The Burnside Ring and Equivariant Cohomotopy
for Infinite Groups
Wolfgang L"uck*
Fachbereich Mathematik
Universit"at M"unster
Einsteinstr. 62
48149 M"unster
Germany
April 4, 2005
Abstract
After we have given a survey on the Burnside ring of a finite group,
we discuss and analyze various extensions of this notion to infinite (dis-
crete) groups. The first three are the finite-G-set-version, the inverse-
limit-version and the covariant Burnside group. The most sophisticated
one is the fourth definition as the equivariant zero-th cohomotopy of the
classifying space for proper actions. In order to make sense of this defin*
*i-
tion we define equivariant cohomotopy groups of finite proper equivariant
CW -complexes in terms of maps between the sphere bundles associated
to equivariant vector bundles. We show that this yields an equivariant
cohomology theory with a multiplicative structure. We formulate a ver-
sion of the Segal Conjecture for infinite groups. All this is analogous and
related to the question what are the possible extensions of the notion of
the representation ring of a finite group to an infinite group. Here possi*
*ble
candidates are projective class groups, Swan groups and the equivariant
topological K-theory of the classifying space for proper actions.
Key words: Burnside ring, equivariant cohomotopy, infinite groups.
Mathematics Subject Classification 2000: 55P91, 19A22.
0 Introduction
The basic notions of the Burnside ring and of stable equivariant cohomotopy
have been defined and investigated in detail for finite groups. The purpose of
this article is to discuss how these can be generalized to infinite (discrete) *
*groups.
____________________________*
email: lueck@math.uni-muenster.de
www: http://www.math.uni-muenster.de/u/lueck/
FAX: 49 251 8338370
1
The guideline will be the related notion of the representation ring which allows
several generalizations to infinite groups, each of which reflects one aspect o*
*f the
original notion for finite groups. Analogously we will present several possible
generalizations of the Burnside ring for finite groups to infinite (discrete) g*
*roups.
There seems to be no general answer to the question which generalization is the
right one. The answer depends on the choice of the background problem such
as universal additive properties, induction theory, equivariant stable homotopy
theory, representation theory, completion theorems and so on. For finite groups
the representation ring and the Burnside ring are related to all these topics
simultaneously and for infinite groups the notion seems to split up into differ*
*ent
ones which fall together for finite groups but not in general.
The following table summarizes in the first column the possible generaliza-
tions to infinite groups of the representation ring RF (G) with coefficients in*
* a
field F of characteristic zero. In the second column we list the analogous gene*
*r-
alizations for the Burnside ring. In the third column we give key words for its
main property, relevance or application. Explanations will follow in the main
body of the text.
_____________________________________________________________________
| | RF (G) A(|G) |key words | |
|_|________________________|___________________|____________________|_|||||||
| | K0(F G) A_|(G) |universal additive in-||
| | | | | |
| | | |variant, equivariant||
| | | |Euler characteristic| |
|_|________________________|___________________|____________________| |
| | f __| | | |
| | Sw (G; F ) A |(G) |induction theory,||
| | | |Green functors | |
|_|________________________|___________________|____________________| |
| | | | | |
| | Rcov,F(G) := Ac|ov(G) := |collecting all values||
| | colimH2Sub (G)RF (H)c|olimH2Sub (G)A(H)|for finite subgroups||
| | FIN | FIN | | |
| | | |with respect to induc-||
| | | |tion | |
|_|________________________|___________________|____________________| |
| | | | | |
| | Rinv,F(G) := Ai|nv(G) := |collecting all values||
| | invlimH2Sub (G)RF (H)|invlimH2Sub (G)A(H)|for finite subgroups||
| | FIN | FIN | | |
| | | |with respect to re-||
| | | |striction | |
|_|________________________|___________________|____________________||||||||
| | K0G(E_G) Ah|o(G) := ss0G(E_G) |completion theorems,| |
| | | | | |
| | | |equivariant vector||
| | | |bundles | |
|_|________________________|___________________|____________________||||||||
| | KG0(E_G) ss|G0(E_G) |representation theo-||
| | | | | |
| | | |ry, Baum-Connes | |
| | | |Conjecture, equi-||
| | | | | |
| | | |variant homotopy | |
| | | |theory, | |
|_|________________________|___________________|____________________|_|
2
The various rings are linked by the following diagram of ring homomorphisms
edgeG __
Aho(G) = ss0G(E_G)____//Ainv(G)oo_G___A (G)
| T |
| PGinv| |__PG
| fflffl|| fflffl|
_0G(E_G)|| Rinv,Q(G)oo___Swf(G; Q)
| SG,Q
| | |
| c| |c
fflffl|edgeG fflffl| fflffl|
K0G(E_G)_________//Rinv,C(G)SG,CSwf(G;oC)o_
where c denotes the obvious change of coefficients homomorphisms and the other
maps will be explained later.
We will also define various pairings which are summarized in the following
diagram which reflects their compatibilities.
__ ~GA
A(G) x A_(G)_________________//_KA_(G) (0.1)
KK
__PGxPG| KKKK G |G KKKK
__| KTKKxid P_| KKK
fflffl| KKKK ~GK fflffl| KKKK
Sw f(G; Q) x K0(QG)____KK______//_K0(QG)eeL KKKQGAK
LLLL KKK LLL KKK
LLLLLL KKK LLL G KKK
SG,Q LLLLLWG,QL KKK LLQKL KK
LLLLLLLL K%%K LLL GA KKK%%
LLLLLLL Ainv(G) x A_(G)_____LLL_________//Z
LLL LLL LLLL
LLL LLL PG|x(PG O(V G)-1) LLL |id
LLL L |inv cov LLL |
R LL%% fflffl| GR L%%fflffl|
inv,Q(G) x Rcov,Q(G)______________//Z
In Section 1 we give a brief survey about the Burnside ring A(G) of a finite
group G in order to motivate the generalizations. In Sections_2, 3 and 4
we treat the finite-G-set-version of the Burnside Ring A(G), the inverse-limit-
version of the Burnside ring Ainv(G) and the covariant Burnside group A_(G).
These definitions are rather straightforward. The most sophisticated version
of the Burnside ring for infinite groups is the equivariant zero-th cohomotopy
ss0G(E_G) of the classifying space E_G for proper G-actions. It will be constru*
*cted
in Section 6 after we have explained the notion of an equivariant cohomology
theory with multiplicative structure in Section 5. One of the main result of th*
*is
paper is
Theorem 6.5 Equivariant Cohomotopy ss*?defines an equivariant coho-
mology theory with multiplicative structure for finite proper equivariant CW -
complexes. For every finite subgroup H of the group G the abelian groups
ssnG(G=H) and ssnH are isomorphic for every n 2 Z and the rings ss0G(G=H)
and ss0H= A(H) are isomorphic.
An important test in the future will be whether the version of the Segal
Conjecture for infinite groups discussed in Section 8 is true.
3
The papers is organized as follows:
1. Review of the Burnside Ring for Finite Groups
2. The Finite-G-Set-Version of the Burnside Ring
3. The Inverse-Limit-Version of the Burnside Ring
4. The Covariant Burnside Group
5. Equivariant Cohomology Theories
6. Equivariant Stable Cohomotopy in Terms of Real Vector Bundles
7. The Homotopy Theoretic Burnside Ring
8. The Segal Conjecture for Infinite Groups
References
1 Review of the Burnside Ring for Finite Groups
In this section we give a brief review of the definition, properties and applic*
*ations
of the Burnside ring for finite groups in order to motivate our definitions for
infinite groups.
Definition 1.1. (Burnside ring of a finite group). The isomorphism
classes of finite G-sets form a commutative associative semi-ring with unit un-
der the disjoint union and the cartesian product. The Burnside ring A(G) is
the Grothendieck ring associated to this semi-ring.
As abelian group the Burnside ring A(G) is the free abelian group with the
set {G=H | (H) 2 ccs(G)} as basis, where ccs(G) denotes the set of conjugacy
classes of subgroups of G. The zero element is represented by the empty set,
the unit is represented by G=G. The interesting feature of the Burnside ring is
its multiplicative structure.
Given a group homomorphism f :G0 ! G1 of finite groups, restriction
with f defines a ring homomorphism f* :A(G1) ! A(G0). Thus A(G) be-
comes a contravariant functor from the category of finite groups to the category
of commutative rings. Induction defines a homomorphism of abelian groups
f*: A(G0) ! A(G1), [S] 7! [G1 xf S], which is not compatible with the multi-
plication. Thus A(G) becomes a becomes a covariant functor from the category
of finite groups to the category of abelian groups.
1.1 The Character Map and the Burnside Ring Congru-
ences
Let G be a finite group. Let ccs(G) be the set of conjugacy classes (H) of
subgroups H G. Define the character map
Y
charG:A(G) ! Z (1.2)
(H)2ccs(G)
by sending the class of a finite G-set S to the numbers {|SH | | (H) 2 ccs(G)}.
This is an injective ring homomorphism whose image can be described by the
so called Burnside ring congruences which we explain next.
4
In the sequel we denote for a subgroup H G by NG H its normalizer {g 2
G | g-1Hg = H}, by CG H = {g 2 G | gh = hg forh 2 H} its centralizer, by
WG H its Weyl group NG H=H and by [G : H] its index. Let pH :NG H ! WG H
be the canonical projection. Denote for a cyclic group C by Gen(C) the set of
its generators. We conclude from [46, Proposition 1.3.5]
Q
Lemma 1.3. An element {x(H)} 2 (H)2ccs(G)Z lies in the image of the
injective character map charG defined in (1.2)if and only if we have for every
(H) 2 ccs(G)
X
| Gen(C)| . [WG H : NWGH C] . x(p-1H(C)) 0 mod |WG H|.
(C)2ccs(WGH)
C cyclic
Example 1.4 (A(Z=p). Let p be a prime and let G be the cyclic group Z=p of
order p. Then A(G) is the free abelian group generated by [G] and [G=G]. The
multiplication is determined by the fact that [G=G] is the unit and [G] . [G] =
p . [G]. There is exactly one non-trivial Burnside ring congruence, namely the
one for H = {1} which is in the notation of Lemma 1.3
x(1) x(G) mod p.
1.2 The Equivariant Euler Characteristic
Next we recall the notion of a G-CW -complex.
Definition 1.5 (G-CW -complex). Let G be a group. A G-CW -complex X
is a G-space together with a G-invariant filtration
[
; = X-1 X0 X1 . . .Xn . . . Xn = X
n 0
such that X carries the colimit topology with respect to this filtration (i.e. *
*a set
C X is closed if and only if C \ Xn is closed in Xn for all n 0) and Xn
is obtained from Xn-1 for each n 0 by attaching equivariant n-dimensional
cells, i.e. there exists a G-pushout
` n-1 ` i2Inqni
i2InG=Hix?S ------! Xn-1?
?y ?y
` n
i2InG=Hix D -`-----!i2InXn
Qi n
A G-CW -complex X is called finite if it is built by finitely many equivaria*
*nt
cells G=H x Dn and is called cocompact if G\X is compact. The conditions
finite and cocompact are equivalent for a G-CW -complex. Provided that G is
finite, X is compact if and only if X is cocompact A G-map f :X ! Y of
G-CW -complexes is called cellular if f(Xn) Yn holds for all n.
5
Definition 1.6 (Equivariant Euler Characteristic). Let G be a finite group
and X be a finite G-CW -complex. Define its equivariant Euler characteristic
OG (X) 2 A(G)
by
1X X
OG (X) := (-1)n . [G=Hi]
n=0 i2In
after choices of the G-pushouts as in Definition 1.5.
This definition is independent of the choice of the G-pushouts by the next
result. The elementary proofs of the next two results are left to the reader. We
denote by XH and X>H respectively the subspace of X consisting of elements
x 2 X whose isotropy group Gx satisfies H Gx and H ( Gx respectively.
Lemma 1.7. Let G be a finite group.
(i)Let X be a finite G-CW -complex. Then
X
OG (X) = O WH G\(XH , X>H ) . [G=H],
(H)2ccs(G)
where O denotes the classical (non-equivariant) Euler characteristic;
(ii)If X and Y are G-homotopy equivalent finite G-CW -complexes, then
OG (X) = OG (Y );
(iii)If
X0 --i0--!X1
? ?
i2?y ?y
X2 ----! X
is a G-pushout of finite G-CW -complexes such that i1 is an inclusion of
finite G-CW -complexes and i2 is cellular, then
OG (X) = OG (X1) + OG (X1) - OG (X0);
(iv)If X and Y are finite G-CW -complexes, then X x Y with the diagonal
G-action is a finite G-CW -complex and
OG (X x Y ) = OG (X) . OG (Y );
(v)The image of OG (X) under the character map charG of (1.2)is given by
the collection of classical (non-equivariant) Euler characteristics {O(XH *
*) |
(H) 2 ccs(G)}.
6
An equivariant additive invariant for finite G-CW -complexes is a pair (A, a)
consisting of an abelian group and an assignment a which associates to every
finite G-CW -complex X an element a(X) 2 A such that a(;) = 0, G-homotopy
invariance and Additivity holds, i.e. the obvious versions of assertions (ii) a*
*nd
(iii) appearing in Lemma 1.7 are true. An equivariant additive invariant (U, u)*
* is
called universal if for every equivariant additive invariant (A, a) there is pr*
*ecisely
one homomorphism of abelian groups OE: U ! A such that OE(u(X)) = a(X)
holds for every finite G-CW -complex X. Obviously (U, u) is (up to unique
isomorphism) unique if it exists.
Theorem 1.8 (The universal equivariant additive invariant). Let G be a
finite group. The pair (A(G), OG ) is the universal equivariant additive invari*
*ant
for finite G-CW -complexes.
1.3 The Equivariant Lefschetz Class
The notion of an equivariant Euler characteristic can be extended to the notion
of an equivariant Lefschetz class as follows.
Definition 1.9. Let G be a finite group and X be a finite G-CW -complex. We
define the equivariant Lefschetz class of a cellular G-selfmap f :X ! X
G (f) 2 A(G)
by X
G (f) = WG H\(fH , f>H ) . [G=H],
(H)2ccs(G)
where (WG H\(fH , f>H )) 2 Z is the classical Lefschetz number of the endo-
morphism WG H\(fH , f>H ) of the pair of finite CW -complexes WG H\(XH , X>H )
induced by f.
Obviously G (id:X ! X) agrees with OG (X). The elementary proof of the
next result is left to the reader.
Lemma 1.10. Let G be a finite group.
(i)If f and g are G-homotopic G-selfmaps of a finite GW -CW -complex X,
then
G (f) = G (g);
(ii)Let
X0 --i0--!X1
? ?
i2?y ?y
X2 ----! X
is a G-pushout of finite G-CW -complexes such that i1 is an inclusion of
finite G-CW -complexes and i2 is cellular. Let fi:Xi! Xi for i = 0, 1, 2
and f :X ! X be the G-selfmaps compatible with this G-pushout. Then
G (f) = G (f1) + G (f2) - G (f0);
7
(iii)Let X and Y be finite G-CW -complexes and f :X ! X and g :Y ! Y
be G-selfmaps. Then
G (f x g) = OG (X) . G (g) + OG (Y ) . G (f);
(iv)Let f :X ! Y and g :Y ! X be G-maps of finite G-CW -complexes.
Then
G (f O g) = G (g O f);
(v)The image of G (f) under the character map charG of (1.2)is given by
the collection of classical (non-equivariant) Lefschetz numbers { (fH ) |
(H) 2 ccs(G)}.
One can also give a universal property characterizing the equivariant Lef-
schetz class (see [22]).
The equivariant Lefschetz class has also the following homotopy theoretic
meaning.
Definition 1.11. A G-homotopy representation X is a finite-dimensional G-
CW -complex such that for each subgroup H G the fixed point set XH is
homotopy equivalent to a sphere Sn(H) for n(H) the dimension of the CW -
complex XH .
An example is the unit sphere SV in an orthogonal representation V of G.
Denote by [X, X]G the set of G-homotopy classes of G-maps X ! X. The
proof of the next theorem can be found in [24, Theorem 3.4 on page 139] and
is a consequence of the equivariant Hopf Theorem (see for instance [46, page
213],[49, II.4], [21]).
Theorem 1.12. Let X be a G-homotopy representation of the finite group G.
Suppose that
(i)Every subgroup H G occurs as isotropy group of X;
(ii)dim(XG ) 1;
(iii)The group G is nilpotent or for every subgroup H G we have dim(X>H )+
2 dim(XH ).
Then the following map is an bijection of monoids, where the monoid struc-
ture on the source comes from the composition and the one on the target from
the multiplication
~= G G
degG :[X, X]G -! A(G), [f] 7! ( (f) - 1) . (O (X) - 1).
We mention that the image of the element degG(f) for a self-G-map of a
G-homotopy representation under the character map charG of (1.2)is given by
the collection of (non-equivariant) degrees {deg(fH ) | (H) 2 ccs(G)}.
8
1.4 The Burnside Ring and Stable Cohomotopy
Let X and Y be two finite pointed G-CW -complexes. Pointed means that we
have specified an element in its 0-skeleton which is fixed under the G-action. *
*If V
is a real G-representation, let SV be its one-point compactification. We will u*
*se
the point at infinity as base point for SV . If V is an orthogonal representati*
*on,
i.e. comes with an G-invariant scalar product, then SV is G-homeomorphic
to the unit sphere S(V R). Given two pointed G-CW -complexes X and Y
with base points x and y, define their one-point-union X _ Y to be the pointed
G-CW -complex X x {y} [ {x} x Y X x Y and their smash product X ^ Y to
be the pointed G-CW -complex X x Y=X _ Y .
We briefly introduce equivariant stable homotopy groups following the ap-
proach due to tom Dieck [49, II.6].
If V and W are two complex G-representations, we write V W if there
exists a complex G-representation U and a linear G-isomorphism OE: U V ! W .
If OE: U V ! W is a linear G-isomorphism, define a map
bV,W :[SV ^ X, SV ^ Y ]G ! [SW ^ X, SW ^ Y ]G
by the composition
[SV ^ X, SV ^ Y ]G -u1![SU ^ SV ^ X, SU ^ SV ^ Y ]G
u2-![SU V ^ X, SU V ^ Y ]G -u3![SW ^ X, SW ^ Y ]G ,
where the map u1 is given by [f] 7! [idSU^f], the map u2 comes from the
~=
obvious G-homeomorphism SU V -! SU ^ SV induced by the inclusion~V
W ! SV ^ SW and the map u3 from the G-homeomorphism SOE:SU V -=!
SW . Any two linear G-isomorphisms OE0, OE1: V1 ! V2 between to complex G-
representations are isotopic as linear G-isomorphisms. (This is not true for re*
*al
G-representations.) This implies that the map bV,W is indeed independent of
the choice of U and OE. One easily checks that bV2,V1O bV0,V1= bV0,V2holds for
complex G-representations V0, V1 and V2 satisfying V0 V1 and V1 V2.
Let I be the set of complex G-representations with underlying complex
vector space Cn for some n. (Notice that the collections of all complex G-
representations does not form a set.) Define on the disjoint union
a
[SV ^ X, SV ^ Y ]G
V 2I
an equivalence relation by calling f 2 [SV ^X, SV ^Y ]G and g 2 [SW ^X, SW ^
Y ]G equivalent if there exists a representation U 2 I with V U and W U
such that bV,U([f]) = bW,U([g] holds. Let !G0(X, Y ) for two pointed G-CW -
complexes X and Y be the set of equivalence classes.
If V is any complex G-representation (not necessarily in I) and f :SV ^X !
SV ^ Y is any G-map, there exists an element W 2 I with V W and we get
an element in !Gn(X, Y ) by bV,W ([f]). This element is independent of the choi*
*ce
of W and also denoted by [f] 2 !Gn(X, Y ).
9
One can define the structure of an abelian group on the set !G0(X, Y ) as
follows. Consider elements x, y 2 !G0(X, Y ). We can choose an element of
the shape C U in I for C equipped with the trivial G-action and G-maps
f, g :SC U ^ X ! SC U ^ X representing x and y. Now using the standard
pinching map r: SC ! SC ^ SC one defines x + y as the class of the G-map
~= C U r^idSU^ idX C C U
SC U ^ X -! S ^ S ^ X ---------! (S _ S ) ^ S ^ X
~= C U C U ~= C U C U f_g C U
-! (S ^S ^X)_(S ^S ^X) -! (S ^X)_(S ^X) --! S ^X.
The inverse of x is defined by the class of
~= C U d^f C U ~= C U
SC U ^ X -! S ^ S ^ X --! S ^ S ^ X -! S ^ X
where d: SC ! SC is any pointed map of degree -1. This is indeed independent
of the choices of U, f and g.
We define the abelian groups
!Gn(X, Y ) = !G0(Sn ^ X, Y ) n 0;
!Gn(X, Y ) = !G0(X, S-n Y ) n 0;
!nG(X, Y ) = !G-n(X, Y ) n 2 Z;
Obviously !Gn(X, Y ) is functorial, namely contravariant in X and covariant in
Y .
Let X and Y be (unpointed) G-CW -complexes. Let X+ and Y+ be the
pointed G-CW -complexes obtained from X and Y by adjoining a disjoint base
point. Denote by {o} the one-point-space. Define abelian groups
ssGn(Y )= !Gn({o}+ , Y+ ) n 2 Z;
ssnG(X) = !nG(X+ , {o}+ ) n 2 Z;
ssGn = ssGn({o}) n 2 Z;
ssnG = ssG-n n 2 Z.
The abelian group ssG0= ss0Gbecomes a ring by the composition of maps. The
abelian groups ssGn(Y ) define covariant functors in Y and are called the equiv*
*a-
riant stable homotopy groups of Y . The abelian groups ssnG(X) define contravar*
*i-
ant functors in X and are called the equivariant stable cohomotopy groups of
X.
We emphasize that our input in ssnGand ssGnare unpointed G-CW -complexes.
This is later consistent with our constructions for infinite groups, where all *
*G-
CW -spaces must be proper and therefore have empty G-fixed point sets and
cannot have base points.
Theorem 1.12 implies the following result due to Segal [41].
Theorem 1.13. The isomorphism degG appearing in Theorem 1.12 induces an
isomorphism of rings ~
degG: ss0G-=!A(G).
For a more sophisticated and detailed construction of and more information
about the equivariant stable homotopy category we refer for instance to [17],
[23].
10
1.5 The Segal Conjecture for Finite Groups
The equivariant cohomotopy groups ssnG(X) are modules over the ring ss0G=
A(G), the module structure is given by composition of maps. The augmentation
homomorphism fflG :A(G) ! Z is the ring homomorphism sending the class of a
finite set S to |S| which is just the component belonging to the trivial subgro*
*up
of the character map defined in (1.2). The augmentation ideal IG A(G) is
the kernel of the augmentation homomorphism fflG .
For an (unpointed) CW -complex X we denote by ssns(X) the (non-equivariant)
stable cohomotopy group of X+ . This is in the previous notation for equiva-
riant stable cohomotopy the same as ssn{1}(X) for {1} the trivial group. If X
is a finite G-CW -complex, we can consider ssns(EG xG X). Since ssnG(X) is a
A(G)-module, we can also consider its IG -adic completion denoted by ssnG(X)IbG.
The following result is due to Carlsson [8].
Theorem 1.14 (Segal Conjecture for finite groups). The Segal Conjecture
for finite groups G is true, i.e. for every finite group G and finite G-CW -com*
*plex
X there is an isomorphism
~= n
ssnG(X)IbG-! sss(EG xG X).
In particular we get in the case X = {o} and n = 0 an isomorphism
~= 0
A(G)IbG -! sss(BG). (1.15)
Thus the Burnside ring is linked via its IG -adic completion to the stable coho-
motopy of the classifying space BG of a finite group G.
Example 1.16 (Segal Conjecture for Z=p). Let G be the cyclic group Z=p of
order p. We have computed A(G) in Example 1.4. If we put x = [G] - p . [G=G],
then the augmentation ideal is generated by x. Since
x2 = ([G] - p)2 = [G]2 - 2p . [G] + p2 = (-p) . x,
we get xn = (-p)n-1x and hence InG= pn-1 . IG for n 2 Z, n 1. This implies
A(G)IbG= invlimn!1 Z IG =InG = Z x Zbp,
where Zbpdenotes the ring of p-adic integers.
1.6 The Burnside Ring as a Green Functor
Let R be an associative commutative ring with unit. Let FGINJ be the cate-
gory of finite groups with injective group homomorphisms as morphisms. Let
M :FGINJ ! R-MODULES be a bifunctor, i.e. a pair (M*, M*) consisting
of a covariant functor M* and a contravariant functor M* from FGINJ to
R-MODULES which agree on objects. We will often denote for an injective
group homomorphism f :H ! G the map M*(f): M(H) ! M(G) by indf
and the map M*(f): M(G) ! M(H) by resf and write indGH= indf and
resHG= resfif f is an inclusion of groups. We call such a bifunctor M a Mackey
functor with values in R-modules if
11
(i)For an inner automorphism c(g): G ! G we have M*(c(g)) = id:M(G) !
M(G);
~=
(ii)For an isomorphism of groups f :G -! H the composites resfO indfand
indfO resfare the identity;
(iii)Double coset formula
We have for two subgroups H, K G
X H\g-1Kg
resKGO indGH= indc(g):H\g-1Kg!K O resH ,
KgH2K\G=H
where c(g) is conjugation with g, i.e. c(g)(h) = ghg-1.
Let OE: R ! S be a homomorphism of associative commutative rings with
unit. Let M be a Mackey functor with values in R-modules and let N and P
be Mackey functors with values in S-modules. A pairing with respect to OE is a
family of maps
m(H): M(H) x N(H) ! P (H), (x, y) 7! m(H)(x, y) =: x . y,
where H runs through the finite groups and we require the following properties
for all injective group homomorphisms f :H ! K of finite groups:
(x1 + x2) . y = x1 . y + x2f.oyrx1, x2 2 M(H), y 2 N(H);
x . (y1 + y2) = x . y1 + x .fy2orx 2 M(H), y1, y2 2 N(H);
(rx) . y = OE(r)(x . y) forr 2 R, x 2 M(H), y 2 N(H);
x . sy = s(x . y) fors 2 S, x 2 M(H), y 2 N(H);
resf(x . y) = resf(x) . resf(y)forx 2 M(K), y 2 N(K);
indf(x) . y = indf(x . resf(y))forx 2 M(H), y 2 N(K);
x . indf(y) = indf(resf(x) .fy)orx 2 M(K), y 2 N(H).
A Green functor with values in R-modules is a Mackey functor U together
with a pairing with respect to id:R ! R and elements 1H 2 U(H) for each finite
group H such that for each finite group H the pairing U(H) x U(H) ! U(H)
induces the structure of an R-algebra on U(H) with unit 1H and for any mor-
phism f :H ! K in FGINJ the map U*(f): U(K) ! U(H) is a homomorphism
of R-algebras with unit. Let U be a Green functor with values in R-modules
and M be a Mackey functor with values in S-modules. A (left) U-module struc-
ture on M with respect to the ring homomorphism OE: R ! S is a pairing such
that any of the maps U(H) x M(H) ! M(H) induces the structure of a (left)
module over the R-algebra U(H) on the R-module OE*M(H) which is obtained
from the S-module M(H) by rx := OE(r)x for r 2 R and x 2 M(H).
Theorem 1.17. (i) The Burnside ring defines a Green functor with values
in Z-modules;
(ii)If M is a Mackey functor with values in R-modules, then M is in a canon-
ical way a module over the Green functor given by the Burnside ring with
respect to the canonical ring homomorphism OE: Z ! R.
12
Proof. (i) Let f :H ! G be an injective homomorphism of groups. Define
indf: A(H) ! A(G) by sending the class of a finite H-set S to the class of the
finite G-set G xf S. Define resf:A(G) ! A(H) by considering a finite G-set as
an H-set by restriction with f. One easily verifies that the axioms of a Green
functor with values in Z-modules are satisfied.
(ii) We have to specify for any finite group G a pairing m(G): A(G) x M(G) !
M(G). This is done by the formula
_ !
X X
m(G) ni. [G=Hi], x := ni. indGHiO resHiG(x).
i i
One easily verifies that the axioms of a module over the Green functor given_by
the Burnside ring are satisfied. |__|
Theorem 1.17 is the main reason, why the Burnside ring plays an important
role in induction theory. Induction theory addresses the question whether one
can compute the values of a Mackey functor on a finite group by its values on
a certain class of subgroups such as the family of cyclic or hyperelementary
groups. Typical examples of such Mackey functors are the representation ring
RF (G) or algebraic K and L-groups Kn(RG) and Ln(RG) of groups rings.
The applications require among other things a good understanding of the prime
ideals of the Burnside ring. For more information about induction theory for
finite groups we refer to the fundamental papers by Dress [12], [13] and for
instance to [46, Chapter 6]. Induction theory for infinite groups is developed *
*in
[5].
As an illustration we give an example how the Green-functor mechanism
works.
Example 1.18 (Artin's Theorem). Let RQ(G) be the rational representation
ring of the finite group G. For any finite cyclic group C one can construct an
element
`C 2 RQ(C)
which is uniquely determined by the property that its character function sends
a generator of C to |C| and every other element of C to zero.
Let G be a finite group. Let Q be the trivial 1-dimensional rational G-
representation. It is not hard to check by a calculation with characters that
X
|G| . [Q] = indHC`C . (1.19)
CCcGyclic
Assigning to a finite group G the rational representation ring RQ(G) inherits
the structure of a Green functor with values in Z-modules by induction and
restriction. Suppose that M is a Mackey functor with values in Z-modules
which is a module over the Green functor RQ. Then for every finite group G
13
the cokernel of the map
M M
indGC: M(C) ! M(G)
CCcGyclic CCcGyclic
is annihilated by multiplication with |G|. This follows from the following calc*
*u-
lation for x 2 M(G) based on (1.19)and the axioms of a Green functor and a
module over it
X X
|G| . x = (|G| . [Q]) . x = indHC(`C ) . x = indHC(`C . resCHx).
CCcGyclic CCcGyclic
Examples for M are algebraic K- and L-groups Kn(RG) and Ln(RG) for any
ring R with Q R. We may also take M to be RF for any field F of character-
istic zero and then the statement above is Artin's Theorem (see [43, Theorem 26
on page 97].
1.7 The Burnside Ring and Rational Representations
Let RQ(G) be the representation ring of finite-dimensional rational G-represen-
tation. Given a finite G-set S, let Q[S] be the rational G-representation given*
* by
the Q-vector space with the set S as basis. The next result is due to Segal [40*
*].
Theorem 1.20. (The Burnside ring and the rational representation
ring for finite groups). Let G be a finite group. We obtain a ring homomor-
phism
P G:A(G) ! RQ(G), [S] 7! [Q[S]].
It is rationally surjective. If G is a p-group for some prime p, it is surjecti*
*ve.
It is bijective if and only if G is cyclic.
1.8 The Burnside Ring and Homotopy Representations
We have introduced the notion of a G-homotopy representation in Definition 1.11.
The join of two G-homotopy representations is again a G-homotopy represen-
tation. We call two G-homotopy representations X and Y stably G-homotopy
equivalent if for some G-homotopy representation Z the joins X * Z and Y * Z
are G-homotopy equivalent. The stable G-homotopy classes of G-homotopy
representations together with the join define an abelian semi-group. The G-
homotopy representation group V (G) is the associated Grothendieck group. It
may be viewed as the homotopy version of the representation ring. Taking the
unit sphere yields a group homomorphism RR(G) ! V (G).
The dimension function of a G-homotopy representation X
Y
dim(X) 2 Z
(H)
14
associates to the conjugacy class (H)Qof a subgroup H G the dimension of
XH . The question which elements in (H)Z occur as dim(X) is studied for
instance in [47], [48], [49, III.5] and [50]. Define V (G, dim) by the exact se*
*quence
Y
0 ! V (G, dim) ! V (G) dim--! Z.
(H)
Let Pic(A(G)) be the Picard group of the Burnside ring, i.e. the abelian group
of projective A(G)-module of rank one with respect to the tensor product. The
next result is taken from [50, 6.5].
Theorem 1.21 (V (G, dim) and the Picard group of A(G)). There is an
isomorphism ~
V (G, dim) =-!Pic(A(G)).
Further references about the Burnside ring of finite groups are [7], [10],[1*
*1],
[16], [19], [20], [36], [45], [52].
2 The Finite-G-Set-Version of the Burnside Ring
From now on G can be any (discrete) group and needs not to be finite anymore.
Next we give a first definition of the Burnside ring for infinite groups.
Definition 2.1. (The finite-G-set-version of the Burnside ring). The
isomorphism classes of finite G-sets form a commutative associative semi-ring
with unit under the disjoint union_and the cartesian product. The finite-G-set-
version of the Burnside ring A (G) is the Grothendieck ring associated to this
semi-ring.
To avoid any confusion, we emphasize that finite G-set means a finite set
with a G-action. This definition is word by word the same as given for a finite
group in Definition 1.1.
Given a group homomorphism f :G0 ! G1 of groups, restriction with f
defines a ring homomorphism f* :A(G1) ! A(G0). Thus A(G) becomes a con-
travariant functor from the category of groups to the category of commutative
rings. Provided that the image of f has finite index, induction defines a homo-
morphism of abelian groups f*: A(G0) ! A(G1), [S] 7! [G1xf S], which is not
compatible with the multiplication.
2.1 Character Theory and Burnside Congruences for the
Finite-G-Set-Version
The definition of the character map (1.2)makes also sense for infinite groups
and we denote it by
____G __ Y H
char :A (G) ! Z, [S] 7! [|(S |)(H). (2.2)
(H)2ccs(G)
15
Given a group homomorphism f :G0 ! G1, define a ring homomorphism
Y Y
f* : ! Z (2.3)
(H1)2ccs(G1) (H0)2ccs(G0)
by sending {x(H1)} to {x(f(H1))}. One easily checks
Lemma 2.4. The following diagram of commutative rings with unit commutes
for every group homomorphism f :G0 ! G1
__ ___charG1Q
A(G1) - ---! (H1)2ccs(G1)Z
? ?
f*?y ?yf*
__ Q
A(G0) -_---!_ (H0)2ccs(G0)Z
charG0
__
Theorem_2.5 (Burnside ring congruences for A(G)). The character map
charG is an injective ring homomorphism.
Already the composition
__ ___charG Y pr Y
A (G) ----! Z -! Z
(H)2ccs(G) (H)2ccs(G)
[G:H]<1
for pr the obvious projectionQis injective.
An element x = {x(H)} 2 (H)2ccs(G)Z lies in the image of the character
____G
map char defined in (2.2)if and only if it satisfies the following two conditio*
*ns:
(i)There exists a normal subgroup Kx G of finite index such that x(H) =
x(H . Kx) holds for all H G, where H . Kx is the subgroup {hk | h 2
H, k 2 Kx};
(ii)We have for every (H) 2 ccs(G) with [G : H] < 1:
X
| Gen(C)|.[WG H : NWGH C].x(p-1H(C)) 0 mod |WG H|,
(C)2ccs(WGH)
C cyclic
where pH :NG H ! WG H is the obvious projection.
____G
Proof. Obviously char_ is a ring homomorphism. ____
Suppose that x 2 A(G) lies in the kernel of charG. For any finite G-set the
intersection of all its isotropy groups is a normal subgroup of finite index in
G._ Hence can find an epimorphism px: G ! Qx onto_a finite_group Qx and
x 2 A(Qx) such that x lies in the image of p*x:A(Qx) ! A(G). Since the map
Y Y
p*x: Z ! Z
(H)2ccs(Qx) (K)2ccs(G)
16
____Qx
is obviously injective and_the_character map char is injective by Lemma 1.3,
we conclude x = 0. Hence charG is injective._ __ ____
Suppose that y lies in the image of charG. Choose x 2 A(G) with charG(x) =
y. As explained above we can find an epimorphism_px:_G ! Qx onto a finite
group Qx and __x2 A(Qx) such that p*x:A(Qx) ! A(G) maps __xto x. Then
Condition (i) is satisfied by Lemma 2.4 if we take Kx to be the kernel of px.
Condition (ii) holds for x since the proof of Lemma 1.3 carries though word by
word to the case, where G is possibly infinite but H G is required to have
finite index in G and_hence_WG H is finite. ____
We_conclude_that charG(x) = 0 if and only if prOcharG(x) = 0 holds. Hence
pr OcharG is injective. Q
Now suppose that x = {y(H)} 2 (H)2ccs(G)Z satisfies Condition (i) and
Condition (ii). Let Qx = G=Kx and let px: G ! Qx be the projection. In the
sequel we abbreviate Q = Qx and p = px. Then Condition (i) ensures that x
lies in the image of the injective map
Y Y
p*: Z ! Z.
(H)2ccs(Q) (K)2ccs(G)
Q
Let y 2 (H)2ccs(Q)Z be such a preimage. Because of Lemma 2.4 it suffices to
prove that y lies in the image of the character map
Y
charQ: A(Q) ! Z.
(H)2ccs(Q)
By Lemma 1.3 this is true if and only if for every subgroup K Q the congru-
ence
X i Q j
| Gen(C)|.[WQ K : NWQK C].y pK )-1(C) 0 mod |WQ K|
(C)2ccs(WQK)
C cyclic
holds, where pQK:NQ K ! WG K is the projection. Fix a subgroup K Q.
Put H = p-1(K) G. The epimorphism p: G ! Q induces an isomorphism
_p:W ~=
G H -! WQ K. Condition (ii) applied to x and H yields
X
| Gen(C)| . [WG H : NWGH C] . x(p-1H(C)) 0 mod |WG H|.
(C)2ccs(WGH)
C cyclic
For any cyclic subgroup C WG H we obtain a cyclic subgroup _p(C) WQ K
and we have
| Gen(C)| = | Gen(_p(C))|;
[WG H : NWGH C] = [WQ K : NWQK _p(C)];
-1 i Q -1 _ j
x pH (C) = y (pK ) (p(C)).
Now the desired congruence for y follows. This finishes the proof of_Theorem_2.*
*5.
|__|
17
__
Example 2.6 (A of the integers). Consider the infinite cyclic group Z. Any
subgroup of finite_index is of the form nZ for some n 2 Z, n 1. As an
abelian group A(Z) is generated by the classes [Z=nZ] for n 2 Z, n 1. The
condition (ii) appearing in Theorem 2.5 reduces to the condition that for every
subgroup nZ for n 2 Z, n 1 the congruence
X in j
OE __ . x(m) 0 mod n
m2Z,m 1,m|n m
holds, where OE is the Euler function, whose value OE(k) is | Gen(Z=kZ)|. The
condition (i) reduces to the condition that there exists nx 2 Z, nx 1 such
that for all m 2 Z, m 1 we have x(mZ) = x(gcd(m, nx)Z), where gcd(m, nx)
is the greatest common divisor of m and nx.
__
Remark 2.7 (The completion bA(G) of A(G)). We call a G-set almost finite
if each isotropy group has finite index and for every positive integer n the nu*
*mber
of orbits G=H in S with [G : H] n is finite. A G-set S is almost finite if and
only if for every subgroupSH G of finite index the H-fixed point set SH is fi*
*nite
and S is the union (H)2ccs(G)SH . Of course every finite G-set S is almost fin*
*ite.
[G:H]<1
The disjoint union and the cartesian product with the diagonal G-action of two
almost finite G-sets is again almost finite. Define Ab(G) as the Grothendieck
ring of the semi-ring of almost finite G-sets under the disjoint_union and the
cartesian product. There is an obvious inclusion of rings A (G) ! bA(G). We
can define as before a character map
dcharG:bA(G) ! Y Z, [S] 7! (|(SH |)(H). (2.8)
(H)2ccs(G)
[G:H]<1
G
WeQleave it to the reader to check that dcharis injective and that an element x*
* in
(H)2ccs(G)Z lies in its image, if and only if x satisfies condition (ii) appe*
*aring
[G:H]<1
in Theorem 2.5.
Dress and Siebeneicher [14] analyze bA(G) for profinite groups G and put it
into relation with the_Witt vector construction. They also explain that bA(G)
is a completion of A(G). The ring bA(Z) is studied and put in relation to the
necklace algebra, ~-rings and the universal ring of Witt vectors in [15].
2.2 The Finite-G-Set-Version and the Equivariant Euler
Characteristic and the Equivariant Lefschetz Class
__
The results of Sections 1.2 and 1.3 carry over to A(G) if one considers only fi*
*nite
G-CW -complexes X whose isotropy group all have finite index in G. But this is
not really new since for any such G-CW -complex X there is a subgroup H G,
namely the intersection of all isotropy groups, such that H is normal, has fini*
*te
index in G and acts trivially on X. Thus X is a finite Q-CW -complex for the
18
finite group Q = G=H and all these invariant are obtained_from_the one over Q
by the applying the obvious ring homomorphism A(Q) = A(Q) ! A(G) to the
invariants already defined over the finite group Q.
2.3 The Finite-G-Set-Version as a Green Functor
The_notions and results of Subsection 1.6 carry over to the finite-G-set-version
A (G) for an infinite group G, we replace the category FGINJ by the category
GRIFI whose objects are groups and whose morphisms are injective group ho-
momorphisms whose image has finite index in the target. However, for infinite
groups this does not seem to be the right approach to induction theory. The
approach presented in Bartels-L"uck [5] is more useful. It is based on classify*
*ing
spaces for families and aims at reducing the family of subgroups, for instance
from all finite subgroups to all hyperelementary finite subgroups or from all v*
*ir-
tually cyclic subgroups to the family of subgroups which admit an epimorphism
to a hyperelementary group and whose kernel is trivial or infinite cyclic.
2.4 The Finite-G-Set-Version and the Swan Ring
Let R be a commutative ring. Let Sw f(G; R) be the abelian group which is
generated by the RG-isomorphisms classes of RG-modules which are finitely
generated free over R with the relations [M0] - [M1] - [M2] = 0 for any short
exact RG-sequence 0 ! M0 ! M1 ! M2 ! 0 of such RG-modules. It be-
comes a commutative ring, the so called Swan ring Sw f(G; R)), by the tensor
product R . If G is finite and F is a field, then Swf (G; F ) is the same as t*
*he
representation ring RF (G) of (finite-dimensional) G-representations over K.
Let G0(RG) be the abelian group which is generated by the RG-isomorphism
classes of finitely generated RG-modules with the relations [M0]-[M1]-[M2] =
0 for any short exact RG-sequence 0 ! M0 ! M1 ! M2 ! 0 of such RG-
modules. There is an obvious map
OE: Sw f(G; R) ! G0(RG)
of abelian groups. It is an isomorphism if G is finite and R is a principle ide*
*al
domain. This follows from [9, Theorem 38.42 on page 22].
We obtain a ring homomorphism
__G __ f
P :A (G) ! Sw (G; Q), [S] 7! [R[S]], (2.9)
where R[S] is the finitely generated free R-module with the finite set S as bas*
*is
and becomes a RG-module by the G-action_on_S.
Theorem 1.20 does not carry over to A(G) for infinite groups. For instance,
the determinant induces a surjective homomorphism
det:Sw (Z; Q) ! Q*, [V ] 7! det(lt:V ! V ),
where lt is left multiplication with a fixed generator t 2 Z. Given a finite Z-
set S, the map lt:Q[S] ! Q[S] satisfies (lt)n = idfor some n 1 and hence
19
__Z __
P (Q[S]) = - 1. Hence_the_image of the composition detOP Zis contained in
{ 1}. Therefore the map PZ of (2.9)is not rationally surjective.
2.5 Maximal Residually Finite Quotients
Let G be a group. Denote by G0 the intersection of all normal subgroups of
finite index. This is a normal subgroup. Let p: G ! G=G0 be the projection.
Recall that G is called residually finite if for every element g 2 G with g 6= 1
there exists a homomorphism onto a finite group which sends g to an element
different from 1. If G is countable, then G is residually finite if and only if
G0 is trivial. The projection p: G ! Gmrf:= G=G0 is the projection onto the
maximal residually finite quotient of G, i.e. Gmrf is residually finite and eve*
*ry
epimorphism f :G ! Q onto a residually_finite group Q factorizes through p
into a composition G -p!Gmrf f-!Q. If G is a finitely generated subgroup of
GLn(F ) for some field F , then G is residually finite (see [35], [51, Theorem
4.2]). Hence for every finitely generated group G each G-representation V with
coefficients in a field F is obtained by restriction with p: G ! Gmrf from a
Gmrf-representation. In particular every G-representation with coefficient in a
field F is trivial if G is finitely generated and Gmrf is trivial.
One easily checks that
__ ~= __
p*: A(Gmrf) -! A(G)
__
is an isomorphism. In particular we have A(G) = Z if Gmrf is trivial. If G is
finitely generated, then
~= f
p*: Swf(Gmrf; F ) -! Sw (G; F )
is an isomorphism. In particular we have Swf(G; F ) = Z if G is finitely gener-
ated and Gmrf is trivial.
__ f
Example 2.10 (A (Z=p1 ) and Sw (Z=p1 ; Q)). Let Z=p1 be the Pr"ufer group,
i.e. the colimit of the directed system of injections of abelian groups Z=p !
Z=p2 ! Z=p3 ! . ...It can be identified with Q=Z(p)or Z[1=p]=Z. We want to
show that the following diagram is commutative and consists of isomorphisms
__ p* __
A ({1}) = Z ----!~= A (Z=p1 )
? ?
__P{1}?y~= __PZ=p1?y~=
*
Sw f({1}; Q) = Z-p---!~=Swf(Z=p1 ; Q)
where p: Z=p1 ! {1} is the projection. Obviously the diagram commutes and
the left vertical arrow is bijective. Hence it remains to show that the horizon*
*tal
arrows are bijective.
Let f :Z=p1 ! Q be any epimorphism onto a finite group. Since Z=p1 is
abelian, Q is a finite abelian group. Since any element in Z=pn has p-power
20
order, we conclude from the definition of Z=p1 as a colimit that Q is a finite
abelian p-group. Since Q is p-divisible, the quotient Q must be p-divisible.
Therefore Q must be trivial. Hence (Z=p1 )mrfis trivial and the upper horizontal
arrow is bijective.
In order to show that the lower horizontal arrow is bijective, it suffices to
show that every (finite-dimensional) rational Z=p1 -representation V is trivial.
It is enough to show that for every subgroup Z=pm its restriction resimV for the
inclusion im :Z=pm ! Z=p1 is trivial. For this purpose choose a positive integer
n such that dimQ(V ) < (p-1).pn. Consider the rational Z=pm+n -representation
resim+nV . Let pm+nk:Z=pm+n ! Z=pk be the canonical projection. Let Q(pk)
be the rational Z=pk-representation given by adjoining a primitive pk-th root of
unity to Q. Then the dimension of Q(pk) is (p - 1) . pk-1. A complete system
of representatives for the isomorphism classes of irreducible rational Z=pm+n -
representations is {respm+nkQ(pk) | k = 0, 1, 2, . .m.+ n}. Since dimQ (V ) <
(p-1).pn, there exists a rational Z=pn-representation with W with resim+nV ~=
respm+nnW . Hence we get an isomorphism of rational Z=pm -representations
resimV ~=resim,m+nO respm+nnW
where im,m+n :Z=pm ! Z=pm+n is the inclusion. Since the composition pm+nnO
im,m+n is trivial, the rational Z=pm -representation resimV is trivial.
It is not true that
Swf(Z=p1 ; C) ! Swf({1}; C) = Z
is bijective because Sw f(Z=p1 ; C) has as abelian group infinite rank (see Ex-
ample 3.16).
3 The Inverse-Limit-Version of the Burnside Ring
In this section we present the inverse-limit-definition of the Burnside ring for
infinite groups.
The orbit category Or (G) has as objects homogeneous spaces G=H and as
morphisms G-maps. Let Sub (G) be the category whose objects are subgroups
H of G. For two subgroups H and K of G denote by conhom G(H, K) the
set of group homomorphisms f :H ! K, for which there exists an element
g 2 G with gHg-1 K such that f is given by conjugation with g, i.e. f =
c(g): H ! K, h 7! ghg-1. Notice that c(g) = c(g0) holds for two elements
g, g0 2 G with gHg-1 K and g0H(g0)-1 K if and only if g-1g0 lies in
the centralizer CG H = {g 2 G | gh = hg for allh 2 H} of H in G. The
group of inner automorphisms of K acts on conhom G(H, K) from the left by
composition. Define the set of morphisms
morSub(G)(H, K) := inn(K)\ conhomG (H, K).
There is a natural projection pr:Or (G) ! Sub (G) which sends a homo-
geneous space G=H to H. Given a G-map f :G=H ! G=K, we can choose
21
an element g 2 G with gHg-1 K and f(g0H) = g0g-1K. Then pr(f)
is represented by c(g): H ! K. Notice that mor Sub(G)(H, K) can be iden-
tified with the quotient mor Or(G)(G=H, G=K)=CG H, where g 2 CG H acts
on morOr (G)(G=H, G=K) by composition with Rg-1:G=H ! G=H, g0H 7!
g0g-1H. We mention as illustration that for abelian G the set of morphisms
mor Sub(G)(H, K) is empty if H is not a subgroup of K, and consists of precisely
one element given by the inclusion H ! K if H is a subgroup in K.
Denote by OrFIN (G) Or(G) and SubFIN (G) Sub(G) the full subcate-
gories, whose objects G=H and H are given by finite subgroups H G.
Definition 3.1. (The inverse-limit-version of the Burnside ring). The
inverse-limit-version of the Burnside ring Ainv(G) is defined to be the commu-
tative ring with unit given by the inverse limit of the contravariant functor
A(?): SubFIN(G) ! RINGS , H 7! A(H).
Since inner automorphisms induce the identity on A(H), the contravariant
functor appearing in the definition above is well-defined.
Consider a group homomorphism f :G0 ! G1. We obtain a covariant func-
tor SubFIN (f): SubFIN(G0) ! SubFIN (G1) sending an object H to f(H). A
morphism u: H ! K given by c(g): H ! K for some g 2 G with gHg-1 K
is sent to the morphism given by c(f(g)): f(H) ! f(K). There is an obvious
transformation from the composite of the functor A(?): SubFIN(G1) ! RINGS
with Sub FIN(f) to the functor Ainv(?): SubFIN (G0) ! RINGS . It is given
for an object H 2 SubFIN (G0) by the ring homomorphism A(f(H)) ! A(H)
induced by the group homomorphism f|H :H ! f(H). Thus we obtain a ring
homomorphism Ainv(f): Ainv(G1) ! Ainv(G0). So Ainvbecomes a contravari-
ant functor GROUPS ! RINGS .
Definition 3.1 reduces to the one for finite groups presented in Subsection 1
since for a finite group G the object G 2 SubFIN (G) is a terminal object.
There is an obvious ring homomorphism, natural in G,
__
T G:A (G) ! Ainv(G) (3.2)
__ __ __
which is induced from the various ring homomorphisms A(iH ): A(G) ! A(H) =
A(H) for the inclusions iH :H ! G for each finite subgroup H G. The
following examples show that it is neither injective nor surjective in general.
3.1 Some Computations of the Inverse-Limit-Version
Example 3.3 (Ainv(G) for torsionfree G). Suppose that G is torsionfree.
Then SubFIN (G) is the trivial category with precisely one object and one mor-
phisms. This implies that the projection pr:G ! {1} induces a ring isomor-
phism ~
Ainv(pr): Ainv({1}) = Z -=! Ainv(G).
In particular we conclude from Example 2.6 that the canonical ring homomor-
phism __ ~
T Z:A(Z) =-!Ainv(Z)
22
of (3.2)is not injective.
Example 3.4 (Groups with appropriate maximal finite subgroups). Let
G be a discrete group which is not torsionfree. Consider the following assertio*
*ns
concerning G:
(M) Every non-trivial finite subgroup of G is contained in a unique maximal
finite subgroup;
(NM) If M G is maximal finite, then NG M = M.
The conditions (M) and (NM) imply the following: Let H be a non-trivial
finite subgroup of G. Then there is a unique maximal finite subgroup MH with
H MH and the set of morphisms in Sub FIN(G) from H to MH consists
of precisely one element which is represented by the inclusion H ! MH . Let
{Mi | i 2 I} be a complete set of representatives of the conjugacy classes of
maximal finite subgroups of G. Denote by ji:Mi ! G, ki:{1} ! Mi and
k :{1} ! G the inclusions. Then we obtain a short exact sequence
Ainv(j{1})xQ i2IAinv(ji) Y
0 ! Ainv(G) ----------------! Ainv({1}) x Ainv(Mi)
i2I
-Q i2IAinv(ki)Y
-----------! Ainv({1} ! 0,
i2I
Q
where : Ainv({1} ! i2IAinv({1} is the diagonal embedding. If we define
Aginv(G) as the kernel of Ainv(G) ! Ainv({1}), this gives an isomorphism
Q g Y
Aginv(G) --i2IAinv(ji)-------!gAinv(Mi).
i2I
Here are some examples of groups Q which satisfy conditions (M) and (NM):
o Extensions 1 ! Zn ! G ! F ! 1 for finite F such that the conjugation
action of F on Zn is free outside 0 2 Zn.
The conditions (M) and (NM) are satisfied by [33, Lemma 6.3].
o Fuchsian groups F
See for instance [33, Lemma 4.5]). In [33] the larger class of cocompact
planar groups (sometimes also called cocompact NEC-groups) is treated.
o Finitely generated one-relator groups G
Let G = <(qi)i2I| r> be a presentation with one relation. Let F be the
free group with basis {qi | i 2 I}. Then r is an element in F . There
exists an element s 2 F and an integer m 1 such that r = sm , the
cyclic subgroup C generated by the class _s2 Q represented by s has order
m, any finite subgroup of G is subconjugated to C and for any q 2 Q
the implication q-1Cq \ C 6= 1 ) q 2 C holds. These claims follows
23
from [34, Propositions 5.17, 5.18 and 5.19 in II.5 on pages 107 and 108].
Hence Q satisfies (M) and (NM) and the inclusion i: C ! G induces an
isomorphism ~
Ainv(i): Ainv(G) =-!Ainv(C).
Example 3.5 (Olshanskii's group). There is for any prime number p > 1075
an infinite finitely generated group G all of whose proper subgroups are finite
of order p [37]. Obviously G contains no subgroup of finite index. Hence the
inclusion i: {1} ! G induces an isomorphism
__ __
A (i): A(G) ! A({1}) = Z
(see Subsection 2.5). Let H be a finite non-trivial subgroup of G. Then H is
isomorphic to Z=p and agrees with its normalizer. So the conditions appearing
in Example 3.4 are satisfied. Hence we obtain an isomorphism
Y
gAinv(G) ~=-! Aginv(Z=p),
(H)2ccsf(G)
H6={1}
where ccsf(G) is the set of conjugacy classes of finite subgroups. This implies
that the natural map __ ~
T G:A (G) =-!Ainv(G)
of (3.2)is not surjective.
Example 3.6 (Extensions of Zn with Z=p as quotient). Suppose that G
can be written as an extension 1 ! A ! G ! Z=p ! 1 for some fixed prime
number p and for A = Zn for some integer n 0 and that G is not torsionfree.
The conjugation action of G on the normal subgroup A yields the structure of
a Z[Z=p]-module on A. Every non-trivial element g 2 G of finite order G has
order p and satisfies
NG = CG = AZ=px .
In particular the conjugation action of NG on is trivial. There is a
bijection ~
~: H1(Z=p; A) -=! ccsf(G),
where H1(Z=p; A) is the first cohomology of Z=p with coefficients in the Z[Z=p]-
module A. If we fix an element g 2 G of order p and a generator s 2 Z=p, the
bijection ~ sends [u] 2 H1(Z=p; A) to () of the cyclic group of order p
if [u] 2 H1(Z=p; A) is representedPby the element u in the kernel of the second
differential A ! A, a 7! p-1i=0si. a. Hence we obtain an exact sequence
Y Y
0 ! Ainv(G) ! Ainv({1}) x Ainv(Z=p) ! Ainv({1}) ! 0
H1(Z=p;A) H1(Z=p;A)
This gives an isomorphism
Y
gAinv(G) ~=-! eA(Z=p).
H1(Z=p;A)
24
3.2 Character Theory and Burnside Congruences for the
Inverse-Limit-Version
Next we define a character map for infinite groups G and determine its image
generalizing Lemma 1.3.
Let ccsf(G) be the set of conjugacy classes (H) of finite subgroups H G.
Given a group homomorphism f :G0 ! G1, let ccsf(G0) ! ccsf(G1) be the
map sending the G0-conjugacy class of a finite subgroup H G0 to the G1-
conjugacy class of f(H) G1. We obtain a covariant functor
ccs:Sub FIN(G) ! SETS, H 7! ccs(H).
For each finite subgroup H G the inclusion H ! G induces a map ccs(H) !
ccsf(G) sending (K) to (K). These fit together to a bijection of sets
~=
OEG :colimH2SubFIN(G)ccs(H)-! ccsf(G). (3.7)
One easily checks that OEG is well-defined and surjective. Next we show injec-
tivity. Consider two elements x0 and x1 in the source of OEG with OEG (x0) =
OEG (x1). For i = 0, 1 we can choose an object Hi2 SubFIN (G) and an element
(Ki) 2 ccsf(Hi) such that the structure map of the colimit for the object (Hi)
sends (Ki) to xi. Then OEG (x0) = OEG (x1) means that the subgroups K0 and K1
of G are conjugated in G. Hence we can find g 2 G with gK0g-1 = K1. The
morphism K0 ! H0 induced by the inclusion yields a map ccs(K0) ! ccs(H0)
sending (K0) to (K0). The morphism K0 ! H1 induced by the conjugation
homomorphism c(g): K0 ! H1 yields a map ccs(K0) ! ccs(H1) sending (K0)
to (K1). This implies x0 = x1.
By the universal property of the colimit we obtain an isomorphism of abelian
groups
i j
_G :map colimH2SubFIN(G)ccs(H), Z
~=
-! invlim(H)2SubFIN(G)map(ccs(H); Z). (3.8)
Define the character map
Y
charGinv:Ainv(G)! Z (3.9)
(H)2ccsf(G)
to be the map for which the composition with the isomorphism
Y map(OEG,id) i j
Z = map(ccsf(G), Z) -------! map colimH2SubFIN(G)ccs(H), Z
(H)2ccsf(G)
-_G-!invlim
H2SubFIN(G)map (ccs(H), Z)
25
is the map
invlimH2SubFIN(G)charH
Ainv(G) = invlimH2SubFIN(G)map (ccs(H), Z) -----------------!
invlimH2SubFIN(G)map(ccs(H); Z),
where charH:A(H) ! map (ccs(H), Z) is the map defined in (1.2).
Theorem 3.10Q(Burnside ring congruences for Ainv(G)). Let x be an
element in (H)2ccsf(G)Z. Then:
(i)The character map
Y
charGinv:Ainv(G) ! Z
(H)2ccsf(G)
of (3.9)is injective;
(ii)The element x lies in the image of the character map
Y
charGinv:Ainv(G) ! Z
(H)2ccsf(G)
of (3.9)if and only if for every finite subgroup K G the following
condition C(K) is satisfied: The image of x under the map induced by the
inclusion iK :K ! G
Y Y
(iK )* Z = map(ccs(H), Z) ! Z = map(ccs(K), Z)
(H)2ccsf(G) (L)2ccsf(K)
satisfies the Burnside ring congruences for the finite group K appearing
in Lemma 1.3;
(iii)If K0 K1 G are two subgroups, then condition C(K1) implies condi-
tion C(K0).
Proof. This follows from Lemma 1.3 and the fact that the inverse limit_is left
exact. |__|
Example 3.11. (Finitely many conjugacy classes of finite subgroups).
Suppose that G has only finitely many conjugacy classes of finite subgroups.
Then we conclude from TheoremQ3.10 that the cokernel of the injective character
map charGinv:Ainv(G) ! (H)2ccsf(G)Z is finite. Hence Ainv(G) is a finitely
generated free abelian group of rank | ccsf(G)|.
Example 3.12 (Ainv(Z=p1 )). We have introduced in Example 2.10 the Pr"ufer
group Z=p1 as colimn!1 Z=pn. Each Z=pn represents a finite subgroup and
each finite subgroup arises in this way. Hence ccs(Z=p1Q) is on one-to-one-
correspondence with Z 0 = {n 2 Z | n 0}. Thus x 2 (H)2ccs(Z=p1Z)can
26
be written as a sequence {x(n)} = {x(n) | n 2 Z 0}, where x(n) corresponds
to the value of x at Z=pn.
Consider the finite subgroup Z=pm . Its subgroups are given by Z=pk for
k = 0, 1, 2 . .m.. Then condition C(Z=pm ) reduces to the set of congruences for
each k = 0, 1, 2, . .,.m - 1
X
Gen (C) . x(p-1k(C)) 0 mod pm-k ,
C (Z=pm )=(Z=pk)
where pk: Z=pm ! (Z=pm )=(Z=pk) is the projection. More explicitly, the con-
dition C(Z=pm ) reduces to the set of congruences for each k = 0, 1, 2, . .,.m *
*- 1
m-kX
x(k) + pi-1. (p - 1) . x(k + i) 0 mod pm-k ,
i=1
which can be rewritten as
m-k-1X
pi. (x(k + i) - x(k + i + 1)) 0 mod pm-k .
i=0
One can see that C(Z=pm1 ) implies C(Z=pm0 ) for m0 m1 as predicted by
Theorem 3.10 (iii).
Suppose that x satisfies C(Z=pm ) for m = 0, 1, 2, . ... We want to show
inductively for l = 0, 1, 2 . .t.hat x(j) x(j+1) mod plholds for j = 0, 1, 2*
*, . ...
The induction begin l = 0 is trivial, the induction step from l - 1 to l 1 do*
*ne
as follows. The m-th equation appearing in condition C(l + m) yields
Xl-1
pi. (x(m + i) - x(m + i + 1)) 0 mod pl.
i=0
Since by induction hypothesis x(k + i) - x(k + i + 1) 0 mod pl-1 holds, this
reduces to
x(m) - x(m + 1) 0 mod pl.
This finishes the induction step.
Since x(j) x(j + 1) mod pl holds for l = 0, 1, 2, . .,.we conclude x(j) =
x(j + 1) for j = 0, 1, 2, . ...On the other hand, if x(j) = x(j + 1) holds for *
*j =
0, 1, 2, . .,.then x obviously satisfies the conditions C(Z=pm ) for m = 0, 1, *
*2, . ...
Theorem 3.10 (i) and (ii) shows that the character map
1 1 Y
charZ=pinv:Ainv(Z=p ) ! Z
(H)2ccsf(Z=p1 )
is injective and its image consists of the copy of the integers given by the co*
*n-
stant series. This implies that the projection pr:Z=p1 ! {1} induces a ring
isomorphism ~
Ainv(pr): Ainv({1}) = Z -=! Ainv(Z=p1 ).
27
In particular we conclude from Example 2.10 that the canonical ring homomor-
phism 1 __ ~
T Z=p :A(Z=p1 ) =-!Ainv(Z=p1 )
of (3.2)is bijective.
3.3 The Inverse Limit Version of the Burnside Ring and
Rational Representations
Analogously to Ainv(G) one defines Rinv,F(G) for a field F to be the commuta-
tive ring with unit given by the inverse limit of the contravariant functor
Rinv,F(?): SubFIN(G) ! RINGS , H 7! RF (H).
This functor has been studied for F = C for instance in [1], [2]. The system
of maps P H: A(H) ! RQ(H) for the finite subgroups H G appearing in
Theorem 1.20 defines a ring homomorphism
PiGnv:Ainv(G) ! Rinv,Q(G). (3.13)
The system of the restriction maps for every finite subgroup H G induces a
homomorphism
SG,F :Sw f(G; F )! Rinv,F(G). (3.14)
Although each of the maps P H for the finite subgroups H G are rationally
surjective by Theorem 1.20, the map Pinvneed not to be rationally surjective in
general, since inverse limits do not respects surjectivity or rationally surjec*
*tivity
in general.
Example 3.15 (Rinv;Q(Z=1 )). Since every finite subgroup of Z=p1 is cyclic,
we conclude from Theorem 1.20 that the map
1 1 ~= 1
PiZ=pnv:Ainv(Z=p ) -! Rinv,Q(Z=p )
is bijective. We have already seen in Example 3.12 that p*: Ainv({1}) !
Ainv(Z=p1 ) is bijective. We conclude from Example 2.10 that the following
diagram is commutative and consists of isomorphisms
__ TZ=p1
A(Z=p1 ) ----!~= Ainv(Z=p1 )
? ?
__PZ=p1?y~= PZ=p1inv?y~=
SZ=p1i;Qnv 1
Swf(Z=p1 ; Q)-----!~=Rinv,Q(Z=p )
and is isomorphic by the maps induced by the projection p: Z=p1 ! {1} to
the following commutative diagram whose corners are all isomorphic to Z and
28
whose arrows are all the identity under this identification.
__ T{1}
A({1}) ----!~=Ainv({1})
? ?
__P{1}?y~= P{1}inv?y~=
S{1};Qinv
Sw f({1}; Q)----!~=Rinv,Q({1})
Example 3.16 (Sw f(Z=p1 ; C) and Rinv;C(Z=1 )). On the other hand let us
consider C as coefficients. Consider the canonical map
1 ,C f 1 1
SZ=p : Sw (Z=p ; C) ! Rinv,C(Z=p )
which is induced by the restriction maps for all inclusions H ! G of finite sub-
groups. If OEH :Rinv,C(Z=p1 ) ! RC(H) is the structure map of the inverse limit
defining Rinv,C(Z=p1 ) for the finite subgroup H Z=p1 , then the composition
Z=p1 ,C _H
Sw f(Z=p1 ; C) S-----!Rinv,C(Z=p1 ) --! RC(H)
is the map given by restriction with the inclusion of the finite subgroup H in
Z=p1 . We claim that this composition is surjective. Choose n with H = Z=pn.
We have to find for every 1-dimensional complex Z=pn-representation V a 1-
dimensional complex Z=p1 -representation W such that_V is the restriction of
V . If V is given by the homomorphism Z=pn ! S1, k7! exp(2ssik=pn), then
the desired W is given by the homomorphism
Z=p1 = Z[1=p]=Z ! S1, k 7! exp(2ssik).
This implies that both Sw f(Z=p1 ; C) and Rinv,C(Z=p1 ) have infinite rank as
abelian groups.
4 The Covariant Burnside Group
Next we give a third version for infinite groups which however will only be an
abelian group, not necessarily a ring.
Definition 4.1 (Covariant Burnside group). Define the covariant Burnside
group A_(G) of a group G to be the Grothendieck group which is associated to the
abelian monoid under disjoint union of G-isomorphism classes of proper cofinite
G-sets S, i.e. G-sets S for which the isotropy group of each element in S and
the quotient G\S are finite.
The cartesian product of two proper cofinite G-sets with the diagonal action
is proper but not cofinite unless G is finite. So for infinite group G we do no*
*t get
a ring structure on the Burnside group A_(G). If G is finite the underlying abe*
*lian
group of the Burnside ring A(G) is just A_(G). Given a group homomorphism
29
f :G0 ! G1, induction yields a homomorphism of abelian group A_(G0) !
A_(G1) sending [S] to [G1 xf S]. Thus A_ becomes a covariant functor from
GROUPS to Z - MODULES .
In the sequel we denote by R[S] for a commutative ring R and a set S the
free R-module with the set S as R-basis. We obtain an isomorphism of abelian
groups
~=
fiG :Z[ccsf(G)]-! A_(G), (H) 7! [G=H]. (4.2)
The elementary proof of the following lemma is left to the reader.
Lemma 4.3. Let H and K be subgroups of G. Then
(i)G=HK = {gH | g-1Kg H};
(ii)The map
OE: G=HK ! ccs(H), gH 7! g-1Kg
induces an injection
WG K\(G=HK ) ! ccs(H);
(iii)The WG K-isotropy group of gH 2 G=HK is (gHg-1 \ NG K)=K;
(iv)If H is finite, then G=HK is a finite union of WG K-orbits of the_shape
WG K=L for finite subgroups L WG K. |__|
The next definition make sense because of the Lemma 4.3 above.
Definition 4.4 (L2-character map). Define for a finite subgroup K G the
L2-character map at (K)
Xr
char_GK:A_(G) ! Q, [S] 7! |Li|-1
i=1
if WG K=L1, WG K=L2,. . . , WG K=Lr are the WG K-orbits of SK . Define the
global L2-character map by
X
char_G:A_(G) ! Q[ccsf(G)], [S] 7! char_GK([S]) . (K).
(K)2ccsf(G)
Notice that one gets from Lemma 4.3 the following explicit formula for the
value of char_GK(G=H). Namely, define
LK (H) := {(L) 2 ccs(H) | L conjugate toK inG}.
For (L) 2 LK (H) choose L 2 (L) and g 2 G with g-1Kg = L. Then
g(H \ NG L)g-1 = gHg-1 \ NG K;
|(gHg-1 \ NG K)=K|-1 = ___|K|____|H.\ N
G L|
30
This implies
X |K|
char_GK(G=H) = __________. (4.5)
(L)2LK (H)|H \ NG L|
Remark 4.6 (Burnside integrality relations). Let T ccsf(G) be a finite
subset closed under taking subgroups, i.e. if (H) 2 T , then (K) 2 T for every
subgroup K H. Since a finite subgroup contains only finitely many subgroups,
one can write ccsf(G) as the union of such subsets T . The union of two such
subsets is again such a subset. So R[ccsf(G)] is the colimit of the finitely
generated free R-modules R[T ] if T runs to the finite subsets of ccsf(G) closed
under taking subgroups.
Fix a subset T of ccsf(G) closed under taking subgroups. One easily checks
using Lemma 4.3 that the composition
G char_G
Z[ccsf(G)] fi--!A_(G) ----! Q[ccsf(G)]
maps Z[T ] to Q[T ]. We numerate the elements in T by (H1), (H2), . .,.(Hr)
such that Hiis subconjugated to (Hj) only if i j holds. Then the composition
G ZQ char_G
Q[ccsf(G)] fi-----!A_(G) Z Q ----! Q[ccsf(G)]
induces a Q-homomorphism
Q[T ] AT--!Q[T ]
given with respect to the basis {(Hi) | i = 1, 2, . .r.} by a matrix A which is
triangular and all whose diagonal entries are equal to 1. The explicit values of
the entries in AT are given by (4.5). The matrix AT is invertible and one can
actually write down an explicit formula for its inverse matrix BT in terms of
M"obius inversion [3, Chaper IV]. The matrix BT yields an isomorphism
~=
BT :Q[T ] -! Q[T ].
Given an element x 2 Q[ccsf(G)], we can find a finite subset T ccsf(G)
closed under taking subgroups such that x lies already in Q[T ]. Then x lies in
the image of the injective L2-character map
char_G:A_(G) ! Q[ccsf(G)]
of Definition 4.4 if and only if
~=
BT :Q[T ] -! Q[T ]
maps x to an element in Z[T ]. This means that the following rational numbers
Xr
BT(i, j) . x(j)
j=1
31
for i = 1, 2 . .,.r are integers, where BT(i, j) and x(j) are the components of
BT and x belonging to (i, j) and j. We call the condition that these rational
numbers are integral numbers the Burnside integrality relations.
Now suppose that G is finite. Then the global L2-character of Definition 4.4
is related to the classical character map (1.2)by the factors |WK|-1, i.e. we
have for each subgroup K of G and any finite G-set S
chGK(S) = |WK|-1 . |SK |. (4.7)
One easily checks that for finite G under the identification (4.7) the Burnside
integrality relations can be reformulated as a set of congruences, which con-
sists of one congruence modulo |WG H| for every subgroup H G (compare
Subsection 1.1).
4.1 Relation to L2-Euler characteristic and Universal Prop-
erty of the Covariant Burnside Group
The Burnside group A_(G) can be characterized as the universal additive in-
variant for finite proper G-CW -complexes and the universal equivariant Eu-
ler characteristic of a finite proper G-CW -complex is mapped to the L2-Euler
characteristics of the WG H-CW -complexes XH by the character map at (H)
for every finite subgroup H G. In particular it is interesting to investigate
the universal equivariant Euler characteristic of the classifying space for pro*
*per
G-actions E_G provided that there is a finite G-CW -model for E_G. All this is
explained in [26, Section 6.6.2].
The relation of the universal equivariant Euler characteristic to the equi-
variant Euler class which is by definition the class of the Euler operator on a
cocompact proper smooth G-manifold with G-invariant Riemannian metric in
equivariant K-homology defined by Kasparov is analyzed in [31]. Equivariant
Lefschetz classes for G-maps of finite proper G-CW -complexes are studied in
[32].
4.2 The Covariant Burnside Group and the Colimit Ver-
sion of the Burnside Ring Agree
Instead of the inverse-limit-version one may also consider the colimit-version
Acov(G) := colimH2SubFIN(G)A(H)
where we consider A as a covariant functor from SubFIN (G) to the category of
Z-modules by induction.
Theorem 4.8 (Acov(G) and A_(G) agree). There obvious map induced by the
various inclusions of a finite subgroup H G
~=
V G:Acov(G) -! A_(G)
is an bijection of abelian groups.
32
Proof. Recall that A_(G) is the free abelian group with the set ccsf(G) of con-
jugacy classes of finite subgroups as basis. Now the claim follows from_the
bijection (3.7). |__|
The analogue for the representation ring is an open conjecture. Namely if
we define for a field F of characteristic zero
Rcov,F(G) := colimH2SubFIN(G)RF (H)
we can consider
Conjecture 4.9. The obvious map
W G,F:Rcov,F(G) ! K0(F [G])
is an bijection of abelian groups.
This conjecture follows from the Farrell-Jones Conjecture for algebraic K-
theory for F [G] as explained in [30, Conjecture 3.3]. No counterexamples are
known at the time of writing. For a status report about the Farrell-Jones Con-
jecture we refer for instance to [30, Section 5].
Let
PcGov:Acov(G) ! Rcov,F(G). (4.10)
be the map induced by the maps P H:A(H) ! RF (H), [S] 7! [F [S]] for the
various finite subgroups H G.
4.3 The Covariant Burnside Group and the Projective
Class Group
Given a finite proper G-set, the Q-module Q[S] with the set S as basis becomes a
finitely generated projective QG-module by the G-action on S. Thus we obtain
a homomorphism
P_G:A_(G) ! K0(QG). (4.11)
Conjecture 4.12. The map P_G:A_(G) ! K0(QG) is rationally surjective.
This conjecture is motivated by the fact that it is implied by Theorem 1.20
and Theorem 4.8 together with Conjecture 4.9.
4.4 The Covariant Burnside Group as Module over the
Finite-Set-Version
If S is a finite G-set and T is a cofinite proper G-set, then their cartesian p*
*roduct
with the diagonal G-action is a cofinite proper G-set. Thus we obtain a pairing
__
~GA:A(G) x A_(G) ! A_(G), ([S], [T ]) 7! [S x T ] (4.13)
33
Analogously one defines a pairing
~GK: Swf(G; Q) x K0(QG) ! K0(QG), ([M], [P ]) 7! [M Q P ](4.14)
which turns K0(QG) into a Swf(G; Q)-module. These two pairings are compat-
ible in the obvious sense (see (0.1)).
4.5 A Pairing between the Inverse-Limit-Version and the
Covariant Burnside Group
Given a finite group H, we obtain a homomorphism of abelian groups
H :A(H) ! hom Z(A(H), Z), [S] 7! H (S),
where H (S): A(H) ! Z maps [T ] to |G\(S x T )| for the diagonal G-operation
on S x T . A group homomorphism f :H ! K induces a homomorphism of
abelian groups resOE:A(K) ! A(H) by restriction and a homomorphism of
abelian groups indOE:A(H) ! A(K) by induction. The latter induces a homo-
morphism of abelian groups (indOE)*: A(K) ! A(H). One easily checks that the
collection of the homomorphisms H for the subgroups H G induces a natu-
ral transformation of the contravariant functors from SubFIN (G) to Z-modules
given by A(?) and hom Z(A(?), Z). Passing to the inverse limit, the canonical
isomorphism of abelian groups
i j ~
hom Z colimH2SubFIN(G)A(H), Z -=! invlimH2SubFIN(G)homZ(A(H), Z)
and the isomorphism appearing in Theorem 4.8 yield a homomorphism of abelian
groups
GA:Ainv(G) ! hom Z(A_(G); Z)
which we can also write a bilinear pairing
GA:Ainv(G) x A_(G)! Z. (4.15)
For a field F of characteristic zero, there is an analogous pairing
GR:Rinv,F(G) x Rcov,F(G)! Z (4.16)
which comes from the various homomorphisms of abelian groups for each finite
subgroup H G
RF (H) ! hom Z(RF (H); Z), [V ] 7! ([W ] 7! dimF(F FG (V F W ))).
The pairings GAand GRare compatible in_the obvious sense (see (0.1)).
The homomorphism of abelian groups GR:Rinv,F(G) ! hom Z(Rcov,F(G), Z)
associated to the pairing GRis injective. Its cokernel is finite if G has only*
* finitely
many conjugacy classes of finite subgroups. It is rationally surjective if ther*
*e is
an upper bound on the orders of finite subgroups of G.
Define the homomorphism QGA:A_(G) ! Z and QGK:K0(QG) ! Z respec-
tively by sending [S] to |G\S| and [P ] to dimQ(Q QG P ) respectively. Then
the pairings ~GA, GA, ~GKand GRare compatible in the obvious sense (see (0.1)*
*).
34
4.6 Some Computations of the Covariant Burnside Group
Example 4.17 (A_(G) for torsionfree G). Suppose that G is torsionfree.
Then the inclusion i: {1} ! G induces a Z-isomorphism
~=
A_(i): A_({1}) = Z -! A_(G).
Example 4.18 (Extensions of Zn with Z=p as quotient). Suppose that G
can be written as an extension 1 ! A ! G ! Z=p ! 1 for some fixed prime
number p and for A = Zn for some integer n 0 and that G is not torsionfree.
We use the notation of Example 3.6 in the sequel. We obtain an exact sequence
M M
0 ! A_({1}) ! A_({1}) A_(Z=p) ! A_(G) ! 0
H1(Z=p;A) H1(Z=p;A)
If we define eA_(G) as the kernel of A_(G) ! A_({1}), we obtain an isomorphism
M ~
eA(Z=p) =-!eA_(G).
H1(Z=p;A)
Let H0 be the trivial subgroup and H1, H2, . .,.Hr be a complete set of represe*
*n-
tatives of the conjugacy classes of finite subgroups. Then r = |H1(Z=p; A)| and
A_(G) is the free abelian group of rank r + 1 with {[G=H0], [G=H1], . .[.G=Hr]}
as basis. Each Hi is isomorphic to Z=p. We compute using (4.5)
chGH0(G=H0) = 1;
chGH0(G=Hj) = 1_p j = 1, 2, . .,.r;
chGHi(G=Hj) = 1 i = j, i, j = 1, 2, . .,.r;
chGHi(G=Hj) = 0 i 6= j, i, j = 1, 2, . .,.r.
The Burnside integralityLconditions (see Remark 4.6) become in this case for
x = (x(i)) 2 ri=0Q.
Xr
x(0) - 1_p. x(i)2 Z;
i=1
x(i) 2 Z i = 1, 2, . .,.r.
Example 4.19 (Groups with appropriate maximal finite subgroups).
Consider the groups appearing in Example 3.4. In the notation of Example 3.4
we get an isomorphism of Z-modules
M L eA_(ji)
eA(Mi) --i2I-----!eA_(G).
i2I
5 Equivariant Cohomology Theories
In this section we recall the axioms of a (proper) equivariant cohomology theory
of [27]. They are dual to the ones of a (proper) equivariant homology theory as
described in [25, Section 1].
35
5.1 Axiomatic Description of a G-Cohomology Theory
Fix a group G and an commutative ring R. A G-CW -pair (X, A) is a pair
of G-CW -complexes. Recall that a G-CW -complex X is proper if and only
if all isotropy groups of X are finite, and is finite if X is obtained from A by
attaching finitely many equivariant cells, or, equivalently, if G\X is compact.*
* A
G-cohomology theory H*Gwith values in R-modules is a collection of covariant
functors HnGfrom the category of G-CW -pairs to the category of R-modules
indexed by n 2 Z together with natural transformations ffinG(X, A): HnG(X, A) !
Hn+1G(A) := Hn+1G(A, ;) for n 2 Z such that the following axioms are satisfied:
o G-homotopy invariance
If f0 and f1 are G-homotopic maps (X, A) ! (Y, B) of G-CW -pairs, then
HnG(f0) = HnG(f1) for n 2 Z;
o Long exact sequence of a pair
Given a pair (X, A) of G-CW -complexes, there is a long exact sequence
ffin-1Gn HnG(j) n HnG(i) n ffinG
. .-.--!HG (X, A) ----! HG (X) ----! HG (A) --!. .,.
where i: A ! X and j :X ! (X, A) are the inclusions;
o Excision
Let (X, A) be a G-CW -pair and let f :A ! B be a cellular G-map of
G-CW -complexes. Equip (X [f B, B) with the induced structure of a G-
CW -pair. Then the canonical map (F, f): (X, A) ! (X [f B, B) induces
an isomorphism
~= n
HnG(F, f): HnG(X, A) -! HG (X [f B, B);
o Disjoint union axiom
Let`{Xi | i 2 I} be a family of G-CW -complexes. Denote by ji:Xi !
i2IXi the canonical inclusion. Then the map
_ !
Y a ~= Y
HnG(ji): HnG Xi -! HnG(Xi)
i2I i2I i2I
is bijective.
If H*Gis defined or considered only for proper G-CW -pairs (X, A), we call
it a proper G-cohomology theory H*Gwith values in R-modules.
5.2 Axiomatic Description of an Equivariant Cohomology
Theory
Let ff: H ! G be a group homomorphism. Given an H-space X, define the
induction of X with ff to be the G-space indffX which is the quotient of G x X
36
by the right H-action (g, x) . h := (gff(h), h-1x) for h 2 H and (g, x) 2 G x X.
If ff: H ! G is an inclusion, we also write indGHinstead of indff.
A (proper) equivariant cohomology theory H*?with values in R-modules con-
sists of a collection of (proper) G-cohomology theories H*Gwith values in R-
modules for each group G together with the following so called induction struc-
ture: given a group homomorphism ff: H ! G and a (proper) H-CW -pair
(X, A) there are for each n 2 Z natural homomorphisms
indff:HnG(indff(X, A))! HnH(X, A) (5.1)
satisfying
(i)Bijectivity
If ker(ff) acts freely on X, then indff:HnG(indff(X, A)) ! HnH(X, A) is
bijective for all n 2 Z;
(ii)Compatibility with the boundary homomorphisms
ffinHO indff= indffOffinG;
(iii)Functoriality
Let fi :G ! K be another group homomorphism. Then we have for n 2 Z
indfiOff= indffO indfiOHnK(f1): HnH(indfiOff(X, A)) ! HnK(X, A),
~=
where f1: indfiindff(X, A) -! indfiOff(X, A), (k, g, x) 7! (kfi(g), x) is *
*the
natural K-homeomorphism;
(iv)Compatibility with conjugation
For n 2 Z, g 2 G and a (proper) G-CW -pair (X, A) the homomorphism
indc(g):G!G: HnG(indc(g):G!G(X, A)) ! HnG(X, A) agrees with HnG(f2),
where f2 is the G-homeomorphism f2: (X, A) ! indc(g):G!G(X, A), x 7!
(1, g-1x) and c(g)(g0) = gg0g-1.
This induction structure links the various G-cohomology theories for differe*
*nt
groups G.
Sometimes we will need the following lemma whose elementary proof is anal-
ogous to the one in [25, Lemma 1.2].
Lemma 5.2. Consider finite subgroups H, K G and an element g 2 G
with gHg-1 K. Let Rg-1: G=H ! G=K be the G-map sending g0H to
g0g-1K and c(g): H ! K be the homomorphism sending h to ghg-1. Let
pr :(indc(g):H!K{o}) ! {o} be the projection to the one-point space {o}. Then
the following diagram commutes
HnG(Rg-1)
HnG(G=K) -------! HnG(G=H)
? ?
indGK?y~= indGH?y~=
indc(g)OHnK(pr)
HnK({o}) ----------! HnH({o})
37
5.3 Multiplicative Structures
Let H*Gbe a (proper) G-cohomology theory. A multiplicative structure assigns
to a (proper) G-CW -complex X with G-CW -subcomplexes A, B X natural
R-homomorphisms
[: HmG(X, A) R HnG(X, B) ! Hm+nG(X, A [ B). (5.3)
This product is required to be compatible with the boundary homomorphism
of the long exact sequence of a G-CW -pair, namely, for u 2 HmG(A) and v 2
HnG(X) and i: A ! X the inclusion we have ffi(u [ v) = ffi(u [ Hn(i)(v).
Moreover, it is required to be graded commutative, to be associative and to
have a unit 1 2 H0G(X) for every (proper) G-CW -complex X.
Let H*?be a (proper) equivariant cohomology theory. A multiplicative struc-
ture on it assigns a multiplicative structure to the associated (proper) G-coho-
mology theory H*Gfor every group G such that for each group homomorphism
ff: H ! G the maps given by the induction structure of (5.1)
~= n
indff:HnG(indff(X, A))-! HH (X, A)
are in the obvious way compatible with the multiplicative structures on H*Gand
H*H.
Example 5.4. Equivariant cohomology theories coming from non-equi-
variant ones). Let K* be a (non-equivariant) cohomology theory with mul-
tiplicative structure, for instance singular cohomology or topological K-theory.
We can assign to it an equivariant cohomology theory with multiplicative struc-
ture H*?in two ways. Namely, for a group G and a pair of G-CW -complexes
(X, A) we define HnG(X, A) by Kn(G\(X, A)) or by Kn(EG xG (X, A)).
5.4 Equivariant Topological K-Theory
In [29] equivariant topological K-theory is defined for finite proper equivaria*
*nt
CW -complexes in terms of equivariant vector bundles. For finite G it reduces to
the classical notion which appears for instance in [4]. Its relation to equivar*
*iant
KK-theory is explained in [38]. This definition is extended to (not necessarily
finite) proper equivariant CW -complexes in [29] in terms of equivariant spectra
using -spaces. This equivariant cohomology theory K*?has the property that
for any finite subgroup H of a group G we have
ae
KnG(G=H) = KnH({o}) = RC(H){0} nneven;odd.
6 Equivariant Stable Cohomotopy in Terms of
Real Vector Bundles
In this section we give a construction of equivariant cohomotopy for finite pro*
*per
G-CW -complexes for infinite groups G in terms of real vector bundles and
38
maps between the associated sphere bundles. The result will be an equivariant
cohomology theory with multiplicative structure for finite proper equivariant
CW -complexes. It generalizes the well-known approach for finite groups in
terms of representations. We will first give the construction, show why it redu*
*ces
to the classical construction for a finite group and explain why we need to
consider equivariant vector bundles and not only representations in the case of
an infinite group.
6.1 Preliminaries about Equivariant Vector Bundles
We will need the following notation. Given a finite-dimensional (real) vector
space V , we denote by SV its one-point compactification. We will use the point
at infinity as the base point of SV in the sequel. Given two finite-dimensional
vector spaces V and W , the obvious inclusion V W ! SV ^ SW induces a
natural homeomorphism
~= V W
OE(V, W ): SV W-! S ^ S . (6.1)
Let
r: SR ! SR _ SR. (6.2)
be the pinching map, which sends x > 0 to ln(x) 2 R SR in the first summand,
x < 0 to - ln(-x) 2 R SR in the second summand and 0 and 1 to the base
point in SR _SR. This is under the identification SR = S1 the standard pinching
map S1 ! S1=S0 ~=S1 _ S1, at least up to pointed homotopy.
We need some basics about G-vector bundles over proper G-CW -complexes.
More details can be found for instance in [29, Section 1]. Recall that a G-CW -
complex is proper if and only if all its isotropy groups are finite. A G-vector
bundle , :E ! X over X is a real vector bundle with a G-action on E such that
, is G-equivariant and for each g 2 G the map lg: E ! E is fiberwise a linear
isomorphism. Such a G-vector bundle is automatically trivial in the equivariant
sense that for each x 2 X there is a G-neighborhood U, a G-map f :U ! G=H
and a H-representation V such that ,|U is isomorphic as G-vector bundle to the
pullback of the G-vector bundle G xH V ! G=H by f. Denote the fiber ,-1(x)
over a point x 2 X by Ex. This is a representation_of the finite isotropy group*
* Gx
of x 2 X. A map of G-vector_bundles (f, f) from ,0: E0_!_X0 to ,1: E1 ! X1
consists_of G-maps f:E0 ! E1 and f :X0 ! X1 with ,1 O f= f O ,0 such that
f is fiberwise a (not necessarily injective or surjective) linear map.
Given a G-vector bundle , :E ! X, let S,: SE ! X be the locally trivial G-
bundle whose fiber over x 2 X is SEx. Consider two G-vector bundles , :E ! X
and ~: F ! X. Let S, ^X S~ :SE ^X SF ! X be the G-bundle whose fiber
of x 2 X is SEx ^ SFx, in other word it is obtained from S,: SE ! X and
S~ :SF ! X by the fiberwise smash product. Define S, _X S~ analogously.
From (6.1)we obtain a natural G-bundle isomorphism
~= , ~
OE(,, ~): S, ~-! S ^X S . (6.3)
39
The next basic lemma is proved in [29, Lemma 3.7].
Lemma 6.4. Let f :X ! Y be a G-map between finite proper G-CW -complexes
and , a G-vector bundle over X. Then there is a G-vector bundle ~ over Y such
that , is a direct summand in f*~.
6.2 The Definition of Equivariant Cohomotopy Groups
Fix a proper G-CW -complex X. Let SPHB G (X) be the following category.
Objects are G-CW -vector bundles , :E ! X over X. A morphism from , :E !
X to ~: F ! X is a bundle map u: S, ! S~ covering the identity id:X ! X
which fiberwise preserve the base points. (We do not require that u is fiberwise
a homotopy equivalence.)
A homotopy h between two morphisms u0, u1 from , :E ! X to ~: F ! X
is a bundle map h: S, x [0, 1] ! S~ from the bundle S, x id[0,1]:SE x [0, 1] !
Xx[0, 1] to S~ which covers the projection Xx[0, 1] ! X and fiberwise preserve
the base points such that its restriction to X x {i} is ui for i = 0, 1.
Let Rk_be the trivial vector bundle X x Rk ! X. We consider it as a G-
vector bundle using the trivial G-action on Rk. Fix an integer n 2 Z. Given
two objects ,i, two non-negative integers kiwith ki+ n 0 and two morphisms
ki , Rki+n
ui:S,i R__! S i _____
for i = 0, 1, we call u0 and u1 equivalent, if there are objects ~i in SPHB G(X)
~=
for i = 0, 1 and an isomorphism of G-vector bundles v :~0 ,0 -! ~1 ,1 such
that the following diagram in SPHB G(X) commutes up to homotopy
k1 , Rk0 id^X u0 ~ Rk1 , Rk0+n
S~0_R__^X S 0 ___ -----! S_0____^X S 0 _____
? ?
oe1?y oe2?y
S~0 ,0 Rk0+k1_ S~0 ,0 Rk0+k1+n_
?? ??
Sv idy Sv idy
S~1 ,1 Rk0+k1_ S~1 ,1 Rk0+k1+n_
? ?
oe3?y oe4?y
S~1 Rk0_^X S,1 Rk1_id^X-u1----!S~1 Rk0_^X S,1 Rk1+n_
where oei stands for the obvious isomorphism coming from (6.3)and permuta-
tion.
We define ssnG(X) to be the set of equivalence classes of such morphisms
u: S, Rk_! S, Rk+n_under the equivalence relation mentioned above. It be-
comes an abelian group as follows. k
Thekzero+elementnis represented by the class of any morphism c: S, R_ !
S, R___ which is fiberwise the constant map onto the base point.
40
Consider classes [u0] and [u1] represented by two morphisms of the shape
ui:S,i Rki_! S,i Rki+n_for i = 0, 1. Define their sum by the class represented
by the morphism
k0+k1+1oe1 , Rk0 , Rk1 R
S,0 ,1 R______-! S 0 ___^X S 1 ___^X S__
-id^X-id^X-r_-----!S,0 Rk0_^ ,1 Rk1_ R_ R_
X S ^X S _X S
o-!iS,0 Rk0_^ ,1 Rk1_ R_j i ,0 Rk0_ ,1 Rk1_ R_j
X S ^X S _X S ^X S ^X S
-(u0^X-id^X_id)X(id^X-u1^X-id)-----------------!
i k0+n k1 j i k0 k1+n j
S,0 R____^X S,1 R__^X SR_ _X S,0 R__^X S,1 R____^X SR_
-oe5_X-oe6---!iS,0 ,1 Rk0+k1+1+n_j_ i ,0 ,1 Rk0+k1+1+n_j
X S
-id_X-id---!S,0 ,1 Rk0+k1+1+n_
where the isomorphisms oeiare given by permutation and the isomorphisms (6.3)
o is given by the distributivity law for smash and wedge-products and r_ is
defined fiberwise by the map r of (6.2).
Consider a class [u] represented by the morphisms of the shape u: S, Rk_!
S, Rk+n_. Define its inverse as the class represented by the composition
k+1 oe1 , Rk R u^X -_id_, Rk+n R oe2 , Rk+1+n
S, R___ -! S __ ^X S__-----! S ____ ^X S__-! S ______,
where -_id_is fiberwise the map - id:R ! R. The proof that this defines the
structure of an abelian group is essentially the same as the one that the abeli*
*an
group structure on the stable homotopy groups of a space is well-defined.
Next consider a pair (X, A) of proper G-CW -complexes. In order to define
the abelian group ssnG(X, A) we consider morphisms u: S, Rk_! S, Rk+n_with
k + n 0 in SPHB G(X) such that u is trivial over A, i.e. for every point a 2 A
the map ua: S,a Rk ! S,a Rk+n is the constant map onto the base point.
In the definition of the equivalence relation for such pairs we require that the
homotopies of two morphisms are stationary over A. Then define ssnG(X, A) as
the set of equivalence classes of morphism u: S, Rk_! S, Rk+n_in SPHB G(X)
with k + n 0 which are trivial over A using this equivalence relation. The
definition of the abelian group structure goes through word by word.
Notice that in the definition of ssnG(X, A) we cannot use as in the classical
settings cones or suspensions since these contain G-fixed points and hence are
not proper unless G is finite. The properness is needed to ensure that certain
basic facts about bundles carry over to the equivariant setting.
6.3 The Proof of the Axioms of an Equivariant Cohomol-
ogy Theory with Multiplicative Structure
In this subsection we want to prove
41
Theorem 6.5 (Equivariant cohomotopy in terms of equivariant vec-
tor bundles). Equivariant Cohomotopy ss*?defines an equivariant cohomology
theory with multiplicative structure for finite proper equivariant CW -complexe*
*s.
For every finite subgroup H of the group G the abelian groups ssnG(G=H) and
ssnH are isomorphic for every n 2 Z and the rings ss0G(G=H) and ss0H = A(H)
are isomorphic.
Consider a G-map f :(X, A) ! (Y, B) of pairs of proper G-CW -complexes.
Using the pullback construction one defines a homomorphism of abelian groups
ssnG(f): ssnG(Y, B) ! ssGn(X, A).
Thus ssnGbecomes a contravariant functor from the category of proper G-CW -
pairs to the category of abelian groups.
Lemma 6.6. Let f0, f1: (X, A) ! (Y, B) be two G-maps of pairs of proper
G-CW -complexes. If they are G-homotopic, then ssnG(f0) = ssnG(f1).
Proof. By the naturality of ssnGit suffices to prove that ssnG(h) = idholds for*
* the
G-map
h: (X, A) x [0, 1] ! (X, A) x [0, 1], (x, t) 7! (x, 0).
Let the element [u] 2 ssnG((X, A) x [0, 1]) be given by the morphism u: S, Rk_!
S, Rk+n_in SPHB G(X x [0, 1]) with k + n 0 which is trivial over X x {0, 1} [
A x [0, 1]. By [29, Theorem 1.2] there is an isomorphism of G-vector bundles
~=
v :, -! h*, covering the identity id:X x[0, 1] ! X x[0, 1] such that v restrict*
*ed
to Xx{0} is the identity id:,|Xx{0} ! ,|Xx{0}. The composition of morphisms
in SPHB G(X x [0, 1])
*, Rk Sv-1 id , Rk u , Rk+n Sv id h*, Rk+n
u0:Sh __ -----! S __ -! S ____ ----! S ____
has the property that its restriction to X x {0} agrees with the restriction of
h*u to X x {0}. Hence this composite u0 is homotopic to the morphism h*u
itself. Namely, if we write h*, = i*0, x [0, 1] for i0: X ! X x [0, 1], x 7! (x*
*, 0),
then the homotopy is given at time s 2 [0, 1] by the morphism
*, Rk i*, Rk+n 0
Si0 __x [0, 1] ! S 0 ____x [0, 1], (z, t) 7! (prOu (z, st), t)
for pr:Si*0, Rk+n_x [0, 1] ! Si*0, Rk+n_the projection. Obviously this homotopy
is stationary over A. We conclude from the equivalence relation appearing in
the definition of ssnGand the definition of ssnG(h) that ssnG(h)([u])_=_[u0] = *
*[u]
holds. |__|
Next we define the suspension homomorphism
oenG(X, A): ssnG(X,!A) ssn+1G((X, A) x ([0, 1], {0, 1})).(6.7)
For a G-vector bundle , over X let , x [0, 1] be the obvious G-vector bundle
over X x [0, 1], which is the same as the pullback of , for the projection X x
42
[0, 1] ! X. Consider a morphism u: S, ! S~ in SPHB G(X) which is trivial
over A. Let
~ R
oe(u): S,x[0,1]= S, x [0, 1] ! S(~ R_)x[0,1]= S ^X S__x [0, 1]
be the morphism in SPHB G(X x [0, 1]) given by
~ R
(z, t) 2 S, x [0, 1] 7! ((u(z) ^ e(t)), t) 2 S ^X S__x [0, 1],
where e: [0, 1] ! SR_comes from the homeomorphism (0, 1) ! R, t 7! ln(x) -
ln(1 - x). The morphism oe(u) is trivial over X x {0, 1} [ A x [0, 1]. We define
the in (X, A) natural homomorphism of abelian groups oenG(X, A) by sending
the class of u to the class of oe(u).
Lemma 6.8. The homomorphism oenG(X, A) of (6.7)is bijective for all pairs of
proper G-CW -complexes (X, A).
Proof. We want to construct an inverse
on+1G:ssn+1G((X, A) x ([0, 1], {0, 1})) ! ssnG(X, A)
of oenG(X, A). Consider two G-vector bundles , and ~ over X and a morphism
v :S,x[0,1]! S~x[0,1]in SPHB G (X x [0, 1]) which is trivial over A x {0, 1}.
Define a morphism in SPHB G(X) which is trivial over A
o(v): S, R_= S, ^X SR_! S~
by sending (z, (e(t)) 2 S,^X SR_to prOv(z, t) for pr:S~x[0,1]= S~ x[0, 1] ! S~
the projection, e: [0, 1] ! SR_the map defined above and t 2 [0, 1]. Next
consider an element [u] 2 ssn+1G(X, A) represented by a morphism u: S, Rk_!
S, Rk+n+1_in SPHB G(X x [0, 1]) for k + n 0 which is trivial over X x {0, 1} [
~=
A x [0, 1]. Choose an isomorphism of G-vector bundles v :,0x [0, 1] -! , which
covers the identity on X x [0, 1] and is the identity over X x {0},Lwhere ,0 is*
* the
restrictionLofn, to X = Xx{0} (see [29, Theorem 1.2]). Let u0:S(,0 Rk_)x[0,1]!
S(,0 R_)x[0,1]be the composition
L k Sv id L k u k+n+1 Sv-1 id L k+n+1
S(,0 R_)x[0,1]----!S(, R_)-!S, R_____ -----! S(,0 R_____)x[0,1].
Notice that [u] = [u0] holdsLin ssn+1G((X,LA) x ([0, 1], {0, 1})). Define on+1G*
*([u])
by the class of o(u0): S,0 Rk+1_! S,0 Rk+n+1_.
Consider [u] 2 ssnG(X, A) represented by the morphism u: S, Rk_! S, Rk+n_
in SPHB G(X) which is trivial over A. Then on+1GO oenG([u]) is represented by t*
*he
morphism o O oe(u) which can be identified with
k+1 oe1 , Rk R u^X id , L Rk+n R oe2 , Rk+1+n R
S, R___ -! S __ ^X S__----! S ____^X S__-! S ______ _
But the latter morphism represents the same class as u. This shows on+1GO
oenG([u]) = [u] and hence on+1GO oenG= id. The proof of oenG([u]) O on+1G=_idis
analogous. |__|
43
So far we have only assumed that the G-CW -complex X is proper. In the
sequel we will need additionally that it is finite since this condition appears*
* in
Lemma 6.4.
Lemma 6.9. Let (X1, X0) be a pair of finite proper G-CW -complexes and
f :X0 ! X2 be a cellular G-map of finite proper G-CW -complexes. Define the
pair of finite proper G-CW -complexes (X, X2) by the cellular G-pushout
X0 ---f-! X2
?? ?
y ?y
X1 ----!F X
Then the homomorphism
~= n
ssnG(F, f): ssnG(X, X2) -! ssG (X1, X0)
is bijective for all n 2 Z.
Proof. We begin with surjectivity. Consider an element a 2 ssnG(X1, X0) repre-
sented by a morphism k k+n
u: S, R_ ! S, R___
in SPHB G (X1) which is trivial over X0. By Lemma 6.4 there is a G-vector
bundle ~ over X, a~G-vector bundle ,0over X1 and an isomorphism of G-vector
bundles v :, ,0 =-!F *~. Consider the morphism u0 in SPHB G(X1) which is
given by the composition
*~ Rk Sv-1 id , ,0 Rk oe1 ,0 , Rk id^X1u ,0 , Rk+n
SF __ -----! S __-! S ^X1 S __ -----! S ^X1 S ____
oe2-!S, ,0 Rk+n_Sv-id---!SF*~ Rk+n_.
By definition it represents the same element in ssnG(X1, X0) as u. Hence we can
assume without loss of generality for the representative u of a that the bundle
, is of the form F *~ for some G-vector bundle ~ over X. Since the morphism
u is trivial over X0, we can find a morphism SPHB G(X)
__u:S~ Rk_! S~ Rk+n_
which is trivial over X2 and satisfies F *(__u) = u. Hence the morphism __udefi*
*nes
an element in ssnG(X, X1) such that ssnG(F, f)([__u]) = [u] = a holds.
It remains to prove injectivity of ssnG(F, f). Consider an element b in the
kernel of ssnG(F, f). Choose a morphism
k , Rk+n
u: S, R_ ! S ____
44
in SPHB G(X) which is trivial over X2 and represents b. Then F *u: SF*, Rk_!
SF*, Rk+n_represents zero in ssnG(X1, X0). Hence we can find a bundle ~ over
X1 such that the composition
*, Rk oe1 ~ F*, Rk id^X1u ~ F*, Rk+n oe2 ~ F*, Rk+n
S~ F __ -! S ^X1S __ -----! S ^X1S ____ -! S ____
is homotopic relative X0 to the trivial morphism. As in the proof of the sur-
jectivity we can arrange using Lemma 6.4 that ~ is of the shape F *,0 for some
G-vector bundle ,0 over X. By replacing u by idS,0^X u we can achieve that
b = [u] still holds and additionally the morphism in SPHB G(X1)
*(, Rk) F*(, Rk+n)
F *u: SF __ ! S ____
is homotopic relative X0 to the trivial map. Since u is trivial over X2, we can
extend this homotopy trivially from X1 to X to show that u itself is homotopic
relative X2 to the trivial map. But this means b = [u] = 0 in ssnG(X, X2)._This
finishes the proof of Lemma 6.9. |__|
Lemma 6.10. Let A Y X be inclusions of finite proper G-CW -complexes.
Then the sequence
n(j) ssn(i)
ssnG(X, Y ) ssG----!ssnG(X, A) --G-!ssnG(Y, A)
is exact at ssnG(X, A) for all n 2 Z, where i and j denote the obvious inclusio*
*ns.
Proof. The inclusion j O i: (Y, A) ! (X, Y ) induces the zero map ssnG(X, Y ) !
ssGn(Y, A) since an element in a 2 ssnG(X, Y ) is represented by a morphism
u: S, Rk_! S, Rk+n_in SPHB G(X) which is trivial over Y and ssnG(j O i)(a)
is represented by the restriction of u to Y .
Consider an element a 2 ssnG(X, A) which is mapped to zero under ssnG(i).
Choose a morphism u: S, Rk_! S, Rk+n_in SPHB G(X) which is trivial over A
and represents a. Hence we can find a G-vector bundle ,0 over Y such that the
morphism in SPHB G(Y ) given by the composition
0 i*, Rkoe1 ,0 i*, Rk id^Yi*u ,0 i*, Rk+n oe2 ,0 i*, Rk+n
S, __-! S ^Y S __- ----! S ^Y S ____-! S ____
is homotopic to the trivial map relative A.
As in the proof of Lemma 6.9 we can arrange using Lemma 6.4 that ,0is the
of the shape i*~ for some G-vector bundle ~ over X. Hence we can achieve by
replacing u by idS~^X u that a = [u] still holds in ssnG(X, A) and additionally
the morphisms in SPHB G(Y )
*, Rk i*, Rk+n
i*u: Sj __! S ____
is homotopic relative A to the trivial map. One proves inductively over the
number of equivariant cells in X - Y and [29, Theorem 1.2] that this homotopy
45
can be extended tona homotopy relative A of the morphism u to another mor-
phism v :S, ! S, R_ in SPHB G(X) which is trivial over Y . But this implies
that the element [v] 2 ssnG(X, Y ) represented by v is mapped to a = [u]_under
ssnG(i). |__|
In order to define a G-cohomology theory we must construct a connecting
homomorphism for pairs. In the sequel maps denoted by ik are the obvious
inclusions and maps denoted by prk are the obvious projections. Consider a
pair of finite proper G-CW -complexes (X, A) and n 2 Z, n 0. We define the
homomorphism of abelian groups
ffinG(X, A): ssnG(A)!ssn+1G(X, A) (6.11)
to be the composition
n(A)
ssnG(A) oeG----!ssn+1G(A x [0, 1], A x {0, 1})
(ssn+1G(i1))-1n+1
---------! ssG (X [Ax{0} A x [0, 1], X q A x {1})
ssn+1G(i2)n+1
-----! ssG (X [Ax{0} A x [0, 1], A x {1})
(ssn+1G(pr1))-1n+1
---------! ssG (X, A),
where oenG(A) is the suspension isomorphism (see Lemma 6.8), the map ssn+1G(i1)
is bijective by excision (see Lemma 6.9) and ssn+1G(pr1) is bijective by homoto*
*py
invariance (see Lemma 6.6) since pr1is a G-homotopy equivalence of pairs.
Lemma 6.12. Let (X, A) be a pair for proper finite G-CW -complexes. Let
i: A ! X and j :X ! (X, A) be the inclusions. Then the following long
sequence is exact and natural in (X, A):
ffin-1Gn ssnG(j) n ssnG(i)n
. .-.--!ssG (X, A) ----! ssG (X) ---! ssG (A)
-ffinG-!ssn+1 ssn+1G(j)n+1 ssn+1G(i)n+1 ffin-1G
G (X, A) -----! ssG (X) -----! ssG (A) ---! . ...
Proof. It is obviously natural. It remains to prove exactness.
Exactness at ssnG(X) follows from Lemma 6.10.
Exactness at ssnG(X, A) follows from the following commutative diagram
(oenG(A))-1Ossn+1G(i1)
ssn+1G X [Ax{0} A x [0, 1], X q A x {1}-------------!~=ssnG(A)
? ?
ssn+1G(i2)?y ffin+1G(X,A)?y
(ssn+1G(pr1))-1 n+1
ssn+1G X [Ax{0} A x [0, 1], A x {1} ---------!~= ssG (X, A)
? ?
ssn+1G(i3)?y ssn+1G(i)?y
ssn+1G(i4) n+1
ssn+1G(X q A x {1}, A x {1}) ------!~= ssG (X)
46
whose left column is exact at ssn+1G X [Ax{0} A x [0, 1], A x {1}by Lemma 6.10.
Exactness at ssnG(A) is proved analogously by applying Lemma 6.10 to the
inclusions
(X x {0, 1}, X x {0} q A x {1})
(X x [0, 1], X x {0} q A x {1})
(X x [0, 1], X x {0,.1})
|___|
We conclude from Lemma 6.6, Lemma 6.9 and Lemma 6.12 that ss*Gdefines
a G-cohomology theory on the category of finite proper G-CW -complexes.
Consider a finite proper G-CW -complex X with two subcomplexes A, B
X. We want to define a multiplicative structure, i.e. a cup-product
[: ssmG(X, A) x ssnG(X, B)! ssm+nG(X, A [ B). (6.13)
Given elements a 2 ssmG(X, A) and b 2 ssnG(X, B), choose appropriate morphisms
u: S, Rk_! S, Rm+k_and v :Sj Rl_! Sj Rn+l_in SPHB G(X) representing a
and b. Define a [ b by the class of the composition of morphisms in SPHB G(X)
which are trivial over A [ B.
k+l oe1 , Rk j Rl
S, j R___-! S __ ^X S __
u^X-v---!S, Rk+m_^ j Rl+n_oe2 , j Rk+l+m+n_
X S -! S .
Next we deal with the induction structure. Consider a group homomorphism
ff: H ! G. The pullback construction for the ff: H ! G-equivariant map
X ! indffX = G xffX, x 7! 1G xffx defines a functor SPHB G(indffX) !
SPHB H (X) which yields a homomorphism of abelian groups
indff:ssnG(indff(X, A))! ssnH(X). (6.14)
Now suppose that the kernel of H acts trivially on X. Let , :E ! X be a H-
vector bundle. Then G xffX is a proper H-CW -complex. Since H acts freely
on X, we obtain a well-defined G-vector bundle G xff, :G xffE ! G xffX.
Thus we obtain a functor indff:SPHBH (X) ! SPHB G(G xffX). It defines a
homomorphism of abelian groups
indff:ssnH(X)! ssnG(indff(X, A)). (6.15)
which turns out to be an inverse of the induction homomorphism (6.14).
Now we have all ingredients of an equivariant cohomology theory with a
multiplicative structure. We leave it to the reader to verify all the axioms. T*
*his
finishes the proof of Theorem 6.5.
47
6.4 Comparison with the Classical Construction for Finite
Groups
Next we want to show that for a finite group G our construction reduces to the
classical one. We first explain why the finite group case is easier.
Remark 6.16 (Advantages in the case of finite groups). The finite group
case is easier because for finite groups the following facts are true. The firs*
*t fact
is that every G-CW -complex is proper. Hence one can view pointed G-CW -
complexes, where the base point is fixed under the G-action and one can carry
out constructions like mapping cones without loosing the property proper. We
need proper to ensure that certain basic facts about G-vector bundles are true
(see [29, Section 1]). The second fact is that every G-vector bundle over a fin*
*ite
G-CW -complex , is a direct summand in a trivial G-vector bundle, i.e. a G-
vector bundle given by the projection V x X ! X for some G-representation
V . This makes for instance Lemma 6.4 superfluous whose proof is non-trivial
in the infinite group case (see [29, Lemma 3.7]).
Next we identify ssnG(X) defined in Subsection 6.2 with the classical defi-
nitions which we have explained in Subsection 1.4 provided that G is a finite
group.
Consider an element a 2 ssnG(X) with respect to the definition given in
Subsection 1.4. Obviously we can find a positive integer k 2 Z with k + n 0
such that a is represented for some complex G-representation V by a G-map
f :SV ^Sk^X+ ! SV ^Sk+n . Let , be the trivial G-vector bundle V xX ! X.
Define a map
__ , Rk oe1 V k fxprX V k+n oe2 , Rk+n
f :S __ -! (S ^ S ) x X - ---! (S ^ S ) x X -! S ____.
It is a morphism in SPHB G(X) and hence defines an element a02 ssnG(X) with
respect to the definition of Subsection 6.2. Thus we get a homomorphism
of abelian groups a 7! a0 from the definition of Subsection 1.4 to the one of
Subsection 6.2. k k+n
Consider a morphism u: S, R_ ! S, R___ in SPHB G(X) representing an
element in b = [u] in ssnG(X) as defined in Subsection 6.2. Choose a G-vector
bundle ~, a complex G-representation V and an isomorphism of (real) G-vector
~=
bundles OE: ~ , -! V x X. Then the morphism
k (OE id)-1~ , Rk oe1 ~ , Rk
v :SV R x X ------! S __ -! S ^X S __
id^u---!S~ ^ , Rk+n_oe2 ~ , Rk+n_OE id V Rk+n
X S -! S ---! S x X
is equivalent to u and hence b = [v]. Since v covers the identity and is fiberw*
*ise
a pointed map, its composition with the projection SV Rk+nx X ! SV Rk+n
yields a map
__v:SV Rk^ X V Rk+n
+ ! S .
48
It defines an element b0:= [__v] in ssnG(X) with respect to the definition give*
*n in
Subsection 1.4. The map b 7! b0is the inverse of the map a ! a0before.
Remark 6.17 (Why consider G-vector bundles?). One may ask why we
consider G-vector bundles , in Section 6. It would be much easier if we would
only consider trivial G-vector bundles V x X for G-representations V . Then we
would not need Lemma 6.4. The proof that ss*Gis a G-cohomology theory with
a multiplicative structure would go through and for finite groups we would get
the classical notion. The problem is that the induction structure does not exis*
*ts
anymore as the following example shows.
Consider a finitely generated group such that Gmrf is trivial. Then any G-
representation V is trivial (see Subsection 2.5). This implies that a morphism
u: S, Rk_! S, Rk+n_in SPHB G (X) for , = V x X is the same as a (non-
equivariant) map
k V Rk+n
SV R ^ (G\X+ ) ! S .
This yields an identification of ssnG(X) with respect to the definition, where *
*all
G-vector bundles are of the shape XxV , with the (non-equivariant) stable coho-
motopy group ssns(G\X). If G contains a non-trivial subgroup H G, then the
existence of an induction structure would predict for X = G=H that ssnG(G=H)
is isomorphic to ssnH, which is in general different from ssns(G\(G=H)) = ssn{1*
*}.
So we need to consider G-vector bundles in order to get induction struc-
tures and hence an equivariant cohomology theory. In particular our definition
guarantees ssnG(G=H) = ssnHfor every group G with a finite subgroup H G.
Remark 6.18 (The coefficients of equivariant stable cohomotopy). It is
important to have information about the values ssnG(G=H) for a finite subgroup
H G of a group G. By the induction structure and the identification above
ssnG(G=H) agrees with the abelian groups ssnH= ssH-ndefined in Subsection 1.4.
The equivariant homotopy groups ssH-nare computed in terms of the splitting
due to Segal and tom Dieck (see [49, Theorem 7.7 in Chapter II on page 154],
[41, Proposition 2]) by
M
ssnG(G=H) = ssH-n = sss-n(BWH K).
(K)2ccs(H)
The abelian group sssqis finite for q 1 by a result of Serre [42] (see also [*
*18]),
is Z for q = 0 and is trivial for q -1. Since WH K is finite, eHp(BWH K; Z) is
finite for all p 2 Z. We conclude from the Atiyah-Hirzebruch spectral sequence
that sss-n(BWH K) is finite for n -1. This implies |ssnG(G=H)| < 1 for
n -1 and that ssnG(G=H) = 0 for n 1. We know already ss0H= A(H) from
Theorem 1.13. Thus we get
|ssnG(G=H)|< 1 n -1;
ss0G(G=H) = A(H);
ssnG(G=H) = {0} n 1.
49
Remark 6.19. (Equivariant Cohomotopy for arbitrary G-CW -com-
plexes). In order to construct an equivariant cohomology theory or an (equi-
variant homology theory) for arbitrary G-CW -complexes it suffices to construct
a contravariant (covariant) functor from the category of small groupoids to the
category of spectra (see [39], [30, Proposition 6.8]). In a different paper we *
*will
carry out such a construction yielding equivariant cohomotopy and homotopy
for arbitrary equivariant CW -complexes and will identify the result with the
one presented here for finite proper G-CW -complexes.
6.5 Rational Computation of Equivariant Cohomotopy
The cohomotopy theoretic Hurewicz homomorphism yields a transformation of
cohomology theories ~
ss*s(X) =-!H*(X; Z)
from the (non-equivariant) stable cohomotopy to singular cohomology with Z-
coefficients. It is rationally an isomorphism provided that X is a finite CW -
complex. It is compatible with the multiplicative structures. The analogue for
equivariant cohomotopy is described next.
Let G be a group and H G be a finite subgroup. Consider a pair of finite
proper G-CW -complexes (X, A). Lemma 4.3 implies that (XH , AH ) is a pair
of finite proper WG H-CW -complexes and WG H\(XH , AH ) is a pair of finite
CW -complexes. Taking the H-fixed point set yields a homomorphism
ffn(H)(X, A): ssnG(X, A) ! ssnWGH(XH , AH ).
This map is natural and compatible with long exact sequences of pairs and
Mayer-Vietoris sequences.
The induction structure with respect to the homomorphism WG H ! {1}
yields a homomorphism
H H n H H
finWGH(XH , AH ): ssns WG H\(X , A ) ! ssWGH (X , A ).
We claim that finWGH(Z, B) is a rational isomorphism for any pair of finite pro*
*per
WG H-CW -complexes (Z, B). Since fi*WGH is natural and compatible with the
long exact sequences of pairs and Mayer-Vietoris sequences, it suffices to prove
the claim for Z = WG H=L and B = ; for any finite subgroup L WG H.
But then finWGH reduces to the obvious map ssn({o}) ! ssnLwhich is a rational
isomorphism by Remark 6.18.
Let
H H n H H
hn(XH , AH ): ssns WG H\(X , A ) ! H WG H\(X , A ); Z
be the cohomotopy theoretic Hurewicz homomorphism. Let
H H n H H
fln WG H\(X , A ) :H WG H\(X , A ); Z Z Q
H H
! Hn WG H\(X , A ); Q
50
be the natural map. Define a Q-homomorphism by the composition
ffn(H)(X,A) ZidQn H H
inG(X, A)(H):ssnG(X, A) Z Q -----------! ssWGH (X , A ) Z Q
(finWGH(XH ,AH ) ZidQ)-1n H H
-----------------! sss WG H\(X , A ) Z Q
hn(XH-,AH-)-ZidQ---------!Hn W H H
G H\(X , A ); Z Z Q
fln(WGH\(XH ,AH ))n H H
--------------! H WG H\(X , A ); Q.
Define
Y
inG(X, A) = inG(X, A)(H):ssnG(X, A) Z Q
(H)2ccs(G)
Y
! Hn WG H\(XH , AH ); Q. (6.20)
(H)2ccs(G)
Theorem 6.21. The maps
~= Y n H H
inG(X, A): ssnG(X, A) Z Q -! H WG H\(X , A ); Q
(H)2ccs(G)
are bijective for all n 2 Z and all pairs of finite proper G-CW -complexes (X, *
*A).
They are compatible with the obvious multiplicative structures.
Proof. One easily checks that i*Gdefines a transformation of G-homology the-
ories, i.e. is natural in (X, A) and compatible with long exact sequences of
pairs and Mayer-Vietoris sequences. Hence it suffices to show that inG(G=K) is
bijective for all n 2 Z and finite subgroups K G. The source and target of
inG(G=K) are trivial for n 6= 0 (see Remark 6.18). The map i0G(G=K) can be
identified using Lemma 4.3 (ii) with the rationalization of the character map
Y
charH :A(H) ! Z
(H)2ccs(G)
defined in (1.2)which is bijective by Lemma 1.3. |___|
6.6 Relating Equivariant Cohomotopy and Equivariant To-
pological K-Theory
We have introduced two equivariant cohomology theories with multiplicative
structure, namely equivariant cohomotopy (see Theorem 6.5) and equivariant
topological K-theory (see Subsection 5.4).
Let X be a finite proper G-CW -complex with G-CW -subcomplexes A and
B and let a 2 ssmG(X, A) be an element. We want to assign to it for every m 2 Z
a homomorphism of abelian groups
OEm,nG(X, A)(a): KnG(X, B) ! Km+nG(X, A [ B). (6.22)
51
Choosekan+integermk 2 Z with k 0, k + m 0 and a morphism u: S, Rk_!
S, R____in SPHB G(X) which is trivial over A and represents a. Let v be the
morphism in SPHB G(X) which is given by the composite
k oe , , Rk id^X u , , Rk+m oe-1 , , Rk+m
v :S, , R_ -! S ^X S __ ----! S ^X S _____--! S _____.
Then v is another representative of a. The bundle , , carries a canonical
structure of a complex vector bundle and we denote this complex vector bundle
by ,C. ~
Let oek(X, A [ B): Km+nG(X, A [ B) =-!Km+n+kG (X, A [ B) x (Dk, Sk-1)
be the suspension isomorphism. Let prk: X x Dk ! X be the projection
and pr*k,C be the complex vector bundle obtained from ,C by the pull back
construction. Associated to it is a Thom isomorphism
k k-1
Tpm+n+kr*k,C:Km+n+kG (X, A [ B) x (D , S )
~= m+n+k+2.dim(,)i pr*, pr*,C| k-1 k k j
-! KG S k C, S k XxS [(A[B)xD[ (X x D )1,
where (X x Dk)1 is*the copy of X x Dk given by the various points at infinity
in the fibers Sprk,Cand pr*k,C|XxSk-1[(A[B)xDk is the restriction of pr*k,C to
X x Sk-1 [ (A [ B) x Dk (see [29, Theorem 3.14]). Let
i * * j
pk: Sprk,C, Sprk,C|XxSk-1[(A[B)xDk[ (X x Dk)1
i k k j
S, , R_, S, , R_|A[B[ X1
be the obvious projection which induces by excision an isomorphism on K*G.
Define an isomorphism
~m+n,m+n+k+2.dim(,):Km+nG(X, A [ B)
i k k j
! Km+n+k+2.dim(,)GS, , R_, S, , R_|A[B[ X1
by the composite Km+n+k+2.dim(,)G(pk)-1 O Tpn+m+kr*k,CO oek(X, A [ B). Define
~n,m+n+k+2.dim(,):KnG(X, B)
i k+m k+m j
! Km+n+k+2.dim(,)GS, , R____, S, , R____|B[ X1
analogously. Let the desired map OEm,nG(X, A)(a) be the composite
OEm,nG(X; A, B)(a): KnG(X, B)
~n,m+n+k+2.dim(,)------------!Km+n+k+2.dim(,)i , , Rk+m_ , , Rk+m_|B *
* j
G S , S [ X1
Km+n+k+2.dim(,)G(v)m+n+k+2.dim(,)i, , Rk , , Rk| j
-------------! KG S __, S __ A[B[ X1
(~m+n,m+n+k+2.dim(,))-1m+n
-----------------! KG (X, A [ B).
52
We leave it to the reader to check that the definition of OEm,nG(X; A, B)(a) is
independent of the choices of k and u. The maps OEm,nG(X; A, B)(a) for the
various elements a 2 ssmG(X, A) define pairings
OEm,nG(X; A, B): ssmG(X, A) x KnG(X,!B)Km+nG(X, A [ B). (6.23)
The verification of the next theorem is left to the reader.
Theorem 6.24. (Equivariant topological K-theory as graded algebra
over equivariant cohomotopy).
(i)Naturality
The pairings OEm,nG(X; A.B) are natural in (X; A, B);
(ii)Algebra structure
The collection of the pairings OEm,nG(X; ;, A) defines the structure of a
graded algebra over the graded ring ss*G(X) on K*G(X, A);
(iii)Compatibility with induction
Let OE: H ! G be a group homomorphism and (X, A) be a pair of proper
finite G-CW -complexes. Then the following diagram commutes
OEm,nG(indff(X;A,B))m+n
ssmG(indff(X, A))x KnG(indff(X, B))-------------!KG (indff(X, A [ B))
? ?
indffx indff?y indff?y
OEm,nH(X;A,B) m+n
ssmH(X, A) x KnH(X, B) ---------! KH (X, A [ B)
(iv)For a 2 ssm-1G(A) and b 2 KnG(X) we have
i j
OEm,nG(X; ;, ;)(ffi(a), b) = ffi OEm-1,nG(A; ;, ;)(a, KnG(j)(b)),
where ffi :ssm-1 (A) ! ssmG(X) and ffi :Km+n-1G(A) ! Km+nG(X) are bound-
ary operators for the pair (X, A) and j :A ! X is the inclusion.
For every pair (X, A) of finite proper G-CW -complexes define a homomor-
phism
_nG(X, A): ssnG(X, A) ! KnG(X, A),a 7! OEn,0G(X, A, ;)(a,(1X6),.25)
where 1X 2 K0G(X) is the unit element. Then Theorem 6.24 implies
Theorem 6.26 (Transformation from equivariant cohomotopy to equi-
variant topological K-theory). We obtain a natural transformation of equi-
variant cohomology theories with multiplicative structure for pairs of equivari*
*ant
proper finite CW -complexes by the maps
_*?:ss*?! K*?.
53
If H G is a finite subgroup of the group G, then the map
_nG(G=H): ssnG(G=H) ! K0G(G=H)
is trivial for n 1 and agrees for n = 0 under the identifications ss0G(G=H) =
ss0H= A(H) and K0G(G=H) = K0H({o}) = RC(H) with the ring homomorphism
A(H) ! RC(H), [S] 7! [C[S]]
which assigns to a finite H-set the associated complex permutation representa-
tion.
7 The Homotopy Theoretic Burnside Ring
In this section we introduce another version of the Burnside ring which is of
homotopy theoretic nature and probably the most sophisticated and interesting
one.
7.1 Classifying Space for Proper G-Actions
We need the following notion due to tom Dieck [44].
Definition 7.1 (Classifying space for proper G-actions). A model for the
classifying space for proper G-actions is a proper G-CW -complex E_G such that
E_GH is contractible for every finite subgroup H G.
Recall that a G-CW -complex is proper if and only if all its isotropy groups
are finite. If E_G is a model for the classifying space for proper G-actions, t*
*hen
for every proper G-CW -complex X there is up to G-homotopy precisely one
G-map X ! E_G. In particular two models are G-homotopy equivalent and
the G-homotopy equivalence between two models is unique up to G-homotopy.
If G is finite, a model for E_G is G=G. If G is torsionfree, E_G is the same as
EG which is by definition the total space of the universal principal G-bundle
G ! EG ! BG.
Here is a list of groups G together with specific models for E_G with the
property that the model is a finite G-CW -complex.
54
_____________________________________________________________________
|_G__________________________________________|E_G___________________|_
|_word_hyperbolic_groups_____________________|Rips_complex__________|
| discrete cocompact subgroup G L of a Lie |L=K for a maximal com-|
| | |
|_group_L_with_finite_ss0(L)_________________|pact_subgroup_K___L___|
| G acts by isometries properly and cocompactly|X |
| on a CAT(0)-space X, for instance on a tree|or a |
| | |
| simply-connected complete Riemannian manifold| |
| | |
|_with_non-positive_sectional_curvature______|______________________|
|_arithmetic_groups__________________________|Borel-Serre_completion|_
|_mapping_class_groups_______________________|Teichm"uller_space____|
| outer automorphisms of finitely generated free|outer space |
| | |
|_groups_____________________________________|______________________|_
More information and more references about E_G can be found for instance
in [6] and [28].
7.2 The Definition of the Homotopy Theoretic Burnside
Ring
We have introduced the equivariant cohomology theory with multiplicative struc-
ture for proper finite equivariant CW -complexes ss*?in Section 6 and the class*
*i-
fying space E_G for proper G-actions in Subsection 7.1.
Definition 7.2 (Homotopy theoretic Burnside ring). Let G be a (discrete)
group such that there exists a finite model E_G for the universal space for pro*
*per
G-actions. Define the homotopy theoretic Burnside ring to be
Aho(G) := ss0G(E_G).
If G is finite, ss0G(E_G) agrees with ss0Gwhich is isomorphic to the Burnside
ring A(G) by Theorem 1.13. So the homotopy theoretic definition Aho(G) re-
flects this aspect of the Burnside ring which has not been addressed by the oth*
*er
definitions before.
After the program described in Remark 6.19 has been carried out, the as-
sumption in Definition 7.2 that there exists a finite model for E_G can be drop*
*ped
and thus the Homotopy Theoretic Burnside ring Aho(G) can be defined by
ss0G(E_G) and analyzed for all discrete groups G.
If G is torsionfree, Aho(G) agrees with ss0s(BG).
Theorem 6.26 implies that the map (see (6.25))
_0G(E_G): Aho(G) = ss0G(E_G) ! K0G(E_G)
is a ring homomorphism. It reduces for finite G to the ring homomorphism
A(G) ! RC(G) sending the class of a finite G-set to the class of the associated
complex permutation representation.
55
7.3 Relation between the Homotopy Theoretic and the
Inverse-Limit-Version
Suppose there is a finite model for E_G. Then there is an equivariant Atiyah-
Hirzebruch spectral sequence which converges to ssnG(E_G) and whose E2-term
is given in terms of Bredon cohomomology
Ep,q2= HpZSubFIN(G)(E_G; ssq?).
Here ssq?is the contravariant functor
ssq?:SubFIN(G) ! Z - MODULES , H 7! ssqH
and naturality comes from restriction with a group homomorphism H ! K rep-
resenting a morphism in SubFIN (G). Usually the Bredon cohomology is defined
over the orbit category, but in our case we can pass to the category SubFIN (G)
because of Lemma 5.2. Details of the construction of HpZSubFIN(G)(E_G; ss?q)
can be found for instance in [27, Section 3]. We will only need the following
elementary facts. There is a canonical identification
H0ZSubFIN(G)(E_G; ssq?)~=invlimH2SubFIN(G)ssqH. (7.3)
If we combine (7.3)with Theorem 1.13 we get an identification
H0ZSubFIN(G)(E_G; ss0?)~=Ainv(G). (7.4)
The assumption that E_G is finite implies together with Remark 6.18
|HpZSubFIN(G)(E_G; ssq?)| < 1ifq -1; (7.5)
HpZSubFIN(G)(E_G; ssq?) = {0}ifp > dim(E_G) orp -1 orq 1.(7.6)
The equivariant Atiyah-Hirzebruch spectral sequence together with (7.4),
(7.5)and (7.6)implies
Theorem 7.7 (Rationally Aho(G) and Ainv(G) agree). Suppose that there
is a finite model for E_G. Then the edge homomorphism
edgeG :Aho(G) = ss0G(E_G) ! Ainv(G)
is a ring homomorphism whose kernel and the cokernel are finite.
The edge homomorphism appearingkinkTheorem 7.7 can be made explicit.
Consider a morphism u: S, R_ ! S, R_ in SPHB G(E_G) representing the ele-
ment a 2 ss0G(E_G). In order to specify edgeG(a) we must define for every finite
subgroup H G an element edgeG(a)H 2 A(H). Choose a point x 2 E_GH .
Then u induces a pointed H-map S,x Rk ! S,x Rk. It defines an element in
ss0H. Let edgeG(a)H be the image of this element under the ring isomorphism
~=
degH :ss0H -! A(H) appearing in Theorem 1.13. One easily checks that the
collection of these elements edgeGH(a) does define an element in the inverse li*
*mit
Ainv(G). So essentially edgeG is the map which remembers just the system of
the maps of the various fibers.
56
Remark 7.8. (Rank of the abelian group Aho(G)). A kind of character
map for the homotopy theoretic version would be the composition of edgeand
the character map charGinvof (3.9). Since we assume that E_G has a finite
model, there are only finitely many conjugacy classes of finite subgroups and
the Burnside ring congruences appearing in Theorem 3.10 becomes easier to
handle. In particular we conclude from Example 3.11 and Theorem 7.7 that
Aho(G) is a finitely generated abelian group whose rank is the number | ccsf(G)|
of conjugacy classes of finite subgroups of G.
7.4 Some Computations of the Homotopy Theoretic Burn-
side Ring
Example 7.9 (Groups with appropriate maximal finite subgroups).
Suppose that the group G satisfies the conditions appearing in Example 3.4 and
admits a finite model for E_G. In the sequel we use the notation introduced in
Example 3.4. Then one can construct a G-pushout (see [28, Section 4.11])
` i
i2IG xMi?EMi ----! EG?
?y ?y (7.10)
`
i2IG=Mi ----! E_G
Taking the G-quotient, yields a non-equivariant pushout. There are long exact
Mayer-Vietoris sequence associated to (7.10)and to the G-quotient. (We ignore
the problem that G xE Miand EG may not be finite. It does not really matter
since both are free or because we will in a different paper extend the definiti*
*on
of equivariant cohomotopy to all proper equivariant CW -complexes). These are
linked by the induction maps with respect to the projections G ! {1}. Splicing
these two long exact sequences together, yields the long exact sequence
Y i {1} j
. .!. ker resMi:ss-1Mi! ss-1s! ss0s(G\E_G) ! Aho(G)
i2I Y
! eA(Mi) ! ss1s(G\E_G) ! . . .
i2I
Example 7.11 (Extensions of Zn with Z=p as quotient). Suppose that
G satisfies the assumptions appearing in Example 3.6. Then G admits a finite
model for E_G. In the sequel we use the notation introduced in Example 3.6.
Then variation of the argument above yields a long exact sequence
Y i {1} j
. .!. ker resZ=p:ss-1Z=p(BAZ=p) ! ss-1s(BAZ=p)! ss0s(G\E_G)
H1(Z=p;A)
Y i {1} j
! Aho(G) ! ker resZ=p:ss0Z=p(BAZ=p) ! ss0s(BAZ=p)! ss1s(G\E_G) ! . . .
H1(Z=p;A)
57
where Z=p acts trivially on BAZ=p. If r is the rank of the finitely generated f*
*ree
abelian group AZ=p, then
i j Mr i (rk) j
ker res{1}Z=p:ssnZ=p(BAZ=p) ! ssns(BAZ=p)= ker res{1}Z=p:ssn-kZ=p! ssn-ks.
k=0
8 The Segal Conjecture for Infinite Groups
We can now formulate a version of the Segal Conjecture for infinite groups. Let
fflG :Aho(G) ! Z be the ring homomorphism which sends an element represented
by a morphism u: S, Rk_! S, Rk_in SPHB G(E_G) to the mapping degree of
the map induced on the fiber ux: S,x Rk ! S,x Rk for some x 2 E_G. This is
the same as the composition
G charGinv Y pr{1}
Aho(G) edge----!Ainv(G) ----! Z ---! Z,
(H)2ccsf(G)
where charGinvis the ring homomorphism defined in (3.9)and pr{1}the projec-
tion onto the factor belonging to the trivial group. We define the augmentation
ideal IG of Aho(G) to be the kernel of the ring homomorphism fflG . Recall that
for a finite proper G-CW -complex X the abelian group ssnG(X) is a ss0G(X)-
module. The classifying map f :X ! E_G is unique up to G-homotopy. Sup-
pose that E_G is finite. Then f induces a uniquely defined ring homomorphism
ss0G(f): Aho(G) = ss0G(E_G) ! ss0G(X) and we can consider ssnG(X) is a Aho(G)-
module.
Conjecture 8.1 (Segal Conjecture for infinite groups). Let G be a group
such that there is a finite model for the classifying space of proper G-actions
E_G. Then for every finite proper G-CW -complex there is an isomorphism
~= n
ssns(EG xG X) -! ssG (X)IbG,
where ssnG(X)IbGis the IG -adic completion of the Aho(G)-module ssnG(X).
In particular we get for all n 2 Z an isomorphism
~= n
ssns(BG) -! ssG (E_G)IbG
and especially for n = 0
~=
ss0s(BG) -! Aho(G)IbG.
If G is finite, Conjecture 8.1 reduces to the classical Segal Conjecture (see
Theorem 1.14).
58
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