. Let X and X denote the fixed-point set
respectively the homotopy fixed-point set of this action.
We define
XhG := map_(EG,X)
as the space of G-maps in the category Top of topological
spaces and maps. As model for EG any acyclic G-complex is possible.
(Here complex always means CW-complex.) X is then unique
up to homotopy.
The definition is not given in the category S of semisimplicial
sets, as it happens in [DZ] and [M] for finite groups. For
topological groups the space EG, constructed as nerve over a
category, is not a simplicial set, but a semisimplicial object
over the category Top . Therefore the same is true for the space
map_(EG,X) ,
where X is interpreted as the singular chain complex of the
topological space X. For finite groups both definitions agree
up to weak homotopy [BK; chapter VIII ].
There are two other interpretations of the homotopy fixed-point
set. The first one is as section space
of the fibration
T(EGxG-iBG)
G
EGx_X > BG
the second one is as fixed-point set
map(EG,X)
where G operates canonically on map(EG,X) .
Let p be a prime, for all time fixed. X' denotes the
Z/p-completion in the sense of Bousfield and Kan [BK].
X is called Z/p-good, if X' is p-complete [BK; 1,5].
Especially nilpotent and other "nice" spaces are Z/p-good.
Look at [BK;VII].

254
The unique G-map EG —> * , where * is the one point set
with a trivial G-action, induces a map
XG = map„(*,X) 1 map_(EG,X)
G v*
Functoriality of the composition gives a composite map
xG~ , x'G , x~hG
p p p
which fits into a commutative diagram
XG , xhG
G" ,G
X . X
P P
Definition: A topological group N is called a p-toral group,
iff there exists an exact sequence
1 —,t —> N —>P —> 1
where T is a Torus and P a finite p-group.
Theorem: If N is a p-toral group and X a Z/p-good connected
finite N-complex, then the map
XN~ _ x~hN
P P
is a weak homotopy equivalence.
Remark: The analogue theorem for finite p-groups, but
without the technical condition Z/p-good, is proved
by H. Miller in [M]. It is the foundation of the rest of
the paper. For this result J. Lannes found another proof.
It is a pleasure to thank J. McClure for valuable
discussions about the book of Bousfield and Kan.
2. Proof of the Theorem
We need some remarks:
2.1 Remark: Let
be an exact sequence of topological spaces and assume,
that H is finite. Let X be a G-space. H acts on the

255
fixed-point set X canonically. We have
(XK)H = XG .
As H is finite, EG is an acyclic K-complex of finite
type. We get
X - map(EG,X)
where map(EG,X) is a G-space. Hence using the above equation
and the exponential law for mapping spaces, we get the
analogue:
xhG _ (xhK)hH
2.2 Remark: Let f:X. > X„ be a weak homotopy equivalence
and a G-map between two G-spaces X., X„ . The horizontal
map in the diagram
EGXGX1 ' EGXGX2
BG
is a weak homotopy equivalence. Because BG is a complex, the
two spaces
map(BG,EGx x.) , i=l,2
G l
are weak homotopy equivalent as well.
We denote with
map(BG,BG). ,
id
the connected component of the identity and with
map(BG,EGx x.)
the space of all maps, which are homotopic to a section.
If we look at the two fibrations
XihG 1 map(BG,EGx X±) » map(BG,BG)id ,
it is easy to see, that the two homotopy fixed-point sets
X. are weak homotopy equivalent.

256
Proof of the theorem: i) reduction to the case of a torus.
Let
1 > T > N > P > 1
be the exact sequence belonging to the. p-toral group N.
T
X is a finite P-complex. It is proved in [M] that
XN- = (XT}P- , (xT'}hP
P P P
is a weak homotopy equivalence. Setting
X~hT = mapT(EN,Xp) ,
remark 2.1 implies a weak homotopy equivalence
X~hN - (X"hT)hN
p w p
The map
XT~ _^ hT
P P
is P-equivariant. Together with (2.2) we can reduce therefore
the problem to the case of a torus.
k 1
ii) Let n be the dimension of T. We can think of Z/p c S
as the group of the roots of unity with order p and define
ok := (Z/p )n , o. := liij o]t
The homomorphism a.. —> T induces a mod p-equivalence
Bo„ —> T ,
which is the same as to say that the map
H..(Bo..;Z/p) . HjtBT.-Z/p)
is an isomorphism.
Now let X be a Z/p-good connected finite T-complex. Then
there are the following maps
hT ho.. ho.
X' > X' » lim x: k
P P < P
T o„ o
X » X » lim X
k
As T is a finite o.-complex for all k, X has also the
structure of a finite o.-complex. Using Miller's Theorem
[M] and the following three propositions, the proof will
be finished in a straightforward way.

257
2.3 Proposition; Let X be a finite T complex. Then it is
T o
X = Jam x k
and the sequence of the fixed-point sets is a finite sequence.
2.4 Proposition: Let X be a Z/p-good finite T-complex. Then
the map
x-hT ( x-ho.
P P
is a weak homotopy equivalence.
2.5 Proposition; Let X be a finite T-complex. Then it is
ho,. ho.
tr (X' ) > lia n (X~ k) .
n p F^— n p
3. Proofs of the Propositions 2.3 - 2.5
Proof of 2.3: If X is a finite T-complex, it consists of a
finite number of cells of the form T/A x e , where AcT is a
closed subgroup.
T °\r
T/Axe belongs to X , if A=T , and it belongs to X K ,
n
if o. cA . Because a„ is dense in T and because there is
only a finite number of orbit types T/A, we get
X°k = XT
for k big enough.
3■1 Lemma: Let Y be a p-complete space. Let
Y , E , b
be a fibration, such that the action of ir. B on
H (Y;Z/p) is nilpotent. Then the Z/p-completion
induces a homotopy equivalence
r(E -> B) » r(E~ -> B*)
p p
between the section spaces.
Proof: Under the above assumption the mod R fibre lemma
[BK,- II, 5] is applicable. We get a fibre square

258
(*)
-t B
which induces a commutative diagram
(**)
map (B,E)
map(Bp,Ep)g
map (B,B) .,
map(B*,B~) . ,
P P id
where the rows are fibrations and the columns are given by
the completion. With the universal property of pullback
diagrams, which fibre squares are, you can prove, that (**)
is up to homotopy a fibre square too. The fibres of the
rows in (**) are exactly the section spaces. This implies
the Lemma.
3.2 Lemma:
Let E.
l
B.
i=0,l
be fibrations with
p-complete fibre, in such a way that the diagram
E„ > E,
is a fibre square. Assume that the operation of tr. (B.)
on H.(E..;Z/p) is nilpotent and that the map BQ —> B.
is a mod p equivalence. Then the two section spaces
r(E.
Bi>
r(E
o
V
are weak homotopy equivalent.
Proof; The assumptions of the mod R fibre square lemma
[BK; II, 5.3 ] are satified. We get up to homotopy a
fibre square
E
Op
Jlp
Op
IP
with homotopy equivalences in the rows. This implies that
the associated section spaces of the fibre squares are weak
homotopy equivalent. If you use 3.1, the proof will be
finished.
Proof of 2.4: The diagram

259
Eo-xo„Xp » ETxTXp
Bo„
BT
is a fibre square with a p-complete fibre in the columns.
Moreover BT is 1-connected. Lemma 3.2 applies.
3.3 Lemma: Let G.cG„c... be a ascending sequence of groups
and define G„„
U G. . Let X be a G..-space. Then the map
hG„
hG.
-i holim X K
holim
is a weak homotopy equivalence.
Proof: For the definition of holim see [BK; XI].
We choose the Milnor model for the spaces EG„ and EG,. Then EG«,
aces EG. or
EG„ = liii} EG,
is exactly the union of the spaces EG. or
This implies that
hG„
k '
mapG (EG ,X) = £im mapG (EGk,X)
hG.
X k .
hG. hG. ,
On the other hand the maps X K > X K L are fibrations.
According to [BK; XI] there is a weak homotopy equivalence
hG, . __hG,,
EG„
lim X k 1 holim X
Proof of 2.5: Because of Lemma 3.3 there is an exact sequence
0
1 ho, hOoo
lim1 tr ,, (X* k) . tr (X'
ho.
lim tr (X* k)
< n p
n+1 p n' p
for all base points [BK; XI, 7.4] . By [M] we get
ho. a.-
n (X* k) a tr (X k )
n p n p
Proposition 2.3 implies that the lim -term must vanish.

260
References:
[BK] A.K. Bousfield and D.M. Kan: Homotopy Limits, Completion,
and Localisation; Lecture Notes in Math. 304, Springer 1972.
[DZ] W.G. Dwyer and A. Zabrodsky: Maps between Classifying
Spaces; preprint.
[M] H. Miller: The Fixed-Point Conjecture; to appear.
Dietrich Notbohm
Mathematisches Institut der
Georg-Augus t-Universi ta t
Bunsenstr. 3-5
D-3400 Gottingen
Bundesrepublik Deutschland

Simply connected manifolds
without S'-symmetry
V. Puppe
Several authors have studied the question of existence of manifolds
with little or no symmetry (s .[1],[2],[6],[7],[16],[17]), e.g. E. Blocntberg has
shown (s.[2]) that there exist closed manifolds which do not admit any
effective topological (continuous) action of a compact Lie group. For
his argument the presence of a rather complicated fundamental group is
essential.
From a completely different point of view M. Atiyah and F.Hirzebruch
had proved earlier (s.[1]) that a compact spin manifold M can not admit
an effective differentiable S1-action if the A-genus k(M) is different
from zero. It has been shown, though, that the differentiability
assumption in their result is crucial, i.e. there exist examples of
topological effective s'-actions on spin manifolds with A(M) * 0 (s.[3] VI.
9.6 and [4]).
Here we prove, using the connection between P.A. Smith-theory and
deformation of algebras (s.[12],[13],[14]), that there exist simply
connected, closed, oriented, differentiable manifolds M such that any
closed, orientable manifold M with H*(M?(C) = H*(M?dJ) (as algebras over
f) has no topological S1-symmetry, i.e. does not admit any non trivial
topological S1-action; in fact, there exist examples which admit non
trivial topological 2Z/ -actions only for (at most) finitely many primes
p (compare [11]).
S.Kwasik and R.Schultz have studied topological S1-actions on
4-manifolds and - among other results - they show, by completely different
methods, that there exist many closed simply connected 4-manifolds
without topological S1-symmetries (s.[19]).
I want to thank R. Buchweitz, J. Damon and A. Iarrobino for
illuminating conversations on the deformation theory of Artin algebras. In
fact, what is described in this note is more or less an interpretation
of certain algebraic results about deformations of algebras (s.[9],[10])
in the context of S1-action from the view point of [12],[13],[14].
If X is a paracompact, finitistic S1-space which is totally non
homologous to zero (TNHZ) in the Borel construction XQ := EG * X (G=sM
with respect to £ech cohomology with rational coefficients and if
dim H*(X;(C) < ~, then the cohomology algebra B := H*(X ;(C) of the fix
point set X can be viewed as a deformation of the algebra A := H*(X;dJ).

262
A one parameter family of deformations Aft] = A § (DCt] (where "~"
indicates the twisting of the multiplication)with A =A and A =B
(disregarding the grading is given by the cohomology H*(X_;(D) of X_ considered
as an algebra over H*(BG,-ffl) = Q[t], deg(t) = 2 (s.[12] for details).
The property (TNHZ) is automatically fulfilled if H° (X;(B) = 0
because then the Leray-Serre spectral sequence degenerates already for
degree reasons.
This suggests the following program to exhibit manifolds without S -
symmetry:
1 . Find a rigid graded algebra A* over (C with A° = 0 and dim A* < <*>,
which fulfils Poincare duality ("rigid" means roughly that all
algebras obtained from A* by deformation are isomorphic to A*; s. [20]
for a discussion of different notions of rigidity and how they relate
to each other).
2. Realize A* as the rational cohomology algebra of a manifold M.
3. Check that H*(M;Q) = H*(MG;Q) implies M = MG.
Yet there are several obstacles:
a) First of all it is not known (to me) and seems to be a very difficult
question to decide whether there exist non trivial rigid algebras of
the desired form (s. [20] for examples of non-commutative finite
dimensional rigid algebras, in particular over S/p ).
b) To realize a Poincare algebra as the cohomology of a (simply
connected) compact manifold one needs certain extra conditions to be
satisfied if the dimension is divisible by 4 (s.[18]).
c) Even if one would find a non trivial rigid graded algebra A, the
isomorphism between A and some deformation of A need not respect the
grading (supposing the algebra obtained by deformation has an a
priori grading).
But not every deformation of an algebra A* = H*(X;0J) which is
possible algebraically can be realized by an S1-action on X. The one
parameter families A 5 3)[t] which correspond to S1-actions on X have
certain special properties (s.[12],[13]):
Thereexists a grading on A. (namely the one given by the isomorphism
A. = H*(X ;$)) such that A ® 0}[t] embeds into the trivial family
A. ® 3j[t] as a graded algebra over dj[t] and the cokernel of this
embedding is dj[t]-torsion. In fact, this is just a reformulation of the
localization theorem, which says that the morphism H* (X.,; ¢)->H* ( (X )G;(C)
induced by the inclusion X —► X becomes an isomorphism after
localization at (0). The property (TNHZ) gives that H*(XG;(C) -» H*((XG)G;(C) is
injective. In particular A 5 3)[t] is then a jump deformation (s.[8]) in
the sense that all A := (A 5 3J[t] 9 <&£, where 3Je is dj considered as a
£ «[t]
3)[ t]-module via 3)[t] -» ¢, t ■—► e, are isomorphic for e * 0; moreover

263
A1 has a filtration such that the associated graded algebra is
isomorphic to A (s.[ 13]) .
r o
Hence for the first part of our program we only need to know that A
is rigid with respect to "g-deformations", i.e. deformations of the
special kind described above. If A has that property we will say that A
is g-rigid. It is shown in [5] and [8] that a non trivial g-deformation
of A lowers the dimension of the second Hochscbild cohomology of A with
coefficients in A (the space of infinitesimal deformations), i.e.
dimm H2(A ,A ) > dimm H2(A.,A.). On the other hand there exist non
B o o (C 1 1
smoothable graded Artin algebras (already defined over ¢) (s.[9],[10]),
i.e. algebra which do not admit deformations to <S x...x ¢, in fact not
even to (E x...x (E if one extends the ground field to (E. Therefore, if
one starts with a non smoothable graded Artin algebra and considers all
algebras which can be obtained from A by (iterated) g-deformations,
there must exist non trivial g-rigid graded algebras (among the
components, i.e. direct factors of the algebras obtained). By Quillen's
results in rational homotopy (s.[15] and also [l8])one can realize such
an algebra as the rational cohomology of a simply connected finite CW-
complex and hence obtains:
Proposition 1: There exist simply connected finite CW complex X such that
D).
the strictly commutative graded algebra H*(X;dJ) is g-rigid (H (X;(C)=0) .
For every S1-action on such a space X the rational cohomology of X
is isomorphic as a filtered algebra to H*(X;3J) (s.[5]). Within the
("filtered") isomorphism type of H* (X;(C) one can choose a graded
algebra A* with "minimal degree", i.e. dim ( § A1) should be maximal for
«Vi=o j
each q. Let Y be a simply connected finite CW-complex with H*(Y;(C)=A*,
then one gets the following:
Corollary 1: For any S1-action on Y the inclusion Y -> Y induces an
isomorphism in rational cohomology.
P
Proof: By the choice of Y one has that H*(Y;0J) and H* (Y ;(C) are
isomorphic as graded vector spaces (and as filtered algebras). Therefore
the morphism H*(Y;(C) ® Q[t] -» H*(YG;(C) ® Q[t] induced by YG —» Y, can
only become an isomorphism after localization, if the evalution at t=0
- ®
<B[t]
(i.e. applying - ® (C°) , which gives H*(Y,dJ) -» H* (Y ,0J) , is an
isomorphism, too.
Remark: Of course Corollary 1 does not imply that any S'-action on Y

264
must be trivial. Instead of Y one could as well take YxD2 a Y (D2 :=
{x EM2; |x| < 1}) which clearly admits a non trivial Sl-action with
(YxD2)G = Y x {0}.
To exhibit simply connected manifolds without S1-symmetry we are
looking for connected Poincare algebras over (C which are g-rigid. It is
shown in [9] and [10] that there are non smoothable connected Poincare
algebras A (graded Gorenstein algebras) with A° = 0 and formal
dimension of A equal to 6. This leads to the following
Proposition 2: If A is a non smoothable connected Poincare algebra of
formal dimension 6 with A° = 0, then A is g-rigid.
Proof: Let B be an algebra obtained from A by a g-deformation. Then B =
k
n B. and B. is a connected Poincare algebra of even formal dimension
1=111
< 6 with B? = 0 for i = 1,...,k. It is easy to see that any connected
Poincare algebra C of formal dimension fd(C) = 2 or 4 and with C = 0
is smoothable. (For fd(C) = 2 this is obvious since C = 3)[x] /, 2.; for
fd(C) = 4 a somewhat round about but simple argument is to observe that
C®(E = H*(M;(E) where M is a connected sum of a number of copies of (EP2s
and therefore admits an S1-action with isolated fix points.) Hence there
must be a component, say B], in B with fd(B)) = 6. By the inequalities
® A1 > dim ® B there is precisely one component with formal dimen-
i=q i=q
sion 6.
The top dimensional generator a £ A is mapped to a non zero element
in B, ® 1 <= B, ® Q[t] by the morphism A ® Q[t] -» B ® Q[t] -» B, ® (Cft],
k
where the last map is induced by the projection B = n B. -» Bi.
i=1 1
(Otherwise the first map in the composition would not become an
isomorphism after localization.)
Evaluated at t = 0 one obtains a morphism A -» Bj of connected
Poincare algebras which has non zero degree and hence is an isomorphism.
This implies, of course, that A = B = B,, since dim A = dimffi B.
Theorem 1: There exist simply connected, closed, oriented differentiable
6-dimensional manifolds M such that no closed, orientable manifold with
the same rational cohomology algebra as M admits any non trivial S1-
action.
Proof: Choose a non smoothable connected Poincare algebra A* of formal
dimension 6 with A° = 0. By Sullivan's results (s.[18]) there exists

265
a simply connected, closed, oriented 6-dimensional differentiable
manifold M with H*(M,(B) = A*. By Proposition 2 the algebra A* is g-rigid;
in fact the proof of Proposition 2 shows that for any S1-action on a
manifold M with H*(M;(C) = A* the inclusion of the fix point set M -» M
induces an isomorphism in rational cohomology. Since M is closed and
orientable this implies M = M, i.e. the action is trivial.
Remarks: a) For the above argument it is, of course, essential to assume
H to be closed. The manifold with boundary M x D2 and the open manifold
o
M x D2 are homotopy equivalent to M and clearly admit non trivial S1-
actions.
b) The following example of a graded Gorenstein algebra is due
to A. Iarrobino (s.[9] Ex.7,[10]). It was checked on a computer to admit
only deformations to algebras of the same "type", in particular it is not
smoothable.
A := R/J with R = dj[a,b,c,d,e,f] and the ideal J is generated by
{3ab-4ac-3bd,ad,ae,b2-af, 1 2bc-9af-1 6bd-1 2ce,be, bf, Sc^ac^cd-Sac-'Jce-
Sdf ,cf,3d2-4bd,12de-12bd-16df-9a2,3e2-4ce,ef-ce-a2,3f2-4df}.
If one assigns the degree 2 to all the generators a,b,c,d,e,f then
J is a homogeneous ideal and A is a connected Poincare algebra of
formal dimension 6 with A° = 0. Iarrabino remarks that the above example
should not be considered as a rare exception but one of many similarly
constructed.
By [18] the algebra A can be realized as the rational cohomology of
a closed oriented 6-dimensional differentiable manifold. Such a
manifold does not admit any non trivial S1-action.
We now discuss some implication of the above method to the non
existence of non trivial cyclic group actions. If suffices to consider
groups G = a/ of prime order p.
The following difficulties occur:
a) An analogue of Sullivan's result about realizing a rational Poincare
algebra as the cohomology of a manifold seems completely out of reach
in the S/i-case.
b) Even if one has a simply connected, closed, oriented manifold M with
H (M;2Z/ ) = 0 and such that H*W;Z,L) is g-rigid in an appropriate
p F
sense (note that H* (B TL/ ;7L/) is not just the polynomial ring over
S/ in one variable in case p is an odd prime), M need not be TNHZ
in EG x M with respect to H*(-,S/ ) for all actions of G = TL/ on M.
G p P
There could be actions such that the Leray-Serre spectral sequence
of the fibration M -» EG x M -» BG does not collapse from the E2-term
G

266
on and the action of G on H*(M;E/ ) may be non trivial either.
Hence to prove an analogous result to Theorem 1 above for a given
fixed prime p (G = 7L/ ) by just immitating the proof does not seem to
work. But it is possible to use similar arguments in order to show that
there are simply connected, closed, oriented differentiable manifolds
that do not admit any non trivial E/ -action, for almost all (i.e. all
but finitely many) primes p. This answers a question of P. Loffler and
H. Rauflen (s.[11]) to the negative.
I am grateful to T. Petrie for suggesting this possibility.
We use Example 7 of [9] (s. Remark b) above) to show the following:
Lemma: There exists a connected Poincare algebra A* of fd(A*) = 6 with
A° = 0, defined over 7L , such that A* := A* ® a/ is g-rigid for
almost all primes.
By g-rigid we mean that any embedding (as graded algebras) of an one
parameter family A* S> a/ [t] -» B* % 72,/ [t] (deg(t) = 2) into a trivial
family with cokernel being TL/ [t]-torsion must actually be an
isomorphism.
Proof: The algebra given as Example 7 in [9] can be written as A* ® ¢,
where A* is a connected graded Poincare algebra of fd(A*) = 6 with
A = 0, defined over 7L and free as a ZS-module, To prove that A* is g-
2 —
rigid we show that the part of the Hochschild cohomology H ' (A*;A*)
which classifies the infinitesimal deformations of negative weight (of
commutative algebras) is zero for almost all primes p. An element in
2 -
H ' (A*;A*) is represented by a 2-cycle u : A*®A* -» A*, i.e. a symmetric
cpp r -'pppp
bilinear form of negativ degree (as a map of graded vector spaces) with
6y = 0. Let y: A*®A* -» A* be a symmetric bilinear form such that
y ® id„ , = y . Then 6y = p-£ for some £ £ C3(A*;A*) and 6£ = 0 since
ffi/p p
A* is torsion free,and p[£] = 0 in H3(A*;A*). If p is a prime which does
not occur in the torsion of H3(A*;A*) then [£] must be zero and there
exists an n £ C2(A*,A*) such that 6n = £ and therefore 6(y-pn) = 0, i.e.
y := y-pn £ Z2(A*;A*) and y ® id_ , = y_. We can "symmetrize" y to get
S/p P
y = (\i+\i ) £ Z2(A*,A*), where y (a.,a2) := y(a2,a ), with y ® id^ , =2y .
P
(More conceptually one might use Harrison or Andre-Quillen cohomology
of commutative algebras instead of Hochschild cohomology. But since we
have to exclude a finite number of primes anyway we may as well stick
to odd primes p, making 2 a unit in a/ .) By [9] and [10] [y] = 0 in
2 - p
H ' (A*®(C,A*®(1J) , i.e. there exists a morphism a : A*®dJ -» A*®dJ such that

267
6a = y ® id_ Let %>,v) denote the localization of S at p .
For almost all primes p the morphism a is already defined over TL. , ,
i.e. there is a map a, ,: A* ® E, , -» A* ® TL, . such that 6a, , =
^ (p) (p) (p) (p)
y ® id„ . It follows that 6 (a, , ® id„ . ) = 2y , i.e. [2y ] and hence
ffi(p) (p) ffi/p P p
[y ] = 0 in H2'~ (A*,?.*) for almost all primes. Since H^'~(A*,A*) is fi~
2 -
nitely generated on gets: H ' (A*,A*) = 0 for almost all primes p.
cpp
The argument now proceeds as in the rational case before to give
that A* is g-rigid for almost all primes.
As before we can realize A* ® a) as the rational cohomology of a
closed, oriented, differentiable 6-dimensional manifold M. For almost
all primes p one has H*(M;Z/ ) = H*(M;Z ) ® TL / . It follows from the
isomorphism A*®(C = H*(M;Q) = H*(M;S) ® ¢, that A* ® TL . = H*(M;S)®a. .
and hence A* = H*(M;E) ® 2Z/ for almost all primes p, since the
denominators of the rational coefficient matrix which gives the isomorphism
between A* ® Q and H*(M;S) ® ¢, and the torsion of H*(M;S) involve
only finitely many primes. Let P denote the set of primes, such that A*
is g-rigid, A* = H*(M;B/) = H*(M;Z) ® S/ for p£P and 7L/ must act
trivially on H* (M;TL. . ) (vAiich - for a given M - is the case if p is large enough) .
Assume S/ acts on M for some p £ P.
P F
The localization theorem for the equivariant cohomology given by the
Borel construction works as well if we use coefficients S, . instead of
G (p)
a/ , i.e. the map H*(EG x M; a, ,) -» H* (EG x M ; TL. ,) induced by the in-
P G (P) G (p)
p
elusion M -» M becomes an isomorphism after inverting the multiplication
with the polynomial generator t e H*(BG;Z. .) = TL. At]/p. (t) •
The group TL/ acts trivially on H*(M;B. . ) ; hence the E2-term of the
Leray-Serre spectral sequence of the fibration M -> EG x M -» BG is given
G
by E2 = H*(BG; H*(M;S. .))= H*(BG;E, . ) ® H*(M;S. . ).
Since Hodd(BG;S, ,)=0= Hodd(M;ZZ, ,) the spectral sequence col-
(p) (p)
lapses already for degree reasons (as in theS*-case with rational
coefficients) . One therefore gets a morphism
H*(^;Z5(p)) = H*(BG;Z(p))S H*(M;Z5( }) » H^(MG)G,Z(p)) = H*(BG;Z(p))8i H*(MG;Z(p))
which becomes an isomorphism after localization.
(Note that H*(MG?Z ) can not have p-torsion since Hodd(MG;S/) = 0
s. e.g. [3], VII (2.2))
Tensoring thds morphism with TL/ gives an embedding
H*(M;Z/ ) § TL/ [t] ► H*(MG;Z/ ) ® TL/ [t] such that the cokernel is
TL/.pft]-torsion.
r-~> C
Since A* = H*(M;2Z/ ) is g-rigid it now follows, that M -> M induces

268
an isomorphism H*(M;ZZ/ ) -» H* (M ;ZZ/ ) and hence we get;
p p J
Theorem 2: There exist Simply-connected, orientable, closed 6-dimension-
al differentiable manifolds M such that for any closed orientable
manifold M with H*(M;Q) = M*(M;(C) a non-trivial action of ZZ/ on M is only
possible for at most a finite number of primes p.
References
[1] ATIYAH, M.F. and HIRZEBRUCH, F.: Spin manifolds and group actions,
Essays on Topology and Related Topics (Memoires dedies a G. de
Rham) , 18-28. Berlin-Heidelberg-New York: Springer 1969
[2] BLOOMBERG, E.M.: Manifolds with no periodic homeomorphism.
Trans.Amer.Math.Soc. 202, 67-78 (1975)
[3] BREDON, G.: Introduction to compact transformation groups.
New York-London: Academic Press 1972
[4] BURGHELEA, D.: Free differentiable S1 and S3 actions on homotopy
spheres, Ann.Sci.Ecole Norm.Sup. (4) 5, 183-215 (1972)
[5] COFFEE, J.P.: Filtered and associated graded rings,
Bull.Amer.Math.Soc. 78, 584-587 (1972)
[6] CONNER, P.E. and RAYMOND, F.: Manifolds with few periodic homeomor-
phisms, Proceedings of the Second Conference on Compact
Transformation Groups (Univ. of Massachusetts, Amherst 1971) Part
II. Springer Lecture Notes in Math. 299, 1-75 (1972)
[7] CONNER, P.E., RAYMOND, F. and WEINBERGER, P.: Manifolds with no
periodic maps, Proceedings of the Second Conference on Compact
Transformation Groups (Univ. of Massachusetts, Amherst 1971)
Part II. Springer Lecture Notes in Math. 299, 81-108 (1972)
[8] GERSTENHABER, M.t On the deformation of rings and algebras IV.
Ann. of Math. 99, 257-276 (1974)
[9] IARR0BIN0, A.: Compressed algebras and components of the punctual
Hilbert scheme, Algebraic Geometry, Sitges 1983,
Proceedings. Spinger Lecture Notes in Math. 1124, 146-166 (1985)
[10] IARR0BIN0, A. and EMSALEM, J.: Some zero-dimensional generic
singularities; finite algebras having small tangent space,
Compositio Math. 36, 145-188 (1978)
[11] LOFFLER, P. und RAUSSEN, M.: Symmetrien von M.annigfaltigkeiten und
rationale Homotopietheoiie.
Math.Ann. 271, 549-576 (1985)
[12] PUPPE, v.: Cohomology of fixed point sets and deformation of
algebras, Manuscripta Math. 23, 343-354 (1978)
[13] PUPPE, V.: Deformation of algebras and cohomology of fixed point
sets, Manuscripta Math. 30, 119-136 (1979)
[14] PUPPE, v.: P.A. Smith theory via deformations.
Homotopie algebrique et algebre locale, Luminy, 1982,
Asterisque 113-114, Soc.Math, de France, 278-287 (1984)
[15] QUILLEN, D.: Rational homotopy theory,
Ann. of Math. 90, 205-295 (1969)
[16] SCHULTZ, R.: Group actions on hypertoral manifolds. I.
Topology Symposium Siegen 1979, Proceedings
Springer Lecture Notes in Math. 788, 364-377 (1980)
[17] SCHULTZ, R.: Group actions on hypertoral manifolds.II.
J. Reine Angew. Math. 325, 75-86 (1981)
[18] SULLIVAN, D.: Infinitesimal computations in topology.
Publ. I.H.E.S. 47, 269-331 (1977)
[19] KWASIK,S. and SCHULTZ,R.: Topological circle actions on 4-manifolds.
Preprint (1987)
[20] GERSTENHABER, M. and SCHACK, S.D.: Relative Hochschild cohomology,
rigid algebras, and the Bockstein. J.Pure Appl.Algebra 43,
53-74 (1986)

2x2- MATRICES AND APPLICATION
TO LINK THEORY
by
Pierre VOGEL
In many subjects in topology , particularly in low dimensionnal topology , a
great deal of the difficulty of the theory come from the presence of very big groups
like ; free groups , braid groups , mapping class groups, fundamental groups of
surfaces or 3- dimensionnal manifolds, ... It is very difficult to make direct
computations in such a group G . A possible way to study it is to consider homology
groups Hn(G) . These functors Hn are derived functors of the abelianization functor H,
and the morphism from Z[G] to Z[H,(G)] is the universal representation of the algebra
Z[G] to a commutative algebra . A possible way to construct other invariants is to
consider representations in the algebra of 2x2-matrices with entries in a
commutative ring . This method was already used in some particular cases . In [2]
Culler and Shalen consider representations of the fundamental group of a surface or a
3-dimensionnal manifold in SL2(C) and obtains many interesting results about
3-dimensionnal manifolds . In [1] Casson considers representations of the fundamental
group of a surface in SU2 and constructs an invariant in Z for homology 3-spheres .
In this paper we consider representations from an algebra R in an algebra
M2(A) .where A is a commutative ring . and we construct functors Jt and C satisfying
the following properties : if R is an algebra. C(R) is a commutative ring and <K.(R) is a
C(R)-algebra . Moreover we have a natural representation from R to cH-(R) an this
representation is in some sense the universel representation from R to the algebra of
2x2- matrices with entries in a commutative ring The algebra <K.(R) is not exactly an

270
algebra of 2x2- matrices but we have a trace map t and a determinant map 6 from
cH-(R) to C(R) and if K is a C(R)- algebra which is an algebraicly closed field , cH.(R)®K is
, in almost all cases, isomorphic to M2(K) .
It is well known that a braid with n components acts on the free group Fn on
n letters . But this action on F2 =F(x.y) is not very interesting if the braid is pure and
has only 2 components . On the other hand if we replace the braid by an embedding L
of 2 intervals in I x R which is standard on the boundary , L doesn't act on F2 neither
on the algebra Z[F2] except if L is a braid .
In this paper we will prove that there exists a ring A , algebraic extension of
a polynomial ring of 5 variables . and a morphism from the ring C(Z[F2]) to A , such
that L acts on cK(Z[F2])®a by conjugation by an element on the form u + vxy , where u
and v belongs to A . The pair (u.v) in A is well defined up to a scalar and depends
only on the concordance class of L . This invariant is explicitly computable , as it is
shown in an example, and it is absolutely not trivial .
§1 - Functors <M. and C.
Let R be the algebra of 2x2 matrices with entries in a commutative ring A.
The trace tr and the determinant det are maps from R to A satisfying the following:
i) tr is A-linear and det is A-quadratic,
ii) det is multiplicative,
iii) for every x,y in R:
tr (xy) - tr (x) tr (y) + det (x + y) - det (x) -det (y) = 0
Moreover, for every matrix in R, we have the Cayley-Hamilton formula:
iv) x - tr (x) x + det (x) = 0
On the other hand, we have a map: x •— x from R to R defined by:
x = tr (x) - x
The map: x ->x is an (anti- ) involution of R and satisfies the following:
VxeR x+x = tr (x)
xx = x x = det (x)

271
Definition 1.1 A quasi 2x2 matrix algebra is an algebra R over a
commutative ring A equipped with an involution and maps t and 8 from R to A
satisfying the following:
PO V xe R t(x) = x + F
8(x) = xx = x x
PI t is A-linear and 8 is A-quadratic
P2 8 is multiplicative
P3 Vx.yER t(xy) - t(x) t(y) = 8(x) + 8(y) - 8(x + y)
P4 VxeR x2 - t(x)x + 8(x) = 0
P5 vx.y e R xy + yx - x t(x) - t(x) y - t(xy) + t(x) t(y) = 0
Remark 1.2 Properties P4 and P5 are obvious consequences of property
PO . Moreover , if A is included in R , PI , P2 , P3 are also consequences of PO .
Remark 1,1¾ If A is a commutative ring, then M2(A) is a quasi 2x2 matrix
algebra over R. But a quasi 2x2 matrix algebra over A is generally not isomorphic to
M2(A). For instance, if A is the field 1R and R is the quaternionic skew field H endowed
with the standard involution, it is easy to check that R is a quasi 2x2 matrix algebra
not isomorphic to M2(1R).
Let us denote by s& (resp. s&2) the category of algebras (resp. quasi 2x2
matrix algebras). If R and S are algebras over commutative rings A and B, a morphism
into s& from (R,A) to (S.B) is a couple of compatible homomorphisms from R to S and A
to B. A morphism is a morphism in s&2 if it respects traces, determinants and
involutions.
Theorem 1.4 There exists a functor (cH.,C) from s& to s&2 and a morphism 11
from the identity functor of s& to (cH.,C) satisfying the following:
For each A-algebra R , each morphism p from R to a quasi 2x2 matrix
algebra M over B factorizes uniquely through the C(R)-algebra cH-(R) .

272
Proof Suppose that R is an algebra over a commutative ring A. Denote by A*
the ring A[R R]. If x belongs to R, the two corresponding elements in R Re A[R R] will
be denoted by t(x) and 6(x) respectively.
So, we get maps t and 6 from R to A'. If we force t and 6 to satisfy properties
PI, P2 and P3, we get a quotient A" of A'.
Now we set:
R' = R ® A"
A
We have a A"- linear map from R' to itself defined by:
VxeR x¥T = 1 ® t(x) - x ® 1
Let R" be the quotient of R' by the two- sided ideal generated by the following
elements:
xy-yx .xeR'.yeR*
x x - 8(x) , xeR'
The A"-algebra R" is clearly a quasi 2x2 matrix algebra. Moreover, it is the universal
one. Now we set:
C(R) = A"
cH-(R) = R"
and (<K.,C) is a functor from <s& to s&2 satisfying the desired property.
Let us consider the following example:
R is the group ring Z[F(x,y)], where F(x,y) is the free group generated by x
and y; R is an algebra over Z.
Theorem l.S In this case, we have:
C(R) = Z[ t(x), t(y), t(xy), 6(x), 6(y), 5(xf\ 6(yf! ]
and cH-(R) is a free C(R)- module with basis 1, x, y and xy.
Proof Denote by Cj(R) the subring of C(R) generated by t(x). t(y), t(xy), 8(x),
6(y), 6(x)"' and6(yf\
Claim 1 for every ueF(x.y), t(u) lies in C,(R).

273
The proof is by induction on the length l(u) of the word u in F(x,y).
Suppose that t(u) lies in Cj(R) for every u in F(x,y) of length less that n. and
let u be a word in F(x,y) of length n.
If u contains a power xp, with p*0,l:
by the Cay ley-Hamilton formula (property P4), xp belongs to:
C,(R) 9 x C,(R)
If p is less than -1, vw and vxw have length less than n and t(u) lies in Cj(R) by
induction. If p is - 1, vw has length less than n, so:
t(u) e C,(R) <==* t(vxw) e C,(R)
The same holds if u contains a non trivial power of y. Thus it is enough to consider the
case where u does not contain xpor yp (p*Oou 1). Hence, the word u has the following
form:
u = xyxy...
u = yxyx ...
If n is bigger than 3, u contains (xy)2 or (yx)2 and t(u) belongs to Cj(R). In the other
case, we have:
n«jl =* t(u)eC,(R)
t(xy) = t(yx) eC,(R)
t(xyx) = t(x2y) eC,(R)
t(yxy) = t(y2x) e C,(R)
and the claim is proved.
CMmJ>. For every u in R, t(u) and 5(u) belong toC,(R).
Let u be an element of R . Since t is linear, t(u) lies in Cj(R). Since 6 is
quadratic, 8(u) belongs to Cj(R) for every u in R if and only if:
VueF(x.y) 6(u) e C,(R)
Vu.v e F(x,y) 5(u + v) - 5(u) -5(v) eC,(R)
But that is easy to check because of property P3.
Let cH.j(R) be the subring of cH-(R) generated by R and C/R). An easy

274
consequence of claims 1 and 2 is:
Claim 3 <M,,(R) is a quasi 2x2 matrix algebra over C((R).
By the universal property of <M(R), we have:
C(R) = C,(R)
<M(R) = <M,(R)
On the other hand, it is easy to check that:
C,(R) + x C(R) * y C(R) * xy C(R)
is an algebra. Then:
cH-(R) = C(R) + x C(R) + y C(R) + xy C(R)
Now, consider the representation p from R to M,(C) defined by:
p(x) =
u 0
0 v
P(y) =
b^
c
where u, v, a, b, c are complex numbers, and:
u* 0 , v*0 , ac-b*0
By universal property, we have maps:
p„: <M(R) -♦ M2(C)
px: C(R) -♦ C
and we check:
p (t(x)) = u + v
plc(8(x)) = uv
p.(8(y)) = ac - b
pt(t(y)) = a + c
px(t(xy)) = au + cv
If a,b, c, u, v are chosen to be algebraically independant , p„(t(x)) , p,(t(y)) ,
p,(t(xy)), px(8(x)), px(8(y)) are algebraically independant too and C(R) is the polynomial
ring Z[t(x), t(y), t(xy), 8(x), 8(y), 8(x)"', 8(y)"' ]. Moreover, 1, pJx). pjy), P»(xy) are
linearly independant, so:
<M(R) = C(R) « x C(R) 9 y C(R) 9 xy C(R)

275
§2-Relation with representations.
Definition 2.1 let K be a field and R be a ring. Two representations p and p'
from R to M2(K) are called almost conjugate if either p and p' are conjugate or p (resp
p') is extension of 1-dimensional representations a and p (resp a' and p') and:
a = a' and p = p'
or: a = p' and p = a'
The set of representations from R to M2(K) modulo almost conjugation will be denoted
by R2(R).
Proposition 2 -2 Let K be a field. Two almost conjugate representations from
a ring R to M2(K) induce the same morphism from C(R) to K.
Proof Let p and p' be almost conjugate representations from R to M2(K). If p
and p' are conjugate, we have a commutative diagram:
M2(K)
M2(K)
and by the universal property, we have diagrams:
M2(K)
cH-(K)
C(R)
M2(K)
Then morphisms p and p' from C(R) to K are the same.
If p and p' are not conjugate, p and p' are conjugate to representations p(
and p', from R to the subring M'2(K) of upper triangular matrices in M2(K). Moreover,
the diagonal evaluation gives a map from M'2(K) to K2 and we get a commutative

276
diagram:
M'2(K)
On the other hand, M'2(K) and K are quasi 2x2 matrix algebras over K and
f. f and the inclusion M'2(K) in M2(K) are morphisms in s&2 . Then, if we apply the
functor C, we get : p,. = pw = p1^ - p'^
Theorem 2-3 let R be a ring and K be a field. Let f be a morphism from
C(R) to K. Then there exists an extension L of K and a representation from R to M2(L)
inducing f. Moreover. L can be chosen to be K or a quadratic extension of K or, if K has
characteristic 2, a subfield of /K
Theorem 2-4 Let R be a ring and K a field. Let p and p' be representations
from R to M2(K). Then, p and p' are quasi conjugate if and only if p and p' induce the
same morphism from C(R) to K.
Proof If f is a character from C(R) to K, let us denote by R2(R,f) the almost
conjugacy classes of representations from R to M2(K) inducing f from C(R) to K.
Let a be the following K-algebra :
A - cH-(R) ® K
C(R)
where the C(R)-algebra structure of K is given by f.
The algebra A is a quasi 2x2 matrix algebra over K. by the universal
property of cH-(R) we have:
R2(R,f) = R2(A,Id)
Then, if we want to prove theorem 2-3. it is enough to show that R2(a®L , Id) is not
empty for some algebraic extension L of K. Theorem 2-4 is equivalent to the fact that
R2(a, Id) has at most one element.

277
Case 1 Suppose that the characteristic of K is different from 2 and:
Vxe A t(x)2 = 4 6(x)
In this case, denote by f the map (1 /2)t from A to K. Since t is linear, f is linear too.
On the other hand, f is equal to 6 and f is multiplicative. Then, for every
x,y e K there exists e=±l such that:
f(xy) = cf(x)f(y)
Since f(xy) and f(x) f(y) are bilinear, it is easy to see that c doesn't depend
on x and y, and f is multiplicative. The morphism:
(Hx) 0 N
[o f(x),
belongs to R2(a, Id) and theorem 2-3 is proved in this case (with L = K)
Let p be an element of R2(a, Id). Since t(x) is equal to 4 5(x) for every x
in A , p(x) is either the scalar matrix f(x) or this matrix plus some nilpotent matrix ,
Then , if p is not the scalar representation f there exists some element x0 in A such
that p(x0) is the matrix
p(x ) =
o
in some basis in K .
'f(x )
o
0
f(x )
0 J
Let xe A and
{c d
a + d = t(x) = 2 f(x)
be the matrix p(x). We have :
f(x0) a + c + f(x0) d = t(x0 x) = 2 f(x0 x) = 2 f(x0) f(x)
Then c is zero, and p is the following morphism:
(f ^
P =
0
for some map g from A to K, and p is almost conjugate to:
ft 0^
Therefore, theorem 2-4 is proved in this case.
Case 2 Suppose that K is of characteristic 2 and the trace t is null on A.
In this case, denote by f the map fb from A to /K~and by L the image of f.
It is easy to see that f is an algebraic homomorphim and the scalar representation f
is an element of R2(a®L . Id) . Hence . theorem 2-3 is proved in this case .

278
Let p be an element of R2(a, Id). If p(x) is a scalar matrix , for every x
in A , L is equal to K and p is the scalar representation f .
If L is equal to K and p(x0) is not a scalar matrix for some xqe a, we can show , as in
the first case , that there exists a map g from A to K such that p is conjugate to the
representation
0 f
and then p is almost conjugate to
f 0'
0 f
Suppose that L is different from K. Let xqe a such that f(x0) is not in K.
Then p(x0) is not a scalar matrix and , as above , p®L is conjugate to a representation p':
ft %)
"> 'J
such that g(x0) is not zero.
If x is an element of a, p'(x) is a linear combination of 1 and p'(x0), and
there exist a.bsK such that:
p(x) = a + b p(x0)
=» f(x) = a + b f(x0)
Therefore L is the extension K[f(x0)] of K and there exist unique functions a and p from
L to K such that:
VueL u = a(u) * p(u) f(x0)
and we have:
Vxea p(x) = a(f(x)) * p(f(x))p(x0)
The conjugacy class of p is the conjugacy class of p(x0) which is the conjugacy class of:
f 0 O
6(x ) 0
v o
SoR2(a, Id) has at most 1 element.
Case 3 We suppose that we are not in cases 1 or 2 and that:
Vx.ys a t(xy)2 - t(xy) t(x) t(y) * 6(x) t(y)2 * 5(y) t(x)2 - 4 6(x) 5(y) = 0
Since we are not in cases 1 or 2, there exists x0e a such that:
t(x0)2 - 4 6(x0) * 0
Let X and n be two elements of a quadratic extension L of K such that:
X * \l = t(x0) X\i = 5(x0)
Since t(x0)2 » 4 6(x0) , I is different from \l. Then, for every xea, the following
equations :

279
a + b = t(x)
X a + \i b = t(x0x)
have a unique solution. Define maps f and g from A to L by:
f(x) = a
g(x) = b
Since we have:
t(x0 x)2 - t(x0 x) t(x0) t(x) * 6(x0) t(x)2 * 6(x) [t(x0)2 - 4 6(x0)] = 0
it is not difficult to compute 6(x). After computation, we get:
6(x) = f(x)g(x)
Clearly, f and g are K-linear.
Let x and y be two elements in A. We set:
a = f(x) b = g(x)
If a* b there exist unique elements a and B in L such that:
a + B = t(y) and a a + b B = t(xy)
and we have as above: 6(y) = a B
Consequently, we have:
a = f(y) and B = g(y)
or: a = g(y) and B = f(y)
Suppose that: a = g(y) and B = f(y)
Let u e L such that: uX+a*un+b
Then t(u x0 + x)2 is different from 4 6(u x0 + u) and , as above, there exist unique
elements a' and B' in L such that:
a' + B' = t(y)
(u X * a) a' + (u ^i + b) B' = t( (ux0 + x) y )
a' B' = 6(y)
And we have:
t( (ux0 + x) y ) = (u X + a) a + (u ^i + b) B
or: t( (ux0 + x) y ) = (u X + a) B + (u ^1 + b) a
In other words:
u( XB + ^ia ) + a a + b B = u( Xa + jxp ) * a a * b B
or: u( XB + ^ia ) + a a + b B = u( XB + ^ia ) + a B + b a

280
i.e. u(a-p) tt-^i) = 0 or (a-p)(a-b)*0
and this is impossible,
hence, if f(x) * g(x) , we have:
t(xy) = f(xy) + g(xy) - f(x) f(y) + g(x) g(y)
Of course, the same holds if f(y) * g(y).
If f(x) = g(x) and f(y) = g(y) , we have:
t(xy)2 - 4 t(xy) f(x) f(y) * 4 f(x)2f(y)2 =
0
=> [t(xy) - 2 f(x)f(y)] = 0
=> t(xy) = 2 f(x) f(y) = f(x) f(y) + g(x) g(y)
Therefore for every x and y in A , we have :
f(xy) + g(xy) = f(x)f(y) * g(x)g(y)
and: f(xy) g(xy) = f(x) g(x) f(y) g(y)
Hence we have two possibilities :
f(xy) = f(x)f(y) and g(xy) = g(x)g(y)
or : f(xy) = g(x) g(y) and g(xy) = f(x) f(y)
Suppose that f(x) and g(x) are different . Let ye a . If f(xy) is equal to g(x)
g(y) , we have :
f(x(y*l)) = g(x)g(y) + f(x)
and: f(x(y+1)) = f(x) (f(y)+1) or f(x(y 1)) = g(x) (g(y)+1)
Since f(x) and g(x) are different , we get:
g(x)g(y) = f(x)f(y)
The same holds if f(y) * g(y) and then in any case .
Finally we have :
V x ,y e a f(xy) = f(x) f(y)
g(xy) = g(x) g(y)
and
*)
is a representation in R,(a®L , Id)
Now suppose that p is a representation in Rz(a®L , Id) . The representation
p®L is conjugate to a representation p' such that :
(X 0^
p'(x0)
0
J
Suppose that x and y are two elements in A and that:
'» *^ (a p">
V 6;
p'(x) =
a b
c d
v
P'(y) =
■v

281
We have:
a + d = t(x) = f(x) + g(x)
Xa + (id = t(x0x) = f(x0x) + g(x0x)
=* a = f(x) d = g(x)
and this implies:
8(x) = ad - be = ad =* be = 0
Then p'x) and p'(y) are triangular.
Suppose that: c = 0 b » 0 p = 0 Y * 0
then we have:
f(xy) = a a + b Y = f(x) f(y) = a a
which is impossible. Hence, p' has the following form:
or
10 g
for some map p from Ato L.
1,9 8
if we change f and g. we may as well suppose that:
ft y\
P'= 0 g
Suppose that p is zero. Then, for every xe A. p'(x) is a linear combination of 1 and p'(x0)
and there exist two functions a and P from A to K such that:
p(x) = a(x) + p(x0) p(x)
Moreover, we have:
t(x) = 2 a(x) + p(x) t(x0)
t(x0x) = a(x) t(x0) + p(x) t(x*)
and a(x) and p(x) depend only on t(x), t(x0), 6(x), 8(x0) and t(x0 x).
Since p(x0) is conjugate to:
ft(x ) -\\
o
5(x ) 0
V o
p is conjugate to:
a
0
1
't(x )
0
5(x )
0
-n
0
If p is non zero, there exists x,e a such that p(x,) » 0. We have:
p'(x ) =
o
V-)
p'(x ) =
a u
0
PJ

282
Since p(x0) is conjugate to:
't(x )
0
8(x )
v. 0
there exists a matrix:
f3 b)
[c dj
in GL2(L) such that:
-n
0
}
a bN
c d.
0>
t(x0) - 1
5(i0)
b^
d
'a b^
c d
' a u^
'a b^
and:
After computation we get:
a * 0 b * 0
aX - Pp. au^i
e K
X - jx b(X- (i)
2
a-p auu.
ty : — r £ K
E M (K)
2
X^T
Xp - an
ba-n)
au|i
e K
e K
x-n b(x-(i) x-n b(x-(i)
Suppose that L is different from K. Then Lisa quadratic extension of K and we have a
Galois action on L:
|T = X a" = p
So we get:
au^i
b(X-(i)
and then L is equal to K.
e K and
au
b(X-n)
E K =* n E K
So L is equal to K and p (= p') is quasi conjugate to:
and theorems 2-3 and 2 - 4 are proved in this case.
Case 4 We suppose that we are not in case 1 or 2 or 3.
For any x and y in a quasi 2x2 matrix algebra, set:
A(x,y) = t(xy)2- t(xy)t(x)t(y) + t(x)2S(y) + t(y)26(x) - 4 6(x)6(y)

283
In this case there exist x and y in a such that A(x,y) is not zero. Let A, be the
subalgebra of A generated by x and y. Clearly A, is generated as a K-vector space by
1, x, y, xy. Suppose we have a relation:
a + bx + cy + d xy = 0 a, b, c, d e K
Then we get:
2a + b t(x) + c t(y) + d t(xy) = 0
a t(x) + b t(x2 ) + c t(xy) + d t(x2y) = 0
a t(y) + b t(xy) + c t(y2 ) + d t(xy2 ) = 0
a t(xy) + b t(x2y) + c t(xy2 ) + d t(x2y2 ) = 0
It is not difficult to check the following:
t(x2) = t(x)2- 2 6(x)
t(y2) = t(y)2- 2 6(y)
t(x2y) = t(x)t(xy) - 6(x)t(y)
t(xy2) = t(y)t(xy) - 6(y)t(x)
t(x2y2 ) = t(xy) t(x) t(y) - t(x2 ) 6(y) - t(y2 ) 6(x) * 2 6(x) 6(y)
and the determinant of this system is A(x,y) which is not zero. Therefore.
( 1, x, y, xy ) is a basis of a, .
Let a + bx + cy + dxy be an element of the center of a,. We have:
x ( a + bx + cy + dxy ) = ( a + bx + cy + dxy ) x
=* (c+dx) (xy - yx) = 0
But we have the following formula:
8(xy - yx) = (xy - yx) (xy-yx~) = 5(xy) + 6(yx) + t(xy2x) - t(xy) t(yx)
= 2 6(x) 6(y) + t(x2y2 ) - t(xy)2 = - A(x.y)
Then we get:
(c+dx) A(x,y) =0 =* c = d = 0
and a + bx which commutes with y is a multiple of 1. Therefore, the center of A, is K.
On the other hand, it is not difficult to see there is no character from A, to
K. Then A, is simple and A( ® L is isomorphic to M2(L) for some quadratic extension L
of K. Consequently there exist elements e-in A,®L,i=l,2 j=l,2 such that:
eij erj> = ° if i » i'
= eH, if j = i'

284
Let us define the following maps f„ from A to L:
Vi.j Vxea f..(x) = t(x e1()
Claim. For every x,y in a and every i,j in (1,2} we have:
f„(xy) = f„(x) f„(y) * fi2(x) f2j(y)
Proof of the claim: For every u,v in A we have:
8(u en + v eM) = 6(u e()) + 8(v e,,) + t(u e„) t(v e)f) - t(u e,, v e,,)
but 6 is multiplicative and 6(e()) is zero. Thus We have:
t(u eM) t(v e)f) = t(u eM v e,,)
and this implies:
t(xy en) = t( x (e„ + e22) y en) = t(x eM y e(|) + t(x e22 y e,,
= t(x e„) t(y eM) + t( X e?l e„ e,2 y efl)
= t(x ef|) t(y eM) + t(x e2| e,,) t(e,2 y en)
= t(x en) t(y en) + t(x e21) t(y eJ2)
So we have
fjj(xy) = t(xy e..) = t(e(. x y e(1 e,,)
= t(eH x eM) t(y ejf en) + t(eH x e21) t(y e^ e,2)
= fi,(x)f„(y) + f12(x)f2,(y)
and the claim is proved.
As a consequence of the claim, we have a morphism f from A to M (L):
((
f =
11
12
21 22,
and it is not difficult to see that f is a morphism in the category <s&2 (i.e. it preserves
trace and determinant).
Now let u and v be two elements of A such that f(u) = 0. We have:
t(u) = t(f(u)) = 0
t(uv) = t(f(uv)) = 0
=> u + u = 0
uv + vu = 0
and this implies: uv = v u
Let a,b in A. We have:

285
u (ab-ba) = aub-bau=a bua-bau = 0
In particular:
u(e12 e21 " e2. e«) = u (en - e22> = °
But en-e22 is invertible . Then u is trivial and f induces a monomorphism from A to
M2(L).
That proves theorem 2-3.
If we have a representation from a to M2(K), a is not a skew field and a is
isomorphic to M2(K). Hence two representations from A to M2(K) are conjugate and
theorem 2-4 is proved.
S3- An invariant for links.
A link of n intervals is an embedding of Ix (1, 2, ... , n) to Ix 1R which is
standard on the boundary.
2
Two links are concordant if there is an embedding F from I x(l n)
toIxIxR standard on dlx Ix(l n) and inducing f( on Ix(i}x(l n) fori =0,1.
The set of concordance of links of n intervals is a set Cn , which is actually a
group for the juxtaposition law [3],
Let L be a link of n intervals. Denote by X the complement of L and by XQ
and Xjthe top part and the bottom part of dX. Let x. (resp x'j) be the element of Jt,(X0)
(resp JtjtX,)) which turns around the i component of L in XQ (resp X,). The
fundamental group Jt,(X0) is a free group with basis x, xn. The same holds for
n(X,). But Jtj(X) is generally not free. We only know the following [31:
There exists a universal group Gn depending only on n and a morphism e
from nt(X0) to Gn such that for any link L, e extends uniquely on Jt^X). Moreover, there
exists a unique automorphism \ depending on the concordance class of a link L on Gn
such that:
Vi= 1 n xL(Xj) = x'j

286
This automorphism satisfies the following:
for every i, \(.x.) is conjugate to xt and \(^^2 ... x„) = x,x2 ... xn
In fact. Gn is the algebraic closure of ^(Xq) in the sense of Levine [4],
The problem is that Gn is completely unknown and it is therefore difficult
to give a description of some automorphism of G .
From now on, we will suppose that L is a link with 2 components. We set:
x, = x x2 = y
Then Jtf(X0) is the free group F(x.y).
Notation 3-2 We set the following in the ring C(Z[F(x.y)]):
a = t(x) b = t(y) c = t(xy)
a = 6(x) p = 5(y)
A denotes the ring C(Z[F(x,y)]) = Z[a. b. c. a, a"', P, p~'] and cH. is the algebra
<M,(Z[F(x,y)]). A is the element of A defined by:
A = c - abc + a p + b a - 4ap
S is the multiplicative subset of A which consists of polynomials
P(a2aH , b2p"')of Z[aV ,b2p"']c A such that P(4,4) = 1.
A is the completion of S A with respect to the ideal generated by A;
A = li m S A n
/ A
Ais the subring of A which consists of all elements of A algebraic over A.
Theorem %-^ let L be a link of 2 intervals. Then the morphism from
ZfotjtXg)] to cH. extends uniquely to a morphism from ZtJt^X)] to cH.®A. Moreover
there exists a unique automorphism pL from <K.®A to itself such that:
pL(x) = x' pL(y) = y'
Furthermore there exist elements u,v in A. unique up to multiplication by a scalar in A
such that:
VzecH. pL(z) = (u + vxy) z (u + vxy)"
The automorphism pL depends only on the concordance class of L and the
correspondance L -*pL is a representation of the group C2 to Aut(<K.®A).

287
Remark 3-4 In fact the morphism from Z[nt(X0)] to cH. extends uniquely
to a morphism from Z[G2] to cH.®a, and we have a canonical representation from G2 to
(.m^a)?
The proof of theorem 3-3 is quite long and will be divided in several
lemmas.
Lemma 3-"> Let (K, K0) be a pair of finite complexes. We suppose that K0 is
homotopy equivalent to a bouquet of two cercles and that K/ is contractible. Let x
Ko
and y be the generators of ^(Kg). Let c be the augmentation map:
e: Z[x, x~ , y, y" ] —» Z e(x) = e(y) = 1
Then we have:
VU2 H.Ot/K), Jt/K,,); r/'df' Z[x, x"', y, y"'] ) = 0
Proof: Let B be the ring: B = eH(l)"' Z[x, x"', y, y"']
We have an augmentation map: B -»Z
Since B is noetherian, and (K, K0) is finite, Ht(K, K0 ; B) is finitely generated.
Let H (K, K0 ; B) be the first non trivial homology group of (K, KQ) , if it exists . Since
HJK, K0 : Z) vanishes, Hp(K, K0 ; B) is killed by some element of B going to 1 in Z
Therefore Hp(K, K : B) vanishes too and (K, K0) is B- acyclic. But, for U2
H,(n,(K), n,(K0); B) is a quotient of Hj(K, K0 : B). That proves the lemma.
Lemma 3-6 Let X be the complement of a link of 2 intervals. Let M be a
Z[x, x , y, y" ]- module such that every element of e (1) acts bijectively on M. Then:
Vi^2 h'Oi/X), n,(X0); M) = H!(G2 , F(x,y); M) = 0
Pfpof: The module M is a module over the ring:
B = r/'of' Ztx.x"1, y,y"']
Then, by the universal coefficient spectral sequence, it is enough to prove:
Vi^2 H.( n,(X). n,(X0); B) = Hj(G2 , P(x,y): M) = 0

288
The first part of that is proved in lemma 3-5.
There exists a sequence of finite complexes [31:
X0 c K, c K2 c ...
such that:
Vi KV is contractible
Xo
G2 = n,(u Kn)
So we have: Vi H.(u Kn , XQ ; B) = lim H,(KB , XQ ; B) = 0
and the lemma can be easily deduced.
Lemma 3-7 Let n^ 1 be an integer, and Tn be the group of units of the
algebra S <K./,n , where I is the two- sided ideal of S <K. generated by xy-yx. Let X
be the complement of a link of two intervals. Then the morphism from F(x,y) to r
factorizes uniquely trough G2 and n((X) .
Proof: This lemma will be proved by induction on n.
For n= 1 , T, is commutative and the lemma is obvious since:
H1(F(x,y); Z) = H,(n,(X); Z) = H,(G2 ; Z)
On the other hand, we have an exact sequence (for n^ 1):
Let G be the group nf(X) or G2 . By induction we have a commutative diagram:
i-i* iB/lB-. -rB„ - rB-i
T t
F(x.y) -> G
and we want to prove that there exists a unique morphism from G to rBO which
makes the diagram commute:
r -» r
n*l n
r t
F(x.y) -> G
Since the multiplicative group 1 + In/ n.i iscommutative.it is enough to prove that
1 + In/ „»i is by the conjugation action a B-module.

289
Let 0) be the element xy-yx of S Jt. It is easy to check the following:
X 0) = 0) X X 0) = 0) X
yo)=o)y yci)=G)y
Then I is generated by 0) and I / n.i is additively isomorphic to
S'lM/ « S"'Z[x,x"',y,y1 x,x'', y.y "*]
0)
On the other hand we have:
,, n , -i , n , -I
X(1+0)U)X = 1+0) xux
y(l+o)u)x = 1+0) yuy
where x' and y' are x and y if n is even, and x and y if n is odd.
Then 1 + In/ n.i is isomorphic to S~Z[x, x~, y , y", x , x ", y.y7"] and F(x,y) and G
acts on it (via H,(F(x,y)) = H^G) ) trivially if n is even and in the following way if n is
odd:
x(u) = x x u y(u) = y y" u
If n is even, 1 + I / n*i is a B - module.
Suppose now that n is odd. Let P(x,y) e Z[x, x ,y,y ] such that P( 1,1) is
1 . Then P(x,y) acts on S*'Z[x, x*1, y, y"! x~,x~", y",y"] by multiplication by
P( x x . y y ). It is not difficult to prove the following:
_ -i2 a - 2a_ .,
(x x ) = xx - 1
a
_ -i,2 b'- 2P- -i
(y y ) = —-— y y - i
Therefore P has the following form:
P=U + Vxx" +Wyy" +T xx'yy"
U. V.W.T e Z[aV,b2p"']
and: U(4,4) + V(4,4) + W(4,4) + T(4,4) = 1
Set: P' = U + V x"x_1 + Wyy"'t T xV' y y "'
We have:
P P' =(U*Vix"' )2+(W + T xx'1 )2+(U+V xx'1 KW + T xx'1 )(yy"'( 7y"') Thus we
have: P P' = U' + V'Tx~' U-, V e Z[a2a"', b2p"']
U'(4,4) + V'(4,4) = 1
In the same way, PP'(U'+V x x~ "') is a polynomial U" in Z[a a . b p ] such that:

290
U"(4,4) = 1
Therefore. PP"(U'+V x x~"') belongs to S and P is invertible in
S~Z[x, x", y ,y1 x~, T"'. y\ y7"'] Thus 1 + i"/ n»i is a B- module.
Lemma 3-8 Let X be the complement of a link of two intervals. Then the
morphism from Z[F(x,y)] to cH.®a factorizes uniquely through Z[G2] and Z[jt,(X)].
Proof: We have:
o) = (xy-yx)(xy-yx) = - 6(xy-yx)
= - 8(xy) - 6(yx) - t(xyyx) + t(xy) t(yx) = A
Therefore we have:
lim (S"'cH./ „ )= lim ( S"'cH./ „ ) = cH.®A
and the map from Z[F(x,y)] to cH.® A. factorizes uniquely through Z[G2] and Z[ji,(X)].
Let u be an element of G2 or n,(X). If u lies in G2 , u is contained in a finitely
generated subgroup G of G2 such that F(x,y) c G is normally surjective. If u is in Jt,(X),
set G = Jt,(X).
In all cases F(x.y) -»G is normally surjective. and G is generated by x,y and
elements z, zn in [G.G], and we have:
Vi=l,... ,n Zj e [F(x.y). G]
So there exist words Wjte,, ... . zn) in the subgroup [F(x.y).F(x.y.z zn)] of the free
group F(x.y.z zn) and we have:
Vi= 1, ... ,n Zj = W.(z, zn)
But we have a canonical map from G to <H.® A. Then z zn can be considered as
elements in Jt® A, and W. is a word in Jt® A which involves z z , z , , ... z . We
can replace z j by z. 6(Zj)" , and by multiplying the relation above by a product of
6(z.), we get:
T-r _ °i| _ —
Vi= 1 n z 11 (z z ) = W' (z z , z ,... z )
i I j j i 1 n 1 n
But Z; is congruent to 1 mod u = xy-yx :
Zj = 1 + CO u,
So we get equations:
(E.) <Poi(u) + 0) <t>n(u) + ... + 0)" <Pqi(u) = 0

291
where <Ppjis a polynomial function of degree p depending on u = (u un) with values
in <K.® A and coefficients in A.
On the other hand, there is a unique morphism from the group of
presentation < x , y , z zn ; Zj = W^Zj zn) > to cH.®A which is standard on
F(x,y). Then equations (E,) have a unique solution in (<K.® A)" and this solution has
algebraic coordinates (over A). Therefore z. belongs to <K.®A and the image of G in
<K.® A is included in <K.®a. That proves the lemma.
We are now able to prove the first part of theorem 3-3. Let us consider the
following diagram:
Z[F(x,y)]
Z[n,(X)] -> cH.®a
Z[F(x\y')]
The composition map <p: Z[F(x',y')] -» cH.®a goes to a quasi 2x2 matrix algebra. Then
p induces morphisms p , p' :
p: cH.(Z[F(x',y')]) ->cH.®a
¢- : C(Z[F(x',y')]) - a
On the other hand, we have:
p(x') is conjugate to x in cH.®a
p(y') is conjugate to y in cH.®a
p(x'y') = xy
Therefore we have:
q>' (t(x')) = t((p(x')) = t(x)
$' (t(y')) = t(ip(y')) = t(y)
9'(6(x')) = 8dp(x')) = 5(x)
iji'(8(y')) = 8((p(y')) = 8(y)
¢- (t(x'y')) = tdp(xV')) = 8(xy)
If we identify x' and x. y' and y, p is a map from cH. to cH.®a inducing the inclusion
A c A in the coefficient ring. Thus p extends to an endomorphism <pL of the a-algebra
cH.®A.
Suppose that L and L' are two links of 2 intervals such that L and L' are
concordant. Let X and X' be the complements of L and L' in IxR and Y be the

292
complement of the cobordism in Ix IxR . We have a commutative diagram:
Z[F(x'.y')]
Zfr/Y)] ^cH.®a
Therefore links L and L' induce the same morphism from Z[F(x',y')] to cH.®a and pL
and pL, are the same. Hence, pL depends only on the concordance class of L.
Suppose that L and L' are two links of 2 intervals. Let L" be the
juxtaposition of L and L'. Let X. X', X" be the complements of L, L'. L" in I x IR . We have
the diagram:
Z[F(x,y)]
Z[F(x,y)]
cH-®A
Z[F(x,y)]
Then we have:
pLo(poaof) = pag
pL, o (p oa" o f) =p a' g'
pL„ o (p oa o f) = P a' g'
and this implies:
pL„ = pL,opL
Thus pL is an automorphism of <H.®Aand p is a representation of the group C2 to
Aut(cH.®A)
Now, the last thing to do is to prove that <pL is the conjugation by some
element in A®Axy and that will be a consequence of the following lemma:
jJJ, Let p be an automorphism of the algebra cH.®a which keeps
xy fixed. Then there exists an element c in A®Axy, unique up to a scalar, such that p is
the conjugation by c.
Proof: Let us denote by a' the following algebra:

293
A' = AW/, 2 ,
X - aX +a
We have a Galois action of this extension:
X ~ X = a-X
Let K and K' be the quotient fields of a and a' . Let |i and 6 be the elements of K'
defined by:
(i + ]T= b
X\l+ X (1= c
|i ]T- 8 - p
Actually 6 lies in K. Thus we have a representation p from cH. to M2(K'):
P(x) =
X
P(y) =
Let CI be the matrix
(0 9^
1 0
in M2(K) c M2(K')
It is easy to see the following: VzeA n p(z) = p(z) n
where denotes the Galois action on K' extended to M2(K'). And then, <K.®K is
isomorphic to the subring R of matrices A e M2(K') such that:
n A = A n
The automorphism p induces an automorphism p0 on R and on R®K' <•
M2(K') and this automorphism keeps the center fixed. Therefore there exists a matrix
c0 in GL2(K") such that:
VAeR f0(A)-c0AE0"
That means:
v ae m2(K') n a = a n
n Eo A Eo"' = Eo A Eo"' n
«=> v a e M2(K') n a = a n =* n e0 a e0"' = e0 n a n"' c0H n
=*n"'F0"' nc0A = AD''^ n c0
Then, for any A in R, A commutes with n"' £^1 n c0 . But R®K' is isomorphic to
M2(K'). Hence n"' ~z~0~x n c0 is central and there exists k e K' such that:
n c0 = k Y0 n
On the other hand, <p0(xy) is equal to xy and e0 commutes with xy. Then there exist u,v
e K' such that: £0 = u ♦ v p(xy) and this implies:
n c = n (u ♦ v p(xy)) = (u + v p(xy)) n

294
= (k ~u" + k v p(xy)) n
=*u = ku v = kv
Since e0 is invertible, u (or v) is not zero. But e0 is defined up to multiplication by a
scalar. Therefore we may as well suppose that u (or v) is equal to 1 and u and v belong
to K. After multiplication by some element in A we will get:
c0 = u + v p(xy) u,v e A
Let us set: E=u+vxys <K.®a
We have the following:
Vze cH.®a <p(z) e = e z
and theorem 3-3 is proved.
Remark 3-10 It is not clear that u + v x y can be chosen to be a unit in
cH.®a. But we have the following:
Proposition 3-11 Let L be a link of 2 intervals. Then the automorphism pL
is the conjugation by an element u + v x y e cH.®a such that u + v goes to 1 by the
augmentation map from a to Z sending a, b, c to 2 and a, p to 1.
EraoiiThe automorphism pL is the conjugation by an element
E=u + vxys A®Axy
Since F8(e)" is the inverse of e in cH.®K , where K is the fraction field of A, we have
the following:
e y e - 0 mod 6(e) F x e = 0 mod 8(e)
but we have:
e y F= e t(y F) - e2 y7 = t(Ey)c- Ue) cy + 8(e) y
F x~e = e~ t( x~e) - F2 x = t( x~e) F- t(E) Fx + 8(e) x
This implies:
t(E y*) (u + vxy) - t(E) (uy* pvx) ■ 0 mod 8(e)
t( x~e) (u + vxy) - t(E) (ux + av V ) = 0 mod 8(e)
Since (1, x, y7. xy) and (1, x. y~. xy7) are a-basis of cH.®a, we get:
u t(E) - 0 mod 8(e) v t(E) - 0 mod 8(e)
ut(Ey). 0 mod 8(e) v t(e y ) ■ 0 mod 8(e)

295
u t( Te) = 0 mod 6(e) v t( x~e) = 0 mod 6(e)
For x( xp in a, denote by < x( x > the ideal generated by x( x We have:
<u.v>.<t(x), t(ey ), t(i~e)> c <6(e)>
=* < u,v> .< 2u + cv, bu + apv, au + bav> c <6(e)>
It is easy to check the following:
< 2u + cv, bu + apv, au + bav> d < b2a-a2p, 2ap-bc, 2ba-ac> . < u,v>
Then we have:
< u,v> 2 < bVa2p, 2ap-bc, 2ba-ac> c < 6(e)>
let w be an element of < u,v > . There exist X and Y in a such that:
w(b2a -a2p) = X 6(e) w(2ap - be) = Y 6(e)
Therefore X(2ap - be) is divisible by b a - a p .
Let B be the subring of A defined by:
B= S"fZ[a, b, a. cT'.p.p"1] [[A]] (see notation 3-2)
We have: A = B e cB . Then there exist X0 , X, e B such that: X = X0 + X,c
and we deduce:
2apX0 - b X, (A - a2p - b2a + 4ap) = 0 mod b2a - a2p
-b X0 + 2ap X, - ab2 X, = 0 mod b2a - a^
=» X, b2A = 0 mod b2a - a2p
Then X( is divisible by b a - a p (in B) and X0 also . Therefore X is divisible by
b a - a p in A and then in A. This implies that w itself is divisible by 6(e), and we
have:
<u,v>2 c <8(e)>
Thus there exist three elements r, s, t e a such that:
2 2 2
u = r 6(e) = r (u + cuv + ap v )
uv = s 5(e) v = t 6(e)
It is easy to check that:
s = rt r + cs + ap t = 1
Moreover, r, s, t depend only on the homothety class of e and on the automorphism <p.
Let us denote by r0 , s0 , t the images of r, s, t by the augmentation
morphism from A to Z which sends a. p to 1 and a, b. c to 2. We have:
so2 = so lo
ro + 2 so + ^ = 1

296
Therefore it is easy to see that there exists a unique integer 6 satisfying the following:
r0 = (i-er s0 = e(i -e)
Now, if we consider another automorphism p', we get another integer 6' and it is easy
to check that 6 + 6' is the integer corresponding to p p' . On the other hand, if p is the
conjugation by xy, we have u = 0, v= 1 and the corresponding integer is 6= 1
Therefore, there exists an integer n such that the corresponding integer of c(xy) is
zero. Denote by e'= u' + v' xy this new element of <H.®a and by r', s\ t" the
corresponding elements in a constructed as above. We have:
r'o ' 1 s o = °
and r' + s' xy goes to a unit in <K.®Z . Hence e is a multiple of:
(r' + s' xy)(xy)"n
which is invertible in cH.®Z
§4-An example.
Consider the link L given by the following picture:
The link is oriented from the top to the bottom and x, y, x', y", z are
elements of the fundamental group n of the complement of L corresponding to paths
turning around parts "over" of L (see the picture)
Because L has three crossings, we have the following relations:
12=11¾ xy"=y'z yz=zy'
and we deduce:
x y' = y' x x' x
Thanks to theorem 3-4, there exists an element e in A®Axy such that:
x" = c x c y" = e y e in cH.®a

297
Therefore we have in <K.®a:
-i -i -i -i
xeye = e y e x exe x
If we multiply on the left by e 6(c) , we get:
cxcyc=ycx e x e x
and this implies:
e uyc7 = y F x e x F
Let f be the antiinvolution of <K.®A sending
x to x and y to y (and xy to yx) and let e' = f(c). We have:
e x = n' y e = e' y
e x = x e' y e =c'y
Then we have:
cxcyc x =n'cyE x = xc'cc'yx
yc xexe = y e n t' e = n y e e' c
and we get:
e' e e' y x = x y e e' e = f ( e' e e' y e )
We have: e'e?= t( e e' ) e' - e' e' e = t( t F ) 7 - 5(e') e~
let us set:
U = t( e F) V = 8(e') e = u + v xy
So we have:
E'EE'yx = U c' y i - V c y i
= U(uyx + vx yyF)-V(uyx + vy xyx)
= U(u y x" + p v x"2 )-V(u y x~ + t(y x")v y x -vy x x y)
= U (u y x +p v x"2 )-V(u y x~ + t(y x")v y x-vay )
and the equation:
e" e e' y x = f(E" e e' y x)
gives rise to the following:
U u(y T- x"y) - V u(y x-xy)-Vvt(yx)(y x-x y)=0
=*(Uu-Vu + Vv t(y x"))(y x~- x"y) = 0
and we get:
U u -V u + V v(ab -c) = 0
On the other hand, we have:
U = t(cF) = t((u + v xy)(u + v x y ))

298
= 2 u +2uvc+v t(xyx y)
= 2u2+2uvc+v2(A+2aP)
(with: A = c -abc + a a + b p - 4ap )
V = 6( c') = u +uvc+apv
Now it is easy to obtain the following equation:
(u + (ab-c) v)(u +cuv + apv ) + Auv =0
Modulo the augmentation ideal of A, we get:
(u + 2v)(u2 +2uv+v2)-0
But we know that u and v can be chosen such that u + v is not congruent to zero. Then
2 2
u +cuv + apv is not zero modulo A, and we have:
u = (c-ab) v mod A
In this example, we can choose v to be 1, and u is the unique element in A, congruent
to c- ab modulo A and satisfying the following equation:
(u + ab - c)( u + cu + a p ) + A u = 0
Actually this equation doesn't have any solution in A. The element u belongs to a cubic
extension of A included in A, and it seems to be very difficult to find a subring of a,
smaller than A where we can do all this construction for all links.
Rererences
[1] A . J . CASSON , oral communication . See also :
A . MARIN , L'invariant de Casson , preprint
[2] M . CULLER and P . B . SHALEN , Varieties of group representations and splittings
of 3-manifolds . Ann. of Math. 117, n°l (1983), PP. 109-146
[3] J . Y . LE DIMET , Cobordisme d'enlacements de disques. To appear
[4] J . P . LEVINE , Link concordance and algebraic closure of groups . Preprint
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