Rational Computations of the Topological
K-Theory of Classifying Spaces of Discrete
Groups
Wolfgang L"uck*
Fachbereich Mathematik
Universit"at M"unster
Einsteinstr. 62
48149 M"unster
Germany
July 14, 2005
Abstract
We compute rationally the topological (complex) K-theory of the clas-
sifying space BG of a discrete group provided that G has a cocompact
G-CW -model for its classifying space for proper G-actions. For instance
word-hyperbolic groups and cocompact discrete subgroups of connected
Lie groups satisfy this assumption. The answer is given in terms of the
group cohomology of G and of the centralizers of finite cyclic subgroups
of prime power order. We also analyze the multiplicative structure.
Key words: topological K-theory, classifying spaces of groups.
Mathematics Subject Classification 2000: 55N15.
0. Introduction and Statements of Results
The main result of this paper is:
Theorem 0.1 (Main result). Let G be a discrete group. Denote by K*(BG)
the topological (complex) K-theory of its classifying space BG. Suppose that
there is a cocompact G-CW -model for the classifying space E_G for proper G-
actions.____________________
*email: lueck@math.uni-muenster.de
www: http://www.math.uni-muenster.de/u/lueck/
FAX: 49 251 8338370
1
Then there is a Q-isomorphism
__n n ~=
chG :K (BG) Z Q -!
_ ! _ !
Y Y Y Y
H2i+n(BG; Q) x H2i+n(BCG ; Qbp),
i2Z p prime(g)2conp(G)i2Z
where conp(G) is the set of conjugacy classes (g) of elements g 2 G of order pd
for some integer d 1 and CG is the centralizer of the cyclic subgroup
generated by g.
The classifying space E_G for proper G-actions is a proper G-CW -complex
such that the H-fixed point set is contractible for every finite subgroup H G.
It has the universal property that for every proper G-CW -complex X there is
up to G-homotopy precisely one G-map f :X ! E_G. Recall that a G-CW -
complex is proper if and only if all its isotropy groups are finite, and is fin*
*ite
if and only if it is cocompact. The assumption in Theorem 0.1 that there is
a cocompact G-CW -model for the classifying space E_G for proper G-actions
is satisfied for instance if G is word-hyperbolic in the sense of Gromov, if G
is a cocompact subgroup of a Lie group with finitely many path components,
if G is a finitely generated one-relator group, if G is an arithmetic group, a
mapping class group of a compact surface or the group of outer automorphisms
of a finitely generated free group. For more information about E_G we refer for
instance to [8] and [23]. We will prove Theorem 0.1 in Section 4.
We will also investigate the multiplicative structure on Kn(BG) Z Q in
Section 5. If one is willing to complexify, one can show:
Theorem 0.2 (Multiplicative structure). Let G be a discrete group. Sup-
pose that there is a cocompact G-CW -model for the classifying space E_G for
proper G-actions.
Then there is a C-isomorphism
__n n ~=
chG,C: K (BG) Z C -!
_ ! _ !
Y Y Y Y
H2i+n(BG; C) x H2i+n(BCG ; Qbp Q C) ,
i2Z p prime(g)2conp(G)i2Z
which is compatible with the standard multiplicative structure on K*(BG) and
the one on the target given by
a, up,(g). b, vp,(g)= a . b, (a . vp,(g)+ b . up,(g)+ up,(g). vp,(g))
for
(g) 2 conp(G);
Y
a, b 2 H2i+*(BG; C);
i2ZY
up,(g), vp,(g)2 H2i+*(BCG ; Qbp Q C),
i2Z
2
Q
andQthe structures of a graded commutative ring on i2ZH2i+*(BG; C) and
Q i2ZH2i+*(BCG2;iQbp+Q*C) coming fromQthe2cup-productiand+the*obvious
i2ZH (BG; C)-module structure on i2ZH (BCG ; Qbp Q C) coming
from the canonical maps BCG ! BG and C ! Qbp Q C.
In Section 6 we will prove Theorem 0.1 and Theorem 0.2 under weaker
finiteness assumptions than stated above.
If G is finite, we get the following integral computation of K*(BG). Through-
out the paper R(G) will be the complex representation ring and IG be its aug-
mentation ideal, i.e. the kernel of the ring homomorphism R(G) ! Z send-
ing [V ] to dimC(V ). If Gp G is a p-Sylow subgroup, restriction defines a
map I(G) ! I(Gp). Let Ip(G) be the quotient of I(G) by the kernel of this
map. This is independent of the choice of the p-Sylow subgroup since two p-
Sylow subgroups~of G are conjugate. There is an obvious isomorphism from
Ip(G) =-!im(I(G) ! I(Gp)). We will prove in Section 3
Theorem 0.3. (K-theory of BG for finite groups G). Let G be a finite
group. For a prime p denote by r(p) = | conp(G)| the number of conjugacy
classes (g) of elements g 2 G whose order |g| is pd for some integer d 1.
Then there are isomorphisms of abelian groups
Y Y
K0(BG) ~= Z x Ip(G) Z Zbp~= Z x (Zbp)r(p);
p prime p prime
K1(BG) ~= 0.
~= Q
The isomorphism K0(BG) -! Z x p primeIp(G) Z Zbpis compatible with the
standard ring structure on the source and the ring structure on the target given
by
(m, up ap). (n, vp bp)= (mn, (mvp bp + nup ap + (upvp) (apbp))
for m, n 2 Z, up, vp 2 Ip(G) and ap, bp 2 Zbpand the obvious multiplication in
Z, Ip(G) and Zbp.
The additive version of Theorem 0.3 has already been explained in [16,
page 125]. Inspecting [15, Theorem 2.2] one can also derive the ring struc-
ture. In [18] the K-theory of BG with coefficients in the field Fp of p elements
has been determined including the multiplicative structure. The proof of Theo-
rem 0.3 we will present here is based on the ideas of this paper. We will and n*
*eed
to show a stronger statement about the pro-group {IG =(IG )n+1} in Theorem 3.5
(b).
A version of Theorem 0.1 for topological K-theory with coefficients in the
p-adic integers has been proved by Adem [1], [2] using the Atiyah-Segal comple-
tion theorem for the finite group G=G0 provided that G contains a torsionfree
subgroup G0of finite index. Our methods allow to drop this condition, to deal
with K*(BG) Z Q directly and study systematically the multiplicative struc-
ture for K*(BG) Z C. They are based on the equivariant cohomological Chern
character of [22].
3
For integral computations of the K-theory and K-homology of classifying
spaces of groups we refer to [17].
The paper is organized as follows:
1. Borel Cohomology and Rationalization
2. Some Preliminaries about Pro-Modules
3. The K-Theory of the Classifying Space of a Finite Group
4. Proof of the Main Result
5. Multiplicative Structures
6. Weakening the Finiteness Conditions
7. Examples and Further Remarks
References
The author wants to the thank the Max Planck Institute for Mathematics
in Bonn for its hospitality during his stay from April 2005 until July 2005 when
this paper was written.
1. Borel Cohomology and Rationalization
Denote by GROUPOIDS the category of small groupoids. Let -SPECTRA
be the category of -spectra, where a morphism f :E ! F is a sequence of maps
fn :En ! Fn compatible with the structure maps and we work in the category
of compactly generated spaces (see for instance [11, Section 1]). A contravari-
ant GROUPOIDS - -spectrum is a contravariant functor E: GROUPOIDS !
-SPECTRA .
Let E be a (non-equivariant) -spectrum. We can associate to it a con-
travariant GROUPOIDS - -spectrum
EBor:GROUPOIDS ! -SPECTRA ; G 7! map (BG; E), (1.1)
where BG is the classifying space associated to G and map (BG; E) is the ob-
vious mapping space spectrum (see for instance [11, page 208 and Definition
3.10 on page 224]). In the sequel we use the notion of an equivariant cohomol-
ogy theory H*?with values in R-modules of [22, Section 1]. It assigns to each
(discrete) group G a G-cohomology theory H*Gwith values in the category of
R-modules on the category of pairs of G-CW -complexes, where * runs through
Z. Let H*?(-, EBor) be the to EBor associated equivariant cohomology theory
with values in Z-modules satisfying the disjoint union axiom, which has been
constructed in [22, Example 1.8]. For a given discrete group G and a G-CW -pair
(X, A) and n 2 Z we get a natural identification
HnG(X, A; EBor)= Hn(EG xG (X, A); E), (1.2)
where H*(-; E) is the (non-equivariant) cohomology theory associated to E. It
4
is induced by the following composite of equivalences of -spectra
G G
mapOr (G)map G(G=?, X) , map BG (G=H), E
G
! mapOr (G)map G(G=?, X) , map (EG xG G=?, E)
G
! map mapG (G=?, X) Or(G)EG xG G=?, E ! map (EG xG X, E)
using the notation of [22]. In the literature Hn(EG xG (X, A); E) is called the
equivariant Borel cohomology of (X, A) with respect to the (non-equivariant)
cohomology theory H*(-; E).
Our main example for E will be the topological K-theory spectrum K,
whose associated (non-equivariant) cohomology theory H*(-; K) is topologi-
cal K-theory K*.
There is a functor
Rat: -SPECTRA ! -SPECTRA , E 7! Rat(E) = E(0),
which assigns to an -spectrum E its rationalization E(0). The homotopy groups
ssk(E(0)) come with a canonical structure of a Q-module. There is a natural
transformation
i(E): E ! E(0) (1.3)
which induces isomorphisms
~=
ssk(i(E)): ssk(E) Z Q-! ssk(E(0)). (1.4)
Composing EBor with Rat yields a contravariant Or(G)- -spectrum denoted
by (EBor)(0). iWe obtain anjequivariant cohomology theory with values in Q-
modules by H*? -; (EBor)(0). The map i induces a natural transformation of
equivariant cohomology theories
i j
i*?(-; E): H*?(-; EBor) Z Q! H*? -; (EBor)(0). (1.5)
Lemma 1.6. If G is a group G and (X, A) is a relative finite G-CW -pair, then
i j
inG(X, A; E): HnG(X, A; EBor) Z Q ! HnG X, A; (EBor)(0)
is a Q-isomorphism for all n 2 Z.
Proof. The transformation i*G(-; E) is a natural transformation of G-cohomology
theories since Q is flat over Z. One easily checks that it induces a bijection *
*in
the case X = G=H, since then there is a commutative square with obvious iso-
morphisms as vertical maps and the isomorphism of (1.4)as lower horizontal
arrow
5
*(G=H;E) i j
HkG(G=H; EBor) Z Q -iG------! HkG G=H; (EBor)(0)
? ?
~=?y ?y~=
i j
ss-k (map(BH, E)) Z Q-------------! ss-k (map(BH, E))(0)
ss-k(i(map(BH,E)))
By induction over the number of G-cells using Mayer-Vietoris sequences one
shows that i*G(X, A) is an isomorphism for all relative finite G-CW -pairs_(X, *
*A).
|__|
Remark 1.7. (Comparison of the various rationalizations). Notice that
i*G(X, A, E) of (1.5)is not an isomorphism for all G-CW -pairs (X, A) because
the source does not satisfy the disjoint union axiom for arbitrary index sets in
contrast to the target. The point is that - Z Q is compatible with direct sums
but not withidirect products.j
Since H*? -; (KBor)(0)is an equivariant cohomology theory with values
in Q-modules satisfying the disjoint union axiom, weican use the equivariantj
cohomological Chern character of [22] to compute H*G E_G; (KBor)(0)for all
groups G.
This is also true for the equivariant cohomology theory with values in Q-
modules satisfying the disjoint union axiom H*? -; K(0) Bor. (Here we have
changed the order ofiBor and (0).)j But this a much worse approximation of
Kk(BG) Z Q than H*G BG; (EBor)(0). Namely, i induces using the universal
property of Rat a natural map of contravariant GROUPOIDS - -spectra
(KBor)(0)! K(0)Bor
and thus a natural map
i j
HkG X; (KBor)(0)! HkG X; K(0) Bor
but this map is in general not an isomorphism. Namely, it is not bijective for
X = G=H for finite non-trivial H and k = 0. In this case the source turns out
to be
i j Y
ss0 (map(BH; K))(0) ~= K0(BH) Z Q ~= Q x (Qbp)r(p)
p| |H|
for r(p) the number of conjugacy classes (h) of non-trivial elements h 2 H of p-
power order, and the target is K0(BH; Q) which turns out to be isomorphic to Q
since the rational cohomology of BH agrees with the one of the one-point-space.
As mentioned before we want toiuse the equivariantjcohomological Chern
character of [22] to compute H*G X; (KBor)(0). This requires a careful analysis
of the contravariant functor
i j
FGINJ ! Q -MOD , H 7! HkG G=H; (KBor)(0) = Kk(BH) Z Q,
6
from the category FGINJ of finite groups with injective group homomorphisms
as morphisms to the category Q -MOD of Q-modules. It will be carried out in
Section 3 after some preliminaries in Section 2.
2. Some Preliminaries about Pro-Modules
It will be crucial to handle pro-systems and pro-isomorphisms and not to
pass directly to inverse limits. In this section we fix our notation for handli*
*ng
pro-R-modules for a commutative ring R, where ring always means associative
ring with unit. For the definitions in full generality see for instance [3, App*
*endix]
or [6, x 2].
For simplicity, all pro-R-modules dealt with here will be indexed by the
positive integers. We write {Mn, ffn} or briefly {Mn} for the inverse system
M0 ff1-M1 ff2-M2 ff3-M3 ff4-. ...
and also write ffmn:= ffm+1 O . .O.ffn :Gn ! Gm for n > m and put ffnn= idGn.
For the purposes here, it will suffice (and greatly simplify the notation) to w*
*ork
with "strict" pro-homomorphisms {fn}: {Mn, ffn} ! {Nn, fin}, i.e. a collection
of homomorphisms fn :Mn ! Nn for n 1 such that fin O fn = fn-1 O ffn holds
for each n 2. Kernels and cokernels of strict homomorphisms are defined in
the obvious way.
A pro-R-module {Mn, ffn} will be called pro-trivial if for each m 1, there
is some n m such that ffmn = 0. A strict homomorphism f :{Mn, ffn} !
{Nn, fin} is a pro-isomorphism if and only if ker(f) and coker(f) are both pro-
trivial, or, equivalently, for each m 1 there is some n m such that im(fimn)
im (fm ) and ker(fn) ker(ffmn). A sequence of strict homomorphisms
{Mn, ffn} {fn}---!{M0n, ff0n} gn-!{M00n, ff00n}
will be called exact if the sequences of R-modules Mn fn-!Nn gn-!M00nis exact
for each n 1, and it is called pro-exact if gn O fn = 0 holds for n 1 and t*
*he
pro-R-module {ker(gn)= im(fn) is pro-trivial.
The following results will be needed later.
Lemma 2.1. Let 0 ! {M0n, ff0n} -{fn}--!{Mn, ffn} -{gn}--!{M00n, ff00n} ! 0 be a
pro-exact sequence of pro-R-modules. Then there is a natural exact sequence
lim-nf1n lim-ng1n ffi
0 ! lim-n 1M0n------! lim-n 1Mn ------! lim-n 1M00n-!
lim-1nf1n lim-1ng1n
lim-1nM10n------!lim-1nM1n------! lim-1nM100n! 0.
7
In particular a pro-isomorphism {fn}: {Mn, ffn} ! {Nn, fin} induces isomor-
phisms ~
lim-nf1n: lim-n 1Mn =-! lim-n 1Nn;
~=
lim-1nf1n: lim-1nM1n -! lim-1nN1n.
Proof. If 0 ! {M0n, ff0n} {fn}---!{Mn, ffn} gn-!{M00n, ff00n} ! 0 is exact, the*
* con-
struction of the six-term sequence is standard (see for instance [31, Proposi-
tion 7.63 on page 127]). Hence it remains to show for a pro-trivial pro-R-module
{Mn, ffn} that lim-n 1Mn and lim-1nM1nvanish. This follows directly from the
standard construction for these limits as the kernel and cokernel of the homo-
morphism
Y Y
Mn ! Mn, (xn)n 1 7! (xn - ffn+1(xn+1))n 1.
n 1 n 1
|___|
Lemma 2.2. Fix any commutative Noetherian ring R, and any ideal I R.
Then for any exact sequence M0 ! M ! M00of finitely generated R-modules,
the sequence
{M0=InM0} ! {M=InM} ! {M00=InM00}
of pro-R-modules is pro-exact.
Proof. It suffices to prove this for a short exact sequence 0 ! M0 ! M !
M00! 0. Regard M0 as a submodule of M, and consider the exact sequence
n n 0o
0 ! (I_M)\M_InM0! {M0=InM0} ! {M=InM} ! {M00=InM00} ! 0.
By [5, Theorem 10.11 on page 107], the filtrations {(InM)\ M0} and {InM0}
of M0 have "bounded difference", i.e. there exists k > 0 with the property that
(In+k M)\ M0 InM0 holds for all n 1. The first term in the above exact
sequence is thus pro-trivial, and so the remaining terms define a short sequenc*
*e_
of pro-R-modules which is pro-exact. |__|
3. The K-Theory of the Classifying Space of a
Finite Group
Next we investigate the contravariant functor from the category FGINJ of
finite groups with injective group homomorphisms as morphisms to the category
Z -MOD of Z-modules
FGINJ ! Z -MOD , H 7! Kk(BH).
8
We need some input from representation theory. Recall that R(G) denotes
the complex representation ring. Let IG be the kernel of the ring homomorphism
res{1}G:R(G) ! R({1} which is the same as the kernel of augmentation ring
homomorphism R(G) ! Z sending [V ] to dimC(V ). We will frequently use the
so called double coset formula (see [28, Proposition 22 in Chapter 7 on page 58*
*]).
It says for two subgroups H, K G
X H\g-1Kg
resKGO indGH= indc(g):H\g-1Kg!KO resH , (3.1)
KgH2K\G=H
where c(g) is conjugation with g, i.e. c(g)(h) = ghg-1, and indand resdenote
induction and restriction. One consequence of it is that indGH:R(H) ! R(G)
sends IH to IG . Obviously resHG:R(G) ! R(H) maps IG to IH .
For an abelian group M let M(p)be the localization of M at p. If Z(p)is
the subring of Q obtained from Z by inverting all prime numbers except p, then
M(p)= M Z Z(p). Recall that the functor ? Z Z(p)is exact.
Lemma 3.2. Let G be a finite group. Let p be a prime number and denote by
Gp a p-Sylow subgroup of G. Then the composite
indGGp resGpG
R(Gp)(p)----! R(G)(p)---! R(Gp)(p)
has the same image as
resGpG:R(G)(p)! R(Gp)(p).
Proof. A subgroup H G is called p-elementary if it is isomorphic to CxP for a
cyclic group C of order prime to p and a p-group P . Let {CixPi| i = 1, 2, . .,*
*.r}
be a complete system of representatives of conjugacy classes of p-elementary
subgroups of G. We can assume without loss of generality Pi Gp. Define for
i = 1, 2, . .,.r a homomorphism of abelian groups
X P \g-1G g
OEi := indc(g):Pi\g-1Gpg!GpO resiPi p :R(Pi) ! R(Gp).
Gp.g.(CixPi)2
Gp\G=(CixPi)
Since the order of Ci is prime to p, we have (Cix Pi) \ g-1Gpg = Pi\ g-1Gpg
for g 2 G. Hence the following diagram commutes (actually before localization)
by the double coset formula
L r Lri=1indGpPi
i=1R(Pi)(p) --------! R(Gp)(p)
L r CixPi?? ?? G
i=1indPi y yindGp
L r Lri=1indGpCixPi
i=1R(Cix Pi)(p) ----------! R(G)(p)
L r Pi ?? ?? Gp
i=1resCixPiy yresG
L r L ri=1OEi
i=1R(Pi)(p) -----! R(Gp)(p)
9
L r Gp
The middle horizontal arrow i=1indCixPi is surjective by Brauer's Theo-
rem [28, Theorem 18 in Chapter 10 on page 75]. The compositeLof the left
lowerLvertical arrow and the left upper vertical arrow ri=1resPiCixPiO indCix*
*PiPi
is ri=1|Ci| . idand hence an isomorphism. Now the claim follows from_an easy
diagram chase. |__|
Lemma 3.3. Let p and q be different primes. Then the composition
indGGp resGqG
R(Gp) ----! R(G) ---! R(Gq)
agrees with |Gq\G=Gp| . indGq{1}O res{1}Gp.
Proof. This follows from the double coset formula (3.1)since Gp\g-1Gqg_=_{1}
for each g 2 G. |__|
Lemma 3.4. Let G be a finite group and let IG R(G) be the augmentation
ideal. Then the following sequence of R(G)-modules is exact
" i QpresGpG Y i G j
0 ! (IG )m -! IG ------! im respG:IG ! IGp ! 0,
m 1 p2P(G)
where i is the inclusion and P(G) is the set of primes dividing |G|.
Q Gp Q T
Proof. The kernel of presG :R(G) ! pR(Gp) is m 1 (IG )m by [4, Propo-
sition 6.12 on page 269]. Hence it remains to show that
Y G Y i G j
resGp:IG ! im resGp:IG ! IGp
p p
is surjective. It suffices to show for a each prime number q that its localizat*
*ion
Y G Y i G j
respG:(IG )(q)! im resGp:(IG )(q)! (IGp)(q)
p p
is surjective. Next we construct the following commutative diagram
L Qp6=qresGpGO indGGpQ i Gp j
p6=q(IGp)(q) ------------! p6=qim resG: (IG )(q)! (IGp)(q)
L G ?? ??
p6=qindGpy iy
Q resGp Q i j
(IG )(q) --p--G--! pim resGpG:(IG )(q)! (IGp)(q)
? ?
p1?y p2?y
i L j f i j
coker p6=qindGGp ----! im resGqG:(IG )(q)! (IGq)(q)
10
Here i is the inclusion and p1 and p2 are the obvious projections. Since the
composition
M L p6=qindGGp Q presGpGY i G j
(IGp)(q)--------! (IG )(q)------! im respG:(IG )(q)! (IGp)(q)
p6=q p
p2-!imiresGq j
G :(IG )(q)! (IGq)(q)
agrees with
M G M i G j
resGqO indGGp: (IGp)(q)! im resGq:(IG )(q)! (IGq)(q)
p6=q p6=q
and hence is trivial by Lemma 3.3, there exists a map
0 1
M i G j
f :coker@ indGGpA! im resGq:(IG )(q)! (IGq)(q)
p6=q
such that the diagram above commutes. Since
Y G G i G j
p2 O respG= resqG:(IG )(q)! im resGq:(IG )(q)! (IGq)(q)
p
is by definition surjective, f is surjective. The upper horizontal arrow in the
commutative diagram above is surjective by Lemma 3.2. Now the claim follows_
by an easy diagram chase. |__|
Theorem 3.5 (Structure of {IG =(IG )n+1}). Let G be a finite group. Let
P(G) be the set of primes dividing |G|.
(a)There are positive integers a, b and c such that for each prime p dividing
the order of |G|
pa . IGp I2Gp;
IbGp p . IGp;
IG . IGp I2Gp;
(IGp)c IG . IGp;
(b)For a prime p dividing |G| let im(resGpG) be the image of resGpG:IG ! IGp.
11
We obtain a sequence of pro-isomorphisms of pro-Z-modules
~= Y Gp n Gp
{IG =(IG )n+1} -! {im(resG )=(IG ) . im(resG )}
p2P(G)
~= Y Gp n Gp
-! {im(resG )=(IGp) . im(resG )}
p2P(G)
~= Y Gp bn Gp
- {im(resG )=(IGp) . im(resG )}
p2P(G)
~= Y Gp n Gp
-! {im(resG )=p . im(resG )}.
p2P(G)
(c)There is an isomorphism of pro-Z-modules
~= n
{Z} {IG =(IG )n} -! {R(G)=(IG ) },
where {Z} denotes the constant inverse system Z -idZ -id. ...
Proof. (a)The existence of the integers a, b and c for which the inclusions
appearing in the statement of Theorem 3.5 hold follows from results of [4, The-
orem 6.1 on page 265] and [7, Proposition 1.1 in Part III on page 277].
(b) These inequalities of assertion (a)imply that the second, third and fourth
map of pro-Z-isomorphism appearing in the statement of Theorem 3.5 are in-
deed well-defined pro-isomorphisms. The first map
~= Y Gp n Gp
{IG =(IG )n+1} -! {im(resG )=(IG ) . im(resG )}
p2P(G)
is a well-definednpro-isomorphismiofjpro-Z-modulesibyjLemmao2.2 and Lemma 3.4
T T
provided m 1 (IG )m=InG. m 1 (IG )m is pro-trivial. The latter state-
ment follows from Lemma 2.2 applied to the exact sequence
" "
0 ! (IG )m ! IG ! IG = (IG )m ! 0.
m 1 m 1
(c)Consider the isomorphism of finitely generated free abelian groups
~=
Z IG -! R(G), (m, x) 7! x + m . [C].
It becomes an isomorphism of rings if we equip the source with the multiplicati*
*on
(m, x) . (n, y) = (mn, my + nx + xy). In particular InG. (IG Z) InG 0 for
n 1. This finishes the proof of Theorem 3.5. __
|__|
Now we can give the proof of Theorem 0.3.
12
i j
Proof. In the sequel we abbreviate im(resGpG) = im resGpG:IG ! IGp. Notice
that im(resGpG) R(Gp) is a finitely generated free Z-module. We obtain from
Lemma 2.1 and Theorem 3.5 an isomorphism
Y G G
lim-n 1R(G)=(IG )n ~=Z x lim-n 1im(resGp)=pn . im(resGp).
p2P(G)
Now the Atiyah-Segal Completion Theorem [6] yields an isomorphisms
~= 0 ~= 0
lim-n 1R(G)=(IG )n -! lim-n 1K ((BG)n) - K (BG) (3.6)
and K1(BG) = 0. This implies
M i G j
K0(BG) ~= Z im resGp:IG ! IGp Z Zbp;
p2P(G)
K1(BG) ~= 0.
Nextiwe show thatjthe rank of the finitely generated free abelian group
im resGpG:IG ! IGp R(Gp) is the number r(p) of conjugacy classes (g) of
elements g 2 G whose order |g| is pd for some integer d 1. This follows from
the commutative diagram
resGpG
C Z R(G) ----! C Z R(Gp)
? ?
~=?y ?y~=
classC(G) ----! classC(Gp)
resGpG
where classC(G) denotes the complex vector space of class functions on G, i.e.
functions G ! C which are constant on conjugacy classes of elements, (and
analogous for Gp), the vertical isomorphisms are given by taking the character *
*of
a complex representation, and the lower horizontal arrow is given by restricting
a function G ! C to Gp. i j
Recall that Ip(G) is canonically isomorphic to im resGpG:IG ! IGp.
One easily checks that the isomorphisms obtained from the one appearing
in Theorem 3.5 (b)and (c)by applying the inverse limit and the isomorphism
(3.6)are compatible with the obvious multiplicative structures. __
This finishes the proof of Theorem 0.3. |__|
4. Proof of the Main Result
In this section we want to prove our main Theorem 0.1. We want to apply the
cohomological equivariant Chern character of [22] to the equivariant cohomology
13
i j
theory H*? -; (KBor)(0). This requires to analyze the contravariant functor
i j
FGINJ ! Q -MOD , H 7! HlG G=H; (KBor)(0) . (4.1)
From (1.2)and Lemma 1.6 we conclude that the contravariant functor (4.1)is
naturally equivalent to the contravariant functor
FGINJ ! Q -MOD , H 7! Kl(BH) Z Q. (4.2)
Theorem 0.3 yields the contravariant functor (4.2)is trivial for odd l and is
naturally equivalent to the contravariant functor
Y
FGINJ ! Q -MOD H 7! Q x Ip(H) Z Qbp (4.3)
p
for even l, where the factor Q is constant in H and functoriality for the other
factors is given by restriction.
Given a contravariant functor F :FGINJ ! Q -MOD , define the Q[aut(H)]-
module
0 1
Y Y
TH F (H) := ker@ F (K ,! H): F (H) ! F (K)A . (4.4)
K(H K(H
Next we compute TH K0(BH) Z Q . Since TH is compatible with direct
products, we obtain from (4.3)a canonical Q[aut(H)]-isomorphism
0 Y
TH K (BH) Z Q = TH (Q) x TH (Ip(H) Z Qbp). (4.5)
p
Since Q is the constant functor, we get
ae
TH (Q) := 0Q ifHi6=f{1};H = {1}. (4.6)
Fix a prime number p. Since for any finite group H the map given by restriction
to finite cyclic subgroups
Y
R(H) ! R(C)
C H
C cyclic
is injective, we conclude
Lemma 4.7. For a finite group H
TH (Ip(H)) = 0,
unless H is a non-trivial cyclic p-group.
14
Let C be a non-trivial finite cyclic p-group. Then we get
i 0 j
TC (Ip(C)) = ker resCC:R(C) ! R(C0) , (4.8)
where C0 C is the unique cyclic subgroup of index p in C.
Recall that taking the character of a rational representation of a finite gr*
*oup
H yields an isomorphism
~=
O: RQ(H) Z Q -! classQ(H),
where RQ(C) is the rational representation ring of C and classQ(H) is the ra-
tional vector space of functions f :H ! Q for which f(g1) = f(g2) holds if the
cyclic subgroups generated by g1 and g2 are conjugate in H (see [28, page 68 and
Theorem 29 on page 102]). Hence there is an idempotent `C 2 RQ(C) ZQ which
is uniquely determined by the property that its character sends a generator of C
to 1 and all other elements to 0. Denote its image under the change of coeffici*
*ents
map RQ(C) Z Q ! R(C) Z Q also by `C . Let `C . R(C) Z Qbp R(C) Z Qbp
be the image of the idempotent endomorphism R(C) ZQbp! R(C) ZQbpgiven
by multiplication with `C .
Lemma 4.9. For every non-trivial cyclic p-group C the inclusion induces a
Q[aut(C)]-isomorphism
~=
`C . R(C) Z Q -! TC (Ip(C) Z Q).
Proof. Since the map resC0C:R(C) ZiQ ! R(C0) Z Q sendsj`C to zero,
`C . R(C) Z Q is contained in ker resC0C:R(C) ! R(C0) Z Q. For x 2
i 0 j
ker resCC:R(C) ! R(C0) Z Q one gets `C . x - x = 0 by the calculation
appearing in the proof of [21, Lemma 3.4 (b)]. |___|
Lemma 4.10. For every proper G-CW -complex X and n 2 Z there is an
isomorphism, natural in X,
__n * i j ~=
chG :HG X; (KBor)(0)-!
Y Y Y Y
H2i+n(G\X; Q) x H2i+nWGC(CG C\XC ; `C .R(C) ZQbp),
i2Z p (C)2Cp(G)i2Z
where Cp(C) is the set of conjugacy classes of non-trivial cyclic p-subgroups of
G and WG C = NG C=CG C is considered as a subgroup of aut(C) and thus acts
on `C . R(C) Z Qbp.
Proof. This follows from [22, Theorem 5.5 (c) and Example 5.6] using_(4.6),_
Lemma 4.7 and Lemma 4.9. |__|
15
For a generator t 2 C let Ct be the C-representation with C as underlyingij
complex vector space such that t operates on C by multiplication with exp 2ssi_*
*|C|.
Let Gen(C) be the set of generators. Notice that aut(C) acts in an obvious way
on Gen(C) such that the aut(C)-action is transitive and free, and acts on R(C)
by restriction. In the sequel OV denotes for a complex representation V its
character.
Lemma 4.11. Let C be a finite cyclic group. Then
(a)The map
~= Y
v(C): `C . R(C) Z C -! C, [V ] 7! (OV (t))t2Gen(C)
Gen(C)
is a C[aut(C)]-isomorphism if aut(C) acts on the target by permuting the
factors. The map v(C) is compatible with the ring structure on the source
induced by the tensor product of representations and the product ring stru*
*c-
ture on the target;
(b)There is an isomorphism of Q[aut(C)]-modules
~=
u(C): Q[Gen (C)] -! `C . R(C) Z Q.
Proof. (a)The map
~= Y
R(C) Z C -! C, [V ] 7! (OV (g))g2C
g2C
is an isomorphism of rings. One easily checks that it is compatible with the
aut(C) actions. Now the assertion follows from the fact that the character of
`C sends a generator of C to 1 and any other element of C to 0.
(b) Obviously Q[Gen (C)] is Q[aut(C)]-isomorphic to the regular representa-
tion Q[aut(C)] since Gen (C) is a transitive free aut(C)-set. It remains to
show that `C . R(C) Z Q is Q[aut(C)]-isomorphic to the regular representa-
tion Q[aut(C)]. By character theory it suffices to show that `C . R(C) Z C
is C[aut(C)]-isomorphic to the regular representation C[aut(C)]. This follows_
from assertion (a). |__|
Lemma 4.12. For every proper G-CW -complex X and n 2 Z there is an
isomorphism, natural in X,
___n i j ~=
chG :H*G X; (KBor)(0)-!
Y Y Y
H2i+n(G\X; Q) x H2i+n(CG \X; Qbp).
i2Z p (g)2conp(G)
16
Proof. Fix a prime p. Let C be a cyclic subgroup of G of order pd for some
integer d 1. The obvious NG C-action on C given by conjugation induces an
embedding of groups WG C ! aut(C). The obvious action of aut(C) on Gen(C)
is free and transitive. Thus we obtain an isomorphism of Qbp[WG C]-modules
Y
Qbp[Gen (C)] ~= Qbp[WG C].
WGC\ Gen(C)
This induces a natural isomorphism
~= Y k C
HkWGC(CG C\XC ; Qbp[Gen (C)]) -! H (CG C\X ; Qbp),
WGC\ Gen(C)
which comes from the adjunction (i*, i!) of the functor restriction i* and coin-
duction i!for the ring homomorphism i: Qbp! Qbp[WG C] and the obvious iden-
tification i!(Qbp) = Qbp[WG C]. There is an obvious bijection between the sets
a
WG C\ Gen(C) ~=conp(G).
(C)2Cp
Now the claim follows from Lemma 4.10 and Lemma 4.11 (b). |___|
Theorem 4.13 (Computation of Kn(EG xG X) Z Q). For every finite
proper G-CW -complex X and n 2 Z there is a natural isomorphism
__n n
chG :K (EG xG X) Z Q
~= Y 2i+n Y Y 2i+n
-! H (G\X; Q) x H (CG \X ; Qbp).
i2Z p (g)2conp(G)
Proof. This follows from Lemma 1.6 and Lemma 4.12. |___|
Lemma 4.14. Let Y 6= ; be a proper G-CW -complex such that Hep(Y ; Q)
vanishes for all p. Let f :Y ! E_G be a G-map. Then G\f :G\Y ! G\E_G
induces for all k isomorphisms
~=
Hk(G\f; Q): Hk(G\Y ; Q) -! Hk(G\E_G; Q);
~= k
Hk(G\f; Q): Hk(G\E_G; Q) -! H (G\Y ; Q);
~= k
Hk(G\f; C): Hk(G\E_G; C) -! H (G\Y, C);
~= k
Hk(G\f; Qbp): Hk(G\E_G; Qbp)-! H (G\Y, Qbp);
~= k
Hk(G\f; Qbp Q C): Hk(G\E_G; Qbp Q C) -! H (G\Y, Qbp Q C).
Proof. The map C*(f) Z idQ:C*(G\Y ) Z Q ! C*(E_G) Z Q is Q-chain map
of projective QG-chain complexes and induces an isomorphism on homology.
Hence it is a QG-chain homotopy equivalence. This implies that C*(f) QG M
and hom QG(C*(f), M) are chain homotopy equivalences and induce isomor- __
phisms on homology and cohomology respectively for every Q-module M. |__|
17
Now we can give the proof of Theorem 0.1.
Proof. We conclude from Lemma 4.14 that for any g 2 conp(G) the up to
CG -homotopy unique CG -map fg: ECG ! E_CG and the up to G-
homotopy unique G-map f :EG ! EG__induce isomorphisms
~= k
Hk(G\f; Q): Hk(G\E_G; Q) -! H (BG; Q); (4.15)
~= k
Hk(CG \fg; Qbp): Hk(CG \E_CG ; Qbp)-!H (BCG , Qbp).(4.16)
Now apply Theorem 4.13 to X = E_G and use (4.15)and (4.16)together with __
the fact that E_Gis a model for E_CG . |__|
5. Multiplicative Structures
In this section we want to deal with multiplicative structures and prove
Theorem 0.2.
Remark 5.1. (Ring structures and multiplicative structures). Suppose
that the -spectrum E comes with the structure of a ring spectrum ~: E^E !
E. It induces a multiplicative structure on the (non-equivariant) cohomology
theory H*(-; E) associated to E. Thus the equivariant cohomology theory given
by the equivariant Borel cohomology H*?(E? x? -; E) associated to E inherits
a multiplicative structure the sense of [22, Section 6].
If the contravariant GROUPOIDS - -spectrum F comes with a ring structure
of contravariant GROUPOIDS - -spectra ~: F ^ F ! F, then the associated
equivariant cohomology theory H*?(-; F) inherits a multiplicative structure.
A ring structure on the -spectrum E induces a ring structure of contravari-
ant GROUPOIDS - -spectra on EBor. The induced multiplicative structure on
H*?(-; EBor) and the one on H*?(E? x? -; E) are compatible with the natural
identification (1.2).
A ring structure on the -spectrum E induces in a natural way a ring struc-
ture on its rationalization Rat(E). Thus a ring structure on the contravariant
GROUPOIDS - -spectra on EBor induces a ring structure on the contravariant
GROUPOIDS - -spectra on (EBor)(0). The natural transformation of equiv-
ariant cohomology theories appearing in (1.5)is compatible with the induced
multiplicative structures.
In this discussion we are rather sloppy concerning the notion of a smash
product. Since we are not dealing with higher structures and just want to take
homotopy groups in the end, one can either use the classical approach in the
sense of Adams or the more advanced new constructions such as symmetric
spectra.
Lemma 5.2. The isomorphism appearing in Lemma 4.10 is compatible with the
multiplicative structure on the source and the one on the target given by
(a, up,(C)) . (b, vp,(C)) = (a . b, a . vp,(C)+ b . vp,(C)+ up,(C). vp,(C)*
*),
18
for
(C) 2 Cp(G);
a, b2 H*(BG; Q);
up,(C), vp,(C)2 H*WGC(CG C\XC ; `C . R(C) Z Qbp),
Q
andQthe structures of a graded commutative ring on i2ZH2i+*(BG; Q) and
2i+* C
i2ZHWGC (CG C\X ; `C . R(C) Z Qbp) coming from the cup-productQand the
multiplicative structureQon `C .R(C) ZQbpand the obvious i2ZH2i+*(BG; Q)-
module structure on i2ZH2i+*WGC(CG C\XC ; `C . R(C) Z Qbp) coming from the
canonical maps CG C\X ! G\X and Q ! Qbp.
Proof. The proof consists of a straightforward calculation which is essentially
based on the following ingredients. In the sequel we use the notation of [22].
The equivariant Chern character of [22, Theorem 6.4] is compatible with the
multiplicative structures.
In Theorem 0.3 we have analyzed for every finite group H the multiplicative
structure on Y
K0(BH) ~= Z x Ip(H) Z Zbp.
p
Thus the Bredon cohomologyigroup appearingjin the target of the Chern char-
acter whose source is H*G X; (KBor)(0) can be identified with
_ !
Y Y Y
H*+2i(G\X; Q) x H2i+*QbpSub(G;F)(X; Ip(?) Z Qbp)
i2Z p i2Z
with respect to the multiplicative structure analogously defined to the one ap-
pearing in Theorem 0.3 taking the obvious multiplicativeQstructures on the fac-
tors and the module structures of the factor for p over i2ZH*+2i(G\X; Q)
into account.
Fix a prime p. The Qbp[aut(C)]-map R(C) Z Qbp! `C . R(C) Z Qbpgiven
by multiplication with the idempotent `C is compatible with the multiplicative
structures. Using the identification of Lemma 4.9 we obtain for each cyclic
p-group C a retraction compatible with the multiplicative structures.
aeC :Ip(C) Z Qbp! TC (Ip(C) Z Qbp)
Recall that TK (Ip(K) Z Qbp)is trivial unless K is a non-trivial cyclic p-grou*
*p.
Use these retractions as the maps aeK in the definition of the isomorphism of
QbpSub(G; F)-modules for M = Ip(?) Z Qbpin [22, (5.1)]. Then we obtain using
the identification of Lemma 4.9 an isomorphism of QbpSub(G; F)-modules
~= Y
Ip(?) Z Qbp-! i(C)!(`C . R(C) Z Qbp),
(C)2Cp
which is compatible with the obvious multiplicative structure on the source
and the one on the target given by the product of the multiplicative structures
19
on the factors i(C)!(`C . R(C) Z Qbp)coming from the obvious one on `C .
R(C) Z Qbp. Using the adjunction (i(C)*, i(C)!) this isomorphism induces an
Qbp-isomorphism compatible with the multiplicative structures
~= Y n
HnQbpSub(G;F)(X; Ip(?) Z Qbp) -! HWGC (CG C\X; `C . R(C) Z Qbp).
(C)2Cp(G)
|___|
Because the isomorphism in Lemma 4.11 (a)is compatible with the multi-
plicative structures, it implies together with Lemma 5.2
Lemma 5.3. For every proper G-CW -complex X and n 2 Z there is a C-
isomorphism, natural in X,
__n * ~=
chG,C: HG (X; (KBor)(0)) Q C -!
_ ! _ !
Y Y Y Y
H2i+n(G\X; C) x H2i+n(CG \X; Qbp Q C) ,
i2Z p (g)2conp(G)i2Z
which is compatible with the multiplicative structure on the target given by
a, up,(g). b, vp,(g)= a . b, (a . vp,(g)+ b . up,(g)+ up,(g). vp,(g))
for
(g) 2 conp(G);
Y
a, b2 H2i+*(G\X; C),
i2ZY
up,(g), vp,(g)2 H2i+*(CG \X; Qbp Q C),
i2Z
Q
andQthe structures of a graded commutative ring on i2ZH2i+*(G\X; C) and
Q i2ZH2i+*(CG \X; Qbp Q C) comingQfrom the cup-product and the obvious
i2ZH2i+*(G\X; C)-module structure on i2ZH2i+*(CG \X; Qbp Q C)
coming from the canonical map BCG ! BG.
Now we are ready to prove Theorem 0.2
Proof. The isomorphism appearing in Lemma 1.6 is compatible with the mul-
tiplicative structures. This is also true for the versions of isomorphisms (4.1*
*5)
and (4.16), where the coefficients Q and Qbpare replaced by C and Qbp Q C._
Now put these together with the isomorphism appearing in Lemma 5.3. |__|
Remark 5.4. (Difference between rationalization and complexifica-
tion). First of all we want to emphasize that the isomorphism appearing in
Theorem 0.2 is not obtained from the isomorphism appearing in Theorem 0.1
by applying - QC since the corresponding statement is already false for the two
20
isomorphisms appearing in Lemma 4.11. Moreover, the isomorphism appearing
in Theorem 0.1 is not compatible with the standard multiplicative structures
on the source and the multiplicative structure on the target which is defined
analogously to the one on the complexified target in Theorem 0.2. The reason
is that the isomorphism appearing in Lemma 4.11 (b)cannot be chosen to be
compatible with the obvious multiplicative structure on its target if we use on
the source the multiplicativeQstructure coming from the obviousQidentification
Qbp[Gen (C)] = Gen(C)Qbpand the product ring structure on Gen(C)Qbp.
One can easily check by hand that there is no Qb3[aut(Z=3)]-isomorphism
compatible with the multiplicative structures
~=
`Z=3. R(Z=3) Qb3= IZ=3 Qb3-! Qb3x Qb3
if we equip the target with the aut(Z=3) ~= Z=2-action given by flipping the
factors and the product Q-algebra structure. The point is that Qb3does not
contain a primite 3-rd root of unity (in contrast to C, see Lemma 4.11 (a)).
Example 5.5 (Multiplicative structure over Q). In general we can give
a simple formula for the multiplicative structure only after complexifying as
explained in Remark 5.4. In the following special case this can be done already
after rationalization. Suppose that for any non-trivial cyclic subgroup C of
prime power order Hen(BCG C; Q) = 0 holds for all n 2 Z and that WG C =
aut(C). The latter means that any automorphism of C is given by conjugation
with some element in NG C. Suppose furthermore that there is a finite model
for E_G. Then we obtain Q-isomorphisms
Y Y
K0(BG) Z Q ~= H2i(BG; Q) x (Qbp)rp(G);
i2Z p
Y
K1(BG) Z Q ~= H2i+1(BG; Q),
i2Z
where rp(G) is the number of conjugacy classes of non-trivial cyclic subgroups *
*of
p-power order what is in this situation the same as | conp(G)|. The isomorphisms
above are compatible with the multiplicative structure on the target given by
(a, u) . (b, v)=(a [ b, a0 . v + b0 . u + u . v);
(a, u) . c= a [ c;
c . d = c [ d,
Q Q Q
for a, b 2 i2ZH2i(BG; Q), c, d 2 i2ZH2i+1(BG; Q) and u, v 2 p(Qbp)rp(G),
where a0 2 Q andQb0 2 Q are the components of a and b in H0(BG; Q) = Q . 1
and we equip p(Qbp)rp(G))with the structure of a Q-algebra coming from the
product of the obvious Q-algebra structures on the various factors Qbp. This
follows from Lemma 5.2 and the conclusion from the formula `C . `C = `C and
Lemma 4.11 (b)that (`C . R(C))aut(C)is generated as Q-vector space by `C and
hence is as Q-algebra isomorphic to Q.
21
If we furthermore assume that eHn(BG; Q) = 0 for all n 2 Z, the formula
simplifies to
Y
K0(BG) Z Q ~= Q x (Qbp)rp(G);
p
K1(BG) Z Q ~= 0.
The first isomorphism is compatible with the multiplicative structures if we put
on the target the one given by
(m, a) . (m, b) = (mn, m . b + n . a + a . b)
Q Q
for m, n 2 Q, a, b 2 p(Qbp)rp(G)and we equip p(Qbp)rp(G)with the structure
of a Q-algebra coming from the product of the obvious Q-algebra structures on
the various factors Qbp.
6. Weakening the Finiteness Conditions
In this section we want to weaken the finiteness assumption occurring in
Theorem 0.1 and Theorem 0.2.
A Z-module M is almost trivial if there is an element r 2 Z, r 6= 0 such that
rm = 0 holds for all m 2 M. A Z-module M is almost finitely generated if
M= tors(M) is a finitely generated Z-module and tors(M) is almost trivial. A
Z-homomorphism is an almost isomorphism if its kernel and cokernel are almost
trivial. An almost isomorphism becomes an isomorphism after rationalization.
The full subcategories of the category of Z-modules given by almost trivial
submodules and by almost finitely generated submodules are Serre-subcategories,
i.e. are closed under subobjects, quotients, and extensions. In particular ther*
*e is
a Five-Lemma for almost isomorphisms. These notions and facts are introduced
and proved in [24, Section 4].
The main result of this section is:
Theorem 6.1 (Weakening the finiteness assumption). The conclusions of
Theorem 0.1 and Theorem 0.2 remain true if we replace the condition that there
is a cocompact G-CW -model for the classifying space E_G for proper G-actions
by the following weaker set of conditions:
There exists a G-CW -complex X satisfying:
(a)The G-CW -complex X is proper and finite dimensional. There is an upper
bound on the orders of its isotropy groups. The set of conjugacy classes
(C) of finite cyclic subgroups C G of prime power order with XC 6= ;
is finite;
(b)For all for k 2 Z we have Hk(X; Z) ~=Hk({o}; Z);
(c)For any finite cyclic subgroup of prime power order C G and integer k
the Z-module Hk(XC ; Z) is almost finitely generated;
22
(d)For any finite cyclic subgroup of prime power order C G and integer k
the Z-module Hk(CG C\XC ; Z) is almost finitely generated.
If X satisfies conditions (a), (b) and (c) above, then the condition (d) is
satisfied if and only if for any finite cyclic subgroup of prime power order C *
* G
and integer k the Z-module Hk(BCG C; Z) is almost finitely generated.
Remark 6.2 (Weakening the finiteness conditions for E_G). Notice that
the conditions (a), (b), (c)and (d)in Theorem 6.1 are satisfied, if the set of
conjugacy classes of finite subgroups of G is finite, there is a finite dimensi*
*onal
model for E_G and for any finite cyclic subgroup of prime power order C G
and integer k the Z-module Hk(BCG C; Z) is almost finitely generated.
Remark 6.3 (Virtually torsionfree groups). Suppose that G contains a
torsionfree subgroup H G of finite index. If there is a finite dimensional
model for BH, then there exists a finite dimensional model for E_G [27]).
However, if there is a finite model for BH, this does not implies that G has
only finitely many conjugacy classes of subgroups or that there is a cocompact
model for E_G or that the centralizers CG C of finite cyclic subgroups are fini*
*tely
generated [20, Section 7].
The proof of Theorem 6.1 needs some preparation.
Lemma 6.4. Let X be a proper G-CW -complex. Let pr:EG xG X ! G\X be
the projection. Fix an integer n 2 Z.
(a)Suppose that there exists for each m 0 a positive integer d(m) such
that for any isotropy group H of X multiplication with d(m) annihilates
Hem(BH; Z). Then the induced map
Hn(pr; Z): Hn(EG xG X; Z) ! Hn(G\X; Z).
is an almost isomorphism for all n 2 Z;
(b)The induced map
Hn(pr; Q): Hn(EG xG X; Q) ! Hn(G\X; Q).
is a Q-isomorphism.
Proof. (a)This is proved in [24, Lemma 8.1]. __
(b) The proof is analogous to the one of (b). |__|
The next result is a generalization of Lemma 1.6 in the case E = K.
Lemma 6.5. Let X be a finite dimensional proper G-CW -complex such that
there is a bound on the orders of finite subgroups. Let J be the set of conjuga*
*cy
classes (C) of finite cyclic subgroups C G of prime power order with XC 6= ;.
Suppose that |J| is finite. Furthermore assume that Hk(CG C\XC ; Z) is almost
finitely generated for every k 2 Z and every finite cyclic subgroup of prime po*
*wer
order C G.
23
Then the map
i j
inG(X; K): HnG(X; KBor) Z Q ! HnG X; (KBor)(0)
is bijective for all n 2 Z.
Proof. In the sequel we use the notation of [22]. Let F(X) be the set of
conjugacy classes of subgroups of H G with XH 6= ;. Since X is proper and
has finite orbit type, F(X) is finite and (H) 2 F(X) implies that H is finite.
Since X is proper and finite dimensional, there is a spectral sequence convergi*
*ng
to Hs+tG(X; KBor)whose E2-term is Es,t2=iHsSub(G;F(X))(X;jKt(BH)) and a
spectral sequence converging to Hs+tG X; (KBor)(0)whose E2-term is Es,t2=
HsSub(G;F(X))(X; Kt(BH) Z Q). Since Q is flat over Z, it suffices to show that
the canonical map
HsSub(G;F(X))(X; Kt(BH)) Z Q ! HsSub(G;F(X))(X; Kt(BH) Z Q)
is bijective for all s and t. We have already explained that the contravariant
ZSub (G; F(X))-module sending H to Kt(BH) is zero for odd t and given for
even t by Y
Kt(BH) ~= Z x Ip(H) Z Zbp.
p
One easily checks using Lemma 4.7 and (4.8)for any finite group H
8
>>>Z i j H = {1};
0 < ker resH0:R(H) ! R(H0) Z Zbp H cyclicp-group,
TH K (BH) ~=> H
>>: H0 H, [H : H0] = p;
0 otherwise.
For every finite cyclic subgroup K G of order pr for some prime p and
integer r 1 choose a retraction r0(K): K0(BK) ! TK (K0(BK)) of the
Z-homomorphism j(K): TK (K0(BK)) ! K0(BK) given by inclusion.0 Such
r0(K) exists since R(K0) and hence the image of resKK:R(K)i! R(K0) is a j
finitely generated free Z-module what implies that ker resK0K:R(K) ! R(K0)
is a direct summand of the finitely generated free Z-module Ip(K) = I(K).
Since WG K is finite, we can define a Z[WG K]-map
X
r(K): K0(BK) ! TK (K0(BK)), x 7! g . r0(K)(g-1 . x).
g2WGK
~=
Then r(K) O j(K) = |WG K| . id. For K = {1} let r(K): K0(BK) -! Z be the
obvious isomorphism which is for trivial reasons a WG K-map. Define a map of
contravariant Sub(G; F(X))-modules
Y
:K0(B?) ! i(K)!TK (K0(B?))
(K)2J
24
by requiring that the composite of with the projection onto the factor belong-
ing to (K) 2 J is the adjoint for the pair (i(K)*, i(K)!) of the WG K-map r(K).
Analogously to the proof of [22, Theorem 2.14 (b)] one shows that (H) is injec-
tive for all objects H 2 Sub(G; F(X)). Here we use the fact that r(K) O j(K) is
injective for all K with (K) 2 J since r(K)Oj(K) = |WG K|.idand K0(BK) and
hence TK (K0(BK)) is torsionfree. Then one constructs analogously to the proof
of [22, Theorem 5.2] for each object (H) 2 Sub(G; F(X)) a Z-homomorphism
0 1
Y
~(H): @ i(K)!TK (K0(B?))A(H) ! K0(BH)
(K)2J
and checks that (H) O ~(H) can be written as a diagonal matrix A(H) which
has upper triangular form and has maps of the shape r . idas diagional entry,
where each r divides a certain integer M(|H|) depending only on the order of
|H|. There is an integer N(|H|) depending only on the order of |H| such that
the size of the square matrix A(H) is bounded by N(|H|). The existence of the
numbers M(|H|) and N(|H|) follows from the finiteness of J. Hence for each
object H 2 Sub(G; F(X)) the cokernel of (H) is annihilated by M(|H|)N(|H|).
Since there is an upper bound on the orders of finite subgroups of G, we can
find an integer L such that for each object H 2 Sub(G; F(X)) the cokernel of
(H) is annihilated by L. The short exact sequence of ZSub (G; F(X))-modules
Y pr
0 ! K0(B?) -! i(K)!TK (K0(B?)) -! coker( ) ! 0
(K)2J
induces a long exact sequence
. .!.Hs-1Sub(G;F(X))(X; coker( )) ! HsSub(G;F(X))(X; K0(B?))
Y
! HsSub(G;F(X))(X; i(K)!TK (K0(B?))) ! HsSub(G;F(X))(X; coker( )) ! . . .
(K)2J
Since multiplication with L induces the zero map coker( ) ! coker( ), multipli-
cation with L induces also the zero map on HsSub(G;F(X))(X; coker( )). Hence
HsSub(G;F(X))(X; coker( )) ZQ is trivial. Since J is finite, and - ZQ is an ex-
act functor which commutes with finite products, we obtain from the adjunction
(i(K)*.i(K)!) a natural isomorphism
~= Y s K 0
HsSub(G;F(X))(X; K0(B?)) ZQ -! HWGK (CG K\X ; TK K (B?)) ZQ.
(K)2J
Similarly we get an isomorphism
~= Y s K 0
HsSub(G;F(X))(X; K0(B?) ZQ) -! HWGK (CG K\X ; TK K (B?) ZQ).
(K)2J
Hence it remains to show for each (K) 2 J and s 0 that the canonical map
HsWGK(CG K\XK ; TK K0(B?)) Z Q ! HsWGK(CG K\XK ; TK K0(B?) Z Q)
25
is bijective. Abbreviate C* = C*(CG K\Y K), L = WG K and M = TK K0(B?).
Then L is a finite group, C* is a ZL-chain complex which is free over Z and for
which there exists an integer n 1 such that tors(Hs(C*)) is annihilated by n
and Hs(C*)= tors(Hs(C*)) is a finitely generated Z-module for all s. It remains
to show for the ZL-module M that the canonical map
Hs(hom ZL(C*, M)) Z Q ! Hs(hom ZL(C*, M Z Q))
is bijective for all s. Let i: {1} ! L be the inclusion. Since L is finite, ind*
*uction
i* and coinduction i! agree. Hence we get natural ZL-chain maps a*: C* !
i*i*C* and b*: i*i*C* ! C* such that b* O a* is multiplication with |L|. They
are explicitly given by
X
as: Cs ! ZL Z Cs, x 7! l l-1 . x;
l2L
bs: ZL Z Cs ! Cs, l y 7! l . y.
Hence we obtain a commutative diagram
Hs(hom ZL(C*, M)) Z Q ----! Hs(hom ZL(C*, M Z Q))
? ?
Hs(b*) ZidQ?y Hs(b*)?y
Hs(hom ZL(ZL Z C*, M)) Z Q----! Hs(hom ZL(ZL Z C*, M Z Q))
? ?
Hs(a*) ZidQ?y Hs(a*)?y
Hs(hom ZL(C*, M)) Z Q ----! Hs(hom ZL(C*, M Z Q))
where the horizontal arrows are the canonical maps and the composite of the
two left vertical maps and the composite of the two right vertical maps are
isomorphisms. Hence it suffices to show that the middle horizontal arrow is an
isomorphism. It can be identified with the canonical map
Hs(hom Z(C*, M)) Z Q ! Hs(hom Z(C*, M Z Q)).
Notice for the sequel that C* is Z-free. By the universal coefficient theorem
we get a commutative diagram with exact rows and the canonical maps as
26
horizontal arrows
0 0
?? ?
y ?y
extZ(Hs-1(C*), M) Z Q----! extZ(Hs-1(C*), M Z Q)
?? ?
y ?y
Hs(hom Z(C*, M)) Z Q ----! Hs(hom Z(C*, M Z Q))
?? ?
y ?y
hom Z(Hs(Z), M) Z Q ----! hom Z(Hs(Z), M Z Q))
?? ?
y ?y
0 0
Since Hs(C*)= tors(Hs(C*) is a finitely generated free abelian group and
there is an integer n which annihilates tors(Hs(C*)), the rational vector spaces
extZ(Hs-1(C*), M) Z Q and extZ(Hs-1(C*), M Z Q) vanish and the lower
vertical arrow is bijective. Hence the middle arrow is bijective. This_finish*
*es
the proof of Lemma 6.5. |__|
Lemma 6.6. Let X be a proper G-CW -complex such that for any cyclic sub-
group C G of prime power order and any k 2 Z we have Hk(XC ; Q) ~=
Hk({o}; Q). Then the up to G-homotopy unique G-map f :X ! E_G induces
for every n 2 Z an isomorphism
i j i j~ i j
HnG f; (KBor)(0):HnG E_G; (KBor)(0)=-!HnG X; (KBor)(0).
Proof. Because of Lemma 4.12 it suffices to show for every n 2 Z and every
cyclic subgroup of prime power order that the map
Hn(CG C\fC ; M): Hn(CG C\(E_G)C ; M) ! Hn(CG C\XC ; M)
is bijective for any Q-module M. The Atiyah-Hirzebruch spectral sequence for
the fibration XC ! ECG C xCGC XC ! BCG C together with the vanishing of
He*(XC ; Q) implies that the projection pr:ECG C xCGC XC ! BCG C induces
for all n 2 Z isomorphisms
Hn(pr; Q): Hn(ECG C xCGC XC ; Q) ! Hn(BCG C; Q).
The projection pr0:ECG C xCGC XC ! CG C\XC induces for all n 2 Z isomor-
phisms
Hn(pr0; Q): Hn(ECG C xCGC XC ; Q) ! Hn(CG C\XC ; Q)
by Lemma 6.4 (b). The same is also true for E_G instead of X. This implies
that
Hn(CG C\fC ; Q): Hn(CG C\XC ; Q) ! Hn(CG C\(E_G)C ; Q)
27
is bijective for all n 2 Z. Hence Hn(CG C\fC ; M) is bijective for any_Q-module
M. |__|
Lemma 6.7. Let X be a proper finite dimensional G-CW -complex such that
Hk(X; Z) ~= Hk({o}; Z) holds for any k 2 Z. Let C G be a cyclic group
of prime power order. Suppose that eHn(XC ; Z) is almost finitely generated for
each n 2 Z. Then:
(a)The Z-module eHn(XC ; Z) is almost trivial and the Q-module eHn(XC ; Q)
is trivial for all n 2 Z;
(b)The map Hn(pr; Z): Hn(ECG C xCGC XC ; Z) ! Hn(BCG C; Z) induced
by the projection pr:ECG CxCGC XC ! BCG C is an almost isomorphism
for all n 2 Z.
Proof. (a)Suppose that C has order pk for k 1. Then eHn(X; Fp) = 0 for all
n 2 Z if Fp is the finite field of order p. By Smith theory eHn(XC ; Fp) = 0 for
all n 2 Z [9, Theorem 5.2]. This implies by the Bockstein sequence associated
to 0 ! Z p.id--!Z ! Fp ! 0 that p . id:eHn(XC ; Z) ! eHn(XC ; Z) is bijective f*
*or
n 2 Z. Since eHn(XC ; Z) is almost finitely generated, it must be almost trivial
for n 1. This implies that eHn(XC ; Q) = 0 for all n 2 Z.
(b) This follows from the Lerray-Serre spectral sequence of the fibration XC !
ECG C x XC ! BCG C whose E2-term is Hs(BCG C; eHt(XC ; Z)) and which
converges to Hs+t(pr: ECG C xCGC XC ! BCG C; Z) and the fact that the full
subcategory of almost trivial Z-modules is a Serre subcategory of the abelian_
category of Z-modules. |__|
Now we can give the proof of Theorem 6.1.
Proof. If one goes through the proofs of Theorem 0.1 and Theorem 0.2 one sees
that the finiteness condition about E_G enters only, when we apply Lemma 1.6
to E_G. Hence it suffices to show that under the assumptions appearing in
Theorem 6.1 the map
i j
inG(E_G; K): HnG(E_G; KBor) Z Q ! HnG E_G; (KBor)(0)
is a Q-isomorphism for all n 2 Z.
Let f :X ! E_G be the up to G-homotopy unique G-map. We obtain the
following commutative diagram
n(E_G;K) i j
HnG(E_G; KBor) Z Q -iG------!HnG E_G; (KBor)(0)
? ?
HnG(f;KBor) ZQ?y ?yHnG(E_G;(KBor)(0))
i j
HnG(X; KBor) Z Q ------! HnG X; (KBor)(0)
inG(X;K)
28
Since Hk(f, Z): Hk(X; Z) ! Hk(E_G; Z) is bijective for all k 2 Z, we conclude
from the Lerray-Serre spectral sequence that Hk(EG xG f, Z): Hk((EG xG
X; Z) ! Hk((EG xG E_G; Z) is bijective for all k 2 Z. This implies that the
left vertical arrow in the commutative square above which can be identified
with Kn(EG xG f): Kn(EG xG E_G) ! Kn(EG xG X) is bijective. The lower
horizontal arrow is bijective by Lemma 6.5. The right vertical arrow is bijecti*
*ve
by Lemma 6.6 and Lemma 6.7 (a). Hence the upper horizontal arrow is bijective.
The claim about the equivalent reformulation of condition (d)follows from_
Lemma 6.4 (a)and Lemma 6.7 (b). This finishes the proof of Theorem 6.1. |__|
7. Examples and Further Remarks
Some finiteness conditions such as appearing in Theorem 6.1 are necessary
as the following example shows.
Example 7.1. (Necessity of the finitenessWconditions). Consider G =
*1i=1Z=p for a prime number p. Then BG ' 1i=1BZ=p and we get
1Y 1Y
K0(BG) ~= K0({o}) x eK0(BZ=p) ~= Z x (Zbp)p-1,
i=1 i=1
Q1
if {o} is the one-point-space. Since Hn(BG; M) ~= i=1Hn(BZ=p; M) = 0
for any Q-module M and n 2, the cohomological dimension of G over Q is
1 and hence G acts on a tree T with finite stabilizers [13]. Then T is a 1-
dimensional model for E_G (see [29, page 20] or [12, Proposition 4.7 on page 17*
*]).
By the Kurosh Subgroup Theorem [26, Theorem 1.10 on page 178]) any non-
trivial finite subgroup of G is conjugated to precisely one of the summands Z=p
and is equal to its centralizer. Hence p is an upper bound on the orders of
finite subgroups of G. Obviously Zbp Z Q is canonically isomorphic to Qbp. If
the conclusion of Theorem 0.1 would be true for G, it would predict that the
canonical map
_ 1 !
Y 1Y 1Y
(Zbp)p-1 Z Q ! (Zbp)p-1 Z Q = (Qbp)p-1
i=1 i=1 i=1
is bijective, what is not true. For instance, the element (p-i)1i=1is not conta*
*ined
in its image.
Notice that in this example all conditions appearing in Theorem 6.1 are
satisfied except the condition that the set of conjugacy classes (C) of finite
cyclic subgroups C G of prime power order with T C6= ; is finite;
We emphasize that no restriction (except properness) occur in Lemma 4.12.
The problem is in Lemma 6.6 some additional finiteness assumptions are needed.
Example 7.2 (SL3(Z)). Consider the group G = SL3(Z). It is well-known that
its rational cohomology satisfies eHn(BSL3(Z); Q) = 0 for all n 2 Z. Actually,
29
we conclude from [30, Corollary on page 8] that for G = SL3(Z) the quotient
space G\E_G is contractible and compact. From the classification of finite sub-
groups of SL3(Z) we see that SL3(Z) contains up to conjugacy two elements
of order 2, two elements of order 4 and two elements of order 3 and no further
conjugacy classes of non-trivial elements of prime power order. The rational
homology of all the centralizers of elements in con2(G) and con3(G) agree with
the one of the trivial group (see [2, Example 6.6]). Hence Theorem 0.1 shows
K0(BSL3(Z)) Z Q ~= Q x (Qb2)4 x (Qb3)2;
K1(BSL3(Z)) Z Q ~= 0.
The identification of K0(BSL3(Z)) Z Q above is compatible with the multi-
plicative structure on the target described in Example 5.5.
Actually the computation using Brown-Petersen cohomology and the Conner-
Floyd relation by Tezuka and Yagita [32] gives the integral computation
K0(BSL3(Z)) ~= Z x (Zb2)4 x (Zb3)2;
K1(BSL3(Z)) ~= 0.
Example 7.3 (Groups with appropriate maximal finite subgroups).
Let G be a discrete group. Consider the following assertions 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;
(C) There is a cocompact model for E_G.
The conditions (M) and (NM) imply the following: For any non-trivial finite
subgroup H G we have NG H = NM H if M is a maximal finite subgroup
containing H. Let {Mi | i 2 I} be a complete set of representatives of the
conjugacy classes of maximal finite subgroups of G. Fix a prime p. Then the
obvious map a ~
conp(Mi) =-!conp(G)
i2I
is a bijection. Let rp(Mi) = | conp(Mi)| be the number of conjugacy classes of
elements in Mi of order pk for some k 1.
Theorem 0.1 yields for a group satisfying conditions (M), (NM) and (C)
above rational isomorphisms
Y Y P r (M )
K0(BG) Z Q ~= H2i(BG; Q) x (Qbp)i2I p i
i2Z p
Y
K1(BG) Z Q ~= H2i+1(BG; Q).
i2Z
Here are some examples of groups Q which satisfy conditions (M), (NM) and
(C)
30
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), (NM) are satisfied by [25, Lemma 6.3]. There are
models for E_G whose underlying space is Rn. The quotient G\E_G looks
like the quotient of T nby a finite group.
o Fuchsian groups F
See for instance [25, Lemma 4.5]). The quotients G\E_G are closed ori-
entable surfaces. In [25] 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. We only have to
consider the case, where Q contains torsion. 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 2 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 from [26, Propositions 5.17,
5.18 and 5.19 in II.5 on pages 107 and 108]. Hence Q satisfies (M) and
(NM). There are explicit two-dimensional models for E_G with one 0-cell
G=C x D0, as many free 1-cells G x D1 as there are elements in I and one
free 2-cell G x D2 (see [10, Exercise 2 (c) II. 5 on page 44]).
For the three examples above one can make H*(BG; Q) = H*(G\E_G; Q)
more explicit.
Example 7.4 (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 .
There is a bijection
~=
~: H1(Z=p; A) x (Z=p)x -! conp(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], k) 2 H1(Z=p; A) x (Z=p)x to the conjugacy class (ugk)
of agk 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 and k 2 Z represents k. There is a
cocompact model for E_G with A Z R as underlying space. Hence Theorem 0.1
yields for G as above rational isomorphisms
Y Yr Y i j
Kn(BG) Z Q ~= H2i+n(BA; Q)Z=px H2i+n B(AZ=p); Qbp,
i2Z k=1 i2Z
31
if we put r = (p - 1) . |H1(Z=p; A)|.
Take for instance APto be the cokernel of the inclusion of Z[Z=p]-modules
Z ! Z[Z=p], n 7! n . p-1i=0ti, where Z carries the trivial Z=p-action and t 2*
* Z=p
is a fixed generator. One can identify A with the extension of Z by adjoining a
primitive p-th root of unity. From the long exact cohomology sequence associ-
ated to the short exact sequence of Z[Z=p]-modules 0 ! Z ! Z[Z=p] ! A ! 0
one concludes that H1(Z=p; A) is a cyclic group of order p. One easily checks
that AZ=p= 0. Hence we obtain for the semi-direct product A o Z=p
2-p
K0(B(A o Z=p)) Z Q ~= Q x (Qbp)p ;
K1(B(A o Z=p)) Z Q ~= 0.
The identification of K0(B(AoZ=p)) ZQ is compatible with the multiplicative
structure on the target described in Example 5.5.
Remark 7.5 (Comparison with Adem's work). The results and the exam-
ples appearing in this paper are consistent with the ones by Adem [2]. Adem
needs that G contains a normal torsionfree subgroup G0 G of finite index
and uses the Atiyah-Segal Completion Theorem for the finite group G=G0 to
compute rationally the K-theory with coefficients in the p-adic integers Zbp. H*
*is
condition that in his notation 0\X is compact is precisely the condition that
there is a cocompact model for E_G. We can drop the condition of the existence
of the normal torsionfree subgroup G0 G of finite index with our methods.
One can get Adem's local computations from ours by replacing for a fixed
prime p the cohomology H*(BG; Q) by H*(BG; Qbp) and ignoring in the product
running over all primes all the contributions coming from primes different from*
* p.
For instance Example 7.2 implies that the Qb3-algebra K0(BSL(3, Z); Zb3) Zb3Qb3
is given by
Qb3[u, v]=(u2 = u, v2 = v, uv = 0).
If one makes the change of variables u = (ff - 2)=3 and v = (fi - 2)=3, one
obtains the presentation in [2, Example 6.6]
Qb3[u, v]=(ff2 - ff - 2, fi2 - fi - 2, fffi - 2(ff + fi - 2)).
Recall that after complexification we can determine the multiplicative structure
in general (see Theorem 0.2).
There are interesting discussions about Euler characteristics and maps of
groups inducing isomorphisms on homology in [1] and [2] which also apply to
our setting.
Remark 7.6 (Hodgkin's computation). Let n be the mapping class group
of the sphere S2 with n punctures for n 3. Hodgkin computes rationally the
K-theory of B n with coefficients in the p-adic integers Zbpusing Adem's for-
mula in [1]. The main work done in the paper by Hodgkin [14, Proposition 2.2
and Theorem 2] is to figure out the set of conjugacy classes of elements of or-
der ps for each prime p and integer s 1 and the rank of Kk(BCG ) Z
32
Q
Q ~= i2ZH2i+k(BCG ; Q) for each element g 2 n of prime power or-
der. One can identify K*(BCG ) Z Q with (K*(BKr) Z Q) r or with
(K*(BKr) Z Q) r-2x 2 for appropriate integers r depending only on the or-
der of g, where Kr is the pure mapping class group of S2 with r punctures and
t denotes the symmetric group of permutation of the set consisting of t ele-
ments. Thus one obtains with Theorem 0.1 the precise structure of the Q-vector
spaces Kk(B n) Z Q. It may be worthwhile to investigate the product struc-
ture on the cohomology H*(CG ; C) since this would lead to a computation
of K*(B n) Z C including its multiplicative structure by Theorem 0.2.
Remark 7.7 (Criterion for torsionfree). Let G be a discrete group with a
finite model for E_G. Then the following assertions are equivalent:
(a)G is torsionfree;
(b)The abelian group Kk(BG) is finitely generated for k 2 Z;
(c)The rational vector space Kk(BG) Z Q is finite dimensional for k 2 Z.
An application of the Atiyah-Hirzebruch spectral sequence proves the implica-
tion (a) ) (b). The implication (b) )(c) is obvious. The implication (c) )(a)
follows from Theorem 0.1 since Qbpis an infinite dimensional Q-vector space.
Remark 7.8 (Torsion prime to the order of finite subgroups). Suppose
that there is a finite model for E_G. Let the ring G be the subring of Q obtai*
*ned
by inverting the orders of finite subgroups of G. We state without giving the
details of the proof but referring to [17] that one can improve Theorem 0.1 to
the statement that there is a G -isomorphism
__n n G ~=
chG,~G: K (BG) Z -!
_ !
Y Y Y
Kn(G\E_G) Z G x H2i+n(BCG ; Qbp).
p prime(g)2conp(G)i2Z
Consider a prime q for which there exists no element order qs for some s 1
in G, in other words, q is not invertible in G . Then the projection BG =
EG xG E_G ! G\E_G induces an isomorphism
~= n
torsq(Kn(G\E_G)) -! torsq(K (BG)) ,
if for an abelian group A we denote by torsq(A) the subgroup of elements
a 2 A which are annihilated by some power of q. In particular Kn(BG) con-
tains q-torsion if and only if Kn(G\E_G) contains q-torsion. It can occur that
Kn(G\E_G) contains elements of order qs for some s 1 (see [19].) We will
explain in [17] that the subgroup of torsion elements in Kn(BG) is finite.
33
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