Topological K-(co-)homology of classifying spaces of discrete groups

0.33 MB·English

by  Michael Joachim

TOPOLOGICAL K-(CO-)HOMOLOGY OF CLASSIFYING SPACES
OF DISCRETE GROUPS
MICHAELJOACHIMANDWOLFGANGLU¨CK
2
1 Abstract. LetGbeadiscretegroup. Wegivemethodstocomputeforagen-
0 eralized(co-)homology theory its values onthe Borel construction EG×GX
2 ofaproperG-CW-complexX satisfyingcertainfinitenessconditions. Inpar-
ticular wegive formulascomputing the topological K-(co)homology K∗(BG)
n
and K∗(BG) up to finite abelian torsion groups. They apply for instance to
a
J arithmeticgroups,wordhyperbolicgroups,mappingclassgroupsanddiscrete
cocompact subgroups of almost connected Lie groups. For finite groups G
3 these formulas are sharp. The main new tools we use for the K-theory cal-
2
culation are a Cocompletion Theorem and Equivariant Universal Coefficient
Theorems which are of independent interest. In the case where G is a finite
]
T groupthesetheoremsreducetowell-knownresultsofGreenleesandB¨okstedt.
K
0. Introduction
.
h
t OneofourmaingoalsinthispaperistocomputeforadiscretegroupG,aproper
a G-CW-complex X and a generalized cohomology theory H∗ and a generalized ho-
m
mology theory H the groups H∗(EG× X) and H (EG× X). In particular
∗ G ∗ G
[
the case is interesting,where X can be chosento be non-equivariantlycontractible

1 because then EG×GX is a model for the classifying space BG. The main results
v are Theorem 3.6 and Theorem 4.1. In the introduction we will concentrate on the
3 case, where H∗ and H are topological K-theory K∗ and K and on classifying
∗ ∗
6
spaces BG.
7
ThroughoutthepaperH (Y;M)andH∗(Y;M)denotesingular(co-)-homology
4 ∗
. of Y with coefficients in the abelian group M, and we omit M from the notation
1
in the case M = Z. Let Z be the ring of p-adic integers, which is the inverse
0 p
2 limit of the inverse system of projections Z/p←p−r−2 Z/p2 ←p−r−3 Z/p3 ←p−r−4 .... Denote
1 by Z/p∞ the quotient Z[1/pb]/Z which is isomorphic to the colimit of the directed
v: system of inclusions Z/p−→p Z/p2 −→p Z/p3 −→p .... The proof of the following result
i will be given in Section 5.
X
r Theorem 0.1 (Topological K-theory of classifying spaces). Let G be a discrete
a
group. Let X be a finite proper G-CW-complex. Suppose for every k ∈ Z that
H (X) = 0 vanishes. Given a prime number p and k ∈ Z, define the natural
k
number
e rk(G) := dim Hk+2i(BC hgi;Q) ,
p Q G
(g)∈Xconp(G) Xi∈Z (cid:0) (cid:1)
where con (G) denotes the set of conjugacy classes of non-trivial elements of p-
p
power order and C hgi is the centralizer in G of the cyclic subgroup hgi generated
G
by g. Let P(G) be the set of primes p which divide the order H of some finite
subgroup H ⊆G. Then:
Date:January2012.
2010 Mathematics Subject Classification. 55H20,55N15,19L47, 57S99.
Keywords and phrases. Classifyingspaces, TopologicalK-theory.
1
2 MICHAELJOACHIMANDWOLFGANGLU¨CK
(i) There is an exact sequence
0 → A → Kk(G\X) → Kk(BG) → B × (Z )rpk(G) → C → 0,
p
p∈YP(G)
where A, B and C are finite abelian groups with b
1 1 1
A⊗ Z =B⊗ Z =C⊗ Z =0;
Z Z Z
P(G) P(G) P(G)
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
(ii) Dually there is an exact sequence
0→C′ → (Z/p∞)rpk+1(G)×B′ →Kk(BG)→Kk(G\X)→A′ →0,
p∈aP(G)
where A′, B′ and C′ are finite abelian groups with
1 1 1
A′⊗ Z =B′⊗ Z =C′⊗ Z =0;
Z Z Z
P(G) P(G) P(G)
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
(iii) If we invert all primes in P(G), then we obtain isomorphisms
Kk(BG)⊗ZZ 1 ∼= Kk(G\X)⊗ZZ 1 × (Qp)rpk(EG);
P(G) P(G)
(cid:20) (cid:21) (cid:20) (cid:21) p∈YP(G)
1 1 b
Kk(BG)⊗ZZ ∼= Kk(G\X)⊗ZZ .
P(G) P(G)
(cid:20) (cid:21) (cid:20) (cid:21)
Under the conditions appearing in Theorem 0.1 the sets con (G) and P(G) are
p
finiteandthedimensionofBC hgiisboundedbythefinitedimensionofX. Hence
G
the numbers rk(X) are well-defined.
p
The exact sequences of assertions (i) and (ii) of Theorem 0.1 are in fact dual to
eachotherwhenworkinginthe categoryoftopologicalgroups. Namely, recallthat
skeletalfiltrationonBGimposesthestructureofapro-discretegrouponK∗(BG),
while the p-adic integers can be equipped with the pro-finite topology. With these
topologies the exact sequence of Theorem 0.1 (ii) is the Pontryagin dual of the
exactsequenceofTheorem0.1(i) andvice versa. Moreoverbothexactsequenceof
Theorem 0.1 are also exact sequences in the category of topological groups in the
sense of [39].
A model EG for the classifying space for proper G-actions is a proper G-CW-
complex,whoseH-fixedpointsetsarecontractibleforallfinitesubgroupsH ⊆G. It
isuniqueuptoG-homotopy. ItisagoodcandidateforX inTheorem0.1,provided
that there exists a finite G-CW-complex model for EG. Examples, for which
this is true, are arithmetic groups in a semisimple connected linear Q-algebraic
group[11],[32],mappingclassgroups(see[29]),groupswhicharehyperbolicinthe
sense of Gromov (see[27], [28]), virtually poly-cyclic groups, and groups which are
cocompactdiscretesubgroupsofLiegroupswithfinitelymanypathcomponents[1,
Corollary 4.14]. On the other hand, for any CW-complex Y there exists a group
G such that Y and G\EG are homotopy equivalent (see [17]). More information
about these spaces EG can be found for instance in [8], [19], [36, Section I.6].
In order to apply Theorem0.1 one needs to understand the CW-complex G\X.
This is often possible using geometric input, in particular in the case X =EG for
the groups mentioned above. Notice that q-torsion in Kk(BG) and K (BG) for a
k
prime number q which does not belong to P(G) must come from the q-torsion in
Kk(G\X) and K (G\X).
k
The rational version of our formula for K-cohomology has already been proved
using equivariant Chern charactersin [20, Theorem 0.1], see also [3, Theorem 6.3].
TOPOLOGICAL K-(CO-)HOMOLOGY OF CLASSIFYING SPACES ... 3
If G is finite, a model for EG is {•} and one gets a complete answer integrally,
see for instance [20, Theorem 0.3].
We will recall the Completion Theorem 2.4 of [21, Theorem 6.5]) and deduce
from it in Section 2.
Theorem 0.2 (Cocompletion Theorem). Let G be a discrete group. Let X be
a finite proper G-CW-complex and let L be a finite dimensional proper G-CW-
complex whose isotropy subgroups have bounded order. Fix a G-map f: X → L
and regard K∗(X) as a module over K (L). Moreover, let I = I (L) be the
G G G
augmentation ideal (see Definition 2.1).
Then there is a short exact sequence
0→colim ext1(K∗+1(X)/In·K∗+1(X),Z)→K (EG× X)
−−−→n≥1 Z G G ∗ G
→colim hom (K∗(X)/In·K∗(X),Z)→0.
−−−→n≥1 Z G G
When working in the category of topological abelian groups and continuous
homomorphisms the sequence can be written in the following more compact form
0→ext1 (K∗+1(X) ,Z)→K (EG× X)→hom (K∗(X) ,Z)→0.
cts G I ∗ G cts G I
Theorem 0.2 is closely related to the local cohomology approach to equivariant
K-homology of Greenleesb[15] (see Remark 2.8 below). b
In Section 5 we prove
Theorem 0.3 (Equivariant Universal Coefficient Theorem for K-theory). Let G
be a discrete group. Let X be a finite proper G-CW-complex X.
Then there are short exact sequences, natural in X,
(0.4) 0→ext (KG (X),Z)→K∗(X)→hom (KG(X),Z)→0;
Z ∗−1 G Z ∗
(0.5) 0→ext (K∗+1(X),Z)→KG(X)→hom (K∗(X),Z)→0,
Z G ∗ Z G
where the homomorphism on the right hand sides are given by (5.19) and (5.20)
respectively. The sequence splits unnaturally.
Theorem0.3reducesforfinitegroupstothecorrespondingresultsofB¨okstedt[10]
as in explained in Remark 5.21.
The work was financially supported by Sonderforschungsbereich 878 Groups,
Geometry and Actions in Mu¨nster, and the Leibniz-Preis of the second author.
1. Some preliminaries about pro-modules
It will be crucial to handle pro-systems and pro-isomorphisms and not to pass
directly to inverse limits. Otherwise we would loose important information which
is for instance needed in order to pass from K-cohomology to K-homology using
universal coefficients theorems. In this section we fix our notation for handling
pro-R-modulesforacommutativeringR,whereringalwaysmeansassociativering
with unit. For the definitions in full generality see for instance [5, Appendix] or [7,
Section 2]. This exposition agrees with the one in [20, Section 2] and is repeated
for the reader’s convenience.
Forsimplicity, all pro-R-modulesdealtwith here will be indexedby the positive
integers. We write {M ,α } or briefly {M } for the inverse system
n n n
M ←α−1 M ←α−2 M ←α−3 M ←α−4 ··· .
0 1 2 3
and also write αm := α ◦···◦α : G → G for n > m and put αn = id .
n m+1 n n m n Gn
For the purposes here, it will suffice (and greatly simplify the notation) to work
with “strict” pro-homomorphisms{f }: {M ,α }→{N ,β }, i.e., a collection of
n n n n n
4 MICHAELJOACHIMANDWOLFGANGLU¨CK
homomorphisms f : M → N for n ≥ 1 such that β ◦f = f ◦α holds
n n n n n n−1 n
for each n ≥ 2. Kernels and cokernels of strict homomorphisms are defined in the
obvious way.
A pro-R-module {M ,α } will be called pro-trivial if for each m ≥ 1, there is
n n
some n ≥m such that αm = 0. A strict homomorphism f: {M ,α } →{N ,β }
n n n n n
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(βm)⊆im(f ) and
n m
ker(f )⊆ker(αm). A sequence of strict homomorphisms
n n
{M ,α }−{−f−n→} {M′,α′ }−g→n {M′′,α′′}
n n n n n n
will be called exact if the sequences of R-modules M −f→n N −g→n M′′ is exact
n n n
for each n ≥ 1, and it is called pro-exact if g ◦f = 0 holds for n ≥ 1 and the
n n
pro-R-module {ker(g )/im(f ) is pro-trivial.
n n
The following results will be needed later.
(cid:9)
Lemma 1.1. Let 0 → {M′,α′ } −{−f−n→} {M ,α } −{−g−n→} {M′′,α′′} → 0 be a pro-
n n n n n n
exact sequence of pro-R-modules. Then there is a natural exact sequence
0→lim M′ −l←−im−−n−≥−1−f→n lim M −l←−im−−n−≥−1−g→n lim M′′ −→δ
←−n≥1 n ←−n≥1 n ←−n≥1 n
lim1 M′ −l←−im−−1n−≥−1−f→n lim1 M −l←−im−−1n−≥−1−g→n lim1 M′′ →0.
←−n≥1 n ←−n≥1 n ←−n≥1 n
Inparticularapro-isomorphism{f }: {M ,α }→{N ,β }inducesisomorphisms
n n n n n
lim f : lim M −∼=→ lim N ;
←−n≥1 n ←−n≥1 n ←−n≥1 n
lim1 f : lim1 M −∼=→ lim1 N .
←−n≥1 n ←−n≥1 n ←−n≥1 n
Proof. If 0 → {M′,α′ } −{−f−n→} {M ,α }−g→n {M′′,α′′} →0 is exact, the construc-
n n n n n n
tion of the six-term sequence is obvious (see for instance [34, Proposition 7.63 on
page 127]). Hence it remains to show for a pro-trivial pro-R-module {M ,α }
n n
that lim M and lim1 M vanish. This follows directly from the standard
←−n≥1 n ←−n≥1 n
construction for these limits as the kernel and cokernel of the homomorphism
M → M , (x ) 7→(x −α (x )) .
n n n n≥1 n n+1 n+1 n≥1
n≥1 n≥1
Y Y
(cid:3)
Lemma 1.2. Let 0 → {M′,α′ } −{−f−n→} {M ,α } −{−g−n→} {M′′,α′′} → 0 be a pro-
n n n n n n
exact sequence of pro-R-modules. Then there is a natural exact sequence
0→colim hom (M′′,Z)→colim hom (M ,Z)
−−−→n≥1 Z n −−−→n≥1 Z n
→colim hom (M′,Z)→colim ext1(M′′,Z)
−−−→n≥1 Z n −−−→n≥1 Z n
→colim ext1(M ,Z)→colim ext1(M′,Z)→0.
−−−→n≥1 Z n −−−→n≥1 Z n
Inparticularapro-isomorphism{f }: {M ,α }→{N ,β }inducesisomorphisms
n n n n n
colim hom (N ,Z) −∼=→ colim hom (M ,Z);
−−−→n≥1 Z n −−−→n≥1 Z n
colim ext1(N ,Z) −∼=→ colim ext1(M ,Z).
−−−→n≥1 Z n −−−→n≥1 Z n
Proof. The proofis analogousto the one of the previous Lemma 1.1 using the fact
that colim is an exact functor. (cid:3)
−−−→n≥1
TOPOLOGICAL K-(CO-)HOMOLOGY OF CLASSIFYING SPACES ... 5
2. Completion and cocompletion theorems
Let G be a discrete group. Denote by K∗ equivariant topological K-theory.
G
This is a multiplicative G-cohomology theory for proper G-CW-complexes which
comes with various extra structures such as induction, restriction and inflation. It
is defined in terms of classifying spaces for G-vector bundles over proper G-CW-
complexes (see [21, Theorem 2.7 and Section 3]).
Recall that a G-CW-complex X is proper if and only if its isotropy groups are
finite [18, Theorem 1.23 on page 18] and is finite if and only if X is cocompact,
i.e. G\X is compact. If one considers only finite proper G-CW-complexes, then
K∗(X) has other descriptions which are all equivalent. There is a construction
G
due to Phillips [30] in terms of infinite dimensional vector bundles. In Lu¨ck and
Oliver [22, Theorem 3.2] it is shown that it suffices to use finite dimensional G-
vector bundles and that one can give a definition in terms of the Grothendieck
group K (X) of the monoid of isomorphism classes of G-vector bundles over X.
G
In case where G is the trivial group, we just write K(X). One can also define
for a finite proper G-CW-complex X the equivariant topological K-theory as the
topologicalK-theoryK (C (X)⋊G)ofthecrossedproductC∗-algebraC (X)⋊G
∗ 0 0
(see [30, Theorem 6.7 on page 96]) , while equivariant topological K-homology of
X (by the dual of the Green-Julg theorem [9, Theorem 20.2.7 (b)]) also can be
defined as the equivariant KK-group KKG(C (X),C). Here C (X) denotes the
∗ 0 0
C∗-algebra of continuous function on X vanishing at infinity.
For any proper G-CW-complex X the is a natural ring homomorphism
γ (X): K (X)→K0(X)
G G G
which allows to regard Kp(X) as a K (X)-module in the sequel.
G G
Definition 2.1 (Augmentationideal). The augmentation ideal I (Y)⊆K (Y) is
G G
given by the set of elements in K (Y) represented by virtual G-vector bundles of
G
dimension zero on all components of Y.
If G is trivial we just write I(Y).
We have the following easy but crucial lemma (cf. Lemma 4.2 in [22]).
Lemma 2.2. Let Z be a CW-complex of dimension n−1. Then the n-fold product
of elements in I(Z)⊆K(Z) is zero.
Now fix a finite proper G-CW-complex X, and a map f: X →L to a finite di-
mensionalproperG-CW-complexLwhoseisotropysubgroupshaveboundedorder.
We obtain a ring homomorphism f∗: K (L) → K (X) by the pullback construc-
G G
tion. HencewecanregardK∗(X)asamoduleovertheringK (L). PutI =I (L).
G G G
For any natural number n≥1, consider the composite
In·K∗(X)⊆K∗(X)−p−r→∗ K∗(EG×X)−∼=→K∗(EG× X)
G G G G
−K−−∗(−in−−−1→) K∗((EG× X)n−1),
G
wherepr: EG×X →X istheprojection,i : (EG× X)n−1 →EG× X isthe
n−1 G G
inclusionofthe (n−1)-skeletonandthe isomorphismK∗(EG×X)−∼→= K∗(EG×
G G
X) comes from dividing out the (free proper) G-action [21, Proposition 3.3]. This
composite is trivial,since the image is containedin the idealwhich is generatedby
the set I((EG× X)n−1)n, and the latter is trivial, since I((EG× X)n−1)n = 0
G G
by Lemma 2.2. The composites therefore define a pro-homomorphism
(2.3) λX,f: {K∗(X)/In·K∗(X)} → {K∗((EG× X)n−1)},
G G G
6 MICHAELJOACHIMANDWOLFGANGLU¨CK
where the structure maps on the left side are given by the obvious projections
and on the right side are induced by the various inclusions of the skeletons. The
followingtheoremistakenfromLu¨ck-Oliver[21, Theorem6.5]),[22,Theorem4.3]).
Theorem 2.4 (CompletionTheorem). Let G bea discrete group. Let X bea finite
proper G-CW-complex and let L be a finite dimensional proper G-CW-complex
whose isotropy subgroups have bounded order. Fix a G-map f: X →L and regard
K∗(X) as a module over K (L). Moreover, let I = I (L) be the augmentation
G G G
ideal.
Then
λX,f: {K∗(X)/In·K∗(X)}→{K∗((EG× X)n−1)}
G G G
is a pro-isomorphism of pro-Z-modules.
The inverse system {K∗(X)/In·K∗(X)} satisfies the Mittag-Leffler condition.
G G
In particular
lim1 K∗−1(EG× X)n)=0,
←−n≥1 G
and λX,f and the various inclusions i : (EG× X)n →EG× X induce isomor-
n G G
phisms
K∗(X) −∼→= K∗(EG× X)−∼→= lim K∗((EG× X)n),
G I G ←−n≥1 G
where K∗(X) =lim K∗(X)/In·K∗(X) is the I-adic completion of K∗(X).
G I ←−n≥1 G G G
b
Remark 2.5. In the case where G is finite and L is a point, Theorem 2.4 above
b
coincideswiththe Atiyah-Segalcompletiontheoremforfinite groups(see [7, Theo-
rem2.1]). The classicalAtiyah-Segalcompletiontheoremis statedfor compactLie
groups. However, the theorem above does not hold if G is replaced by a Lie group
of positive dimension (see [22, Section 5]).
We now pass to K-homology.
Lemma 2.6. If X is a finite proper G-CW-complex, then EG× X is homotopy
G
equivalent to a CW-complex of finite type.
Proof. We use induction over the dimension and subinduction over the number
of equivariant cells of top dimension in X. The induction beginning X = ∅ is
trivial. In the induction step we write X as a pushout of a diagram G/H ×Dn ←−i
G/H × Sn−1 → Y, where i is the inclusion, Y a G-CW-subcomplex of X and
n=dim(X). We obtain a pushout of CW-complexes
EG× G/H ×Sn−1 //EG× Y
G G
idEG×Gi
(cid:15)(cid:15) (cid:15)(cid:15)
EG× G/H ×Dn // EG× X
G G
with id × i a cofibration. Hence EG× X has the homotopy type of a CW-
EG G G
complex of finite type if EG× G/H ×Sn−1, EG× Y and EG× G/H ×Dn
G G G
have this property. This is true for the first two by the induction hypothesis and
for the third one since it is homotopy equivalent to BH. (cid:3)
Now we can give the proof of the Cocompletion Theorem 0.2
Proof of Theorem 0.2. Because of Lemma 2.6 we can choose a CW-complex Y of
finite type and a cellular homotopy equivalence f: Y →EG× X. Let fn: Yn →
G
(EG× X)n be the map induced on the n-skeletons. Notice that fn is not nec-
G
essarily a homotopy equivalence and K∗(fn) is not necessarily an isomorphism.
Nevertheless,oneeasilychecksthatweobtainapro-isomorphismofpro-Z-modules
{K∗(fn)}: {K∗((EG× X)n)}→{K∗(Yn)}.
G
TOPOLOGICAL K-(CO-)HOMOLOGY OF CLASSIFYING SPACES ... 7
Thus we obtain from the Completion Theorem 2.4 a pro-isomorphism of pro-Z-
modules
{K∗(fn)}◦λX,f: {K∗(X)/In·K∗(X)}→{K∗(Yn)}.
G G
¿From the K-homology version of the universal coefficient theorem for topologi-
cal K-theory 5.1 for finite CW-complexes and the fact that colim is an exact
−−−→n≥1
functor, we get the exact sequence
0→colim ext1(K∗+1(Yn),Z)→K (Y)
−−−→n≥1 Z ∗
→colim hom (K∗(Yn),Z)→0.
−−−→n≥1 Z
The map f and the pro-isomorphism {K∗(fn)}◦λX,f induce isomorphisms (see
Lemma 1.2)
K (f): K (Y) −∼=→ K (EG× X);
∗ ∗ ∗ G
colim ext1(K∗(X)/In·K∗(X),Z) −∼=→ colim ext1(K∗+1(Yn),Z);
−−−→n≥1 Z G G −−−→n≥1 Z
colim hom (K∗(X)/In·K∗(X),Z) −∼=→ colim hom (K∗(Yn),Z).
−−−→n≥1 Z G G −−−→n≥1 Z
Combining these isomorphisms with the exact sequence above proves the Cocom-
pletion Theorem 0.2. (cid:3)
Remark 2.7. The Cocompletion Theorem0.2 can be formulated elegantly within
the category of abelian topological groups and continuous homomorphisms. If we
equip the completion K∗(X) with the I-adic topology and Z with the discrete
G I
topologythenthe setofcontinuoushomomorphismshom (K∗(X) ,Z)is isomor-
cts G I
phic to c−−ol−i→mn≥1homZ(KG∗(Xb)/In·KG∗(X),Z). On the other hand, although the
category of topological abelian groups is not exact one can introdubce a notion of
exact sequences (in the sense of [39, Section 1.1]) and correspondingly a notion of
a group of isomorphisms classes of extensions (see [39, Corollary on page 537]).
In the case at hand we get that the group of isomorphisms classes of extensions
ext (K∗(X) ,Z), the continuous ext-group in the sense of [39], is isomorphic to
cts G I
colim ext1(K∗(X)/In·K∗(X),Z). Withtheseidentificationstheexactsequence
−−−→n≥1 Z G G
of the Cocompbletion Theorem 0.2 reads
0→ext1 (K∗+1(X) ,Z)→K (EG× X)→hom (K∗(X) ,Z)→0.
cts G I ∗ G cts G I
Remark 2.8. The Cocompletion Theorem 0.2 is closely related to the local coho-
mology approachto equivbariant K-homology due to Greenlees [15].bIf G is a finite
group it follows from [15, (4.2) and (5.1)] that there is a short exact sequence
(2.9) 0→H1(KG(X)) →K (EG× X)→H0(KG(X)) →0,
I ◦ ∗+1 ∗ G I ◦ ∗
where Hk(M ) denotes the k-th local cohomology of the graded R(G)-module M
I ◦ ◦
with respect to the augmentation ideal I ⊂R(G). A precise definition of the local
cohomologygroupsoccuringin(2.9)canbefoundin[15,Section2]. ByaTheorem
of Grothendieck in [16] (quoted in [15] as Theorem 2.5 (ii)) one has
Hn(KG(X)) ∼=colim extn (R(G)/In,KG(X)).
I ◦ ∗ −−−→n≥1 R(G) ∗
Using exact sequence of the Equivariant Universal Coefficient Theorem for K-
homology as stated in Remark 5.21, the adjunction
hom (M,hom (C,N))∼=hom (M ⊗ C,N))
R(G) R(G) R(G) R(G)
for R(G)-modules M,C,N with C being finitely generated, and the R(G)-module
isomorphism exti (M,R(G)) ∼= exti(M,Z) (emphasized in Remark 5.21) one
R(G) Z
can see that the exact sequence of the Cocompletion Theorem 0.2 is exact if and
onlyif (2.9)is. InparticulartheCocompletionTheoremyieldsanalternativeproof
for the exactness of (2.9).
8 MICHAELJOACHIMANDWOLFGANGLU¨CK
3. Borel cohomology
Let H∗ be a (generalized) cohomology theory with values in the category of Z-
modules which satisfies the disjoint union axiom for arbitrary index sets, i.e., for
any family {X |i∈I} the map
i
Hk(j ): Hk( X )−∼=→ Hk(X )
i i i
i∈I i∈I i∈I
Y a Y
is an isomorphism, where j : X → X is the canonical inclusion. Any such
i i i∈I i
theory H∗ is given by an Ω-spectrum E and, vice versa, any cohomology theory
`
givenbyanΩ-spectrumsatisfiesthedisjointunionaxiom. GivenaCW-complexX,
letHk(X)bethecokernelofthemapHk({•})→Hk(X)inducedbytheprojection
X → {•}. Our main example for H∗ will be topological K-theory K∗. If M is
an aebelian group, we define the cohomology theory H∗(−;M) by the Ω-spectrum
which is the fibrant replacement of the smash product of the spectrum associated
withH∗ withtheMoorespectrumassociatedtoM. IfM isaringR,thenH∗(−;R)
takes values in the category of R-modules.
Lemma 3.1. Let X be a CW-complex such that its reduced singular cohomology
Hk(X;Hl({•}))withcoefficientsintheabelian groupHl({•})vanishesfor allk ≥0
and l ∈Z. Then
e (i) The inclusion Xn−1 → Xn of the (n − 1)-skeleton into the n-skeleton
induces the zero-map Hk(Xn)→Hk(Xn−1) for all k ∈Z and n≥2. The
pro-Z-module {Hk(Xn)} is pro-trivial;
(ii) We have Hk(X)=0 foer all k ∈Z.e
Proof. (i) Since Xn is efinite dimensional, the reduced Atiyah-Hirzebruch spec-
tral cohomology seequence converges to Hk+l(Xn). It has as E -term Ek,l(Xn) =
2 2
Hk(Xn;Hl({•})). Since Hk(X;Hl({•})) = 0 for all k, we have Ek,l(Xn) = 0 for
2
k 6= n. This implies Ek,l(Xn) = 0 for ke6= n. We have the descending filtration
∞
Fek,m−kHm(Xn) of Hm(Xen) such that
Fk,lHm(Xn)/Fk+1,l−1Hm(Xn)∼=Ek,l(Xn).
∞
Hence Fk,lHm(Xn) = 0 for k ≥ n and Fk,lHm(Xn) = Hm(Xn) for k < n. Since
the map Hm(Xn)→Hm(Xn−1) respects this filtration, it must be trivial.
(ii) Recall Milnor’s exact sequence [38, Theorem 1.3 in XIII.1 on page 605]
0→lim1 Hk−1(Xn)→Hk(X)→lim Hk(Xn)→0.
←−n→∞ ←−n→∞
Since {Hk(Xn)} is pro-trivial for all k ∈Z, we conclude from Lemma 1.1
e e e
lim1 Hk−1(Xn) = 0;
e ←−n→∞
lim Hk(Xn) = 0.
←−n→∞e
This finishes the proof of Lemma 3.1. (cid:3)
e
Lemma 3.2. Let Y be a finite CW-complex Let R be a commutative associative
ring which is flat over Z. Then the canonical R-map
Hk(Y)⊗ R−∼→= Hk(Y;R)
Z
is an isomorphism.
Proof. SinceRisflatoverZ,themapHk(Y)⊗ R−∼=→Hk(Y;R)isatransformation
Z
of homology theories. It is bijective for Y = {•}. Hence by a Mayer-Vietoris
argument it is bijective for every finite CW-complex Y. (cid:3)
TOPOLOGICAL K-(CO-)HOMOLOGY OF CLASSIFYING SPACES ... 9
Lemma 3.3. Let X be a finite proper G-CW-complex. Let P(X) be the set of
primespwhichdividetheorderofsomeisotropygroupofX. LetZ⊆Z 1 ⊆Q
P(X)
be the ring obtained from Z by inverting the elements in P(X). Let q(Xh ): EiG×G
X →G\X be the projection. Then there is for k ∈Z a R-map, natural in X,
1 1
rk(X): Hk(EG× X)⊗ Z →Hk(G\X)⊗ Z
G Z Z
P(X) P(X)
(cid:20) (cid:21) (cid:20) (cid:21)
suchthatrk(X)◦Hk(q(X))⊗ id: Hk(G\X))⊗ Z 1 →Hk(G\X))⊗ Z 1
Z Z P(X) Z P(X)
is an isomorphism. h i h i
Proof. Put Λ=Z 1 . Consider the following commutative diagram
P(X)
h i
Hk(G\X)⊗ Λ Hk(q(X))⊗ZΛ //Hk(EG× X)⊗ Λ
Z G Z
(cid:15)(cid:15) (cid:15)(cid:15)
Hk(G\X;Λ) Hk(q(X);Λ) //Hk(EG× X;Λ)
G
The left vertical arrow is an isomorphism by Lemma 3.2. Hence it suffices to show
thatthelowerhorizontalmapisanisomorphism. SinceX isfiniteproper,aMayer-
Vietoris argumentshows that it suffices to treat the case X =G/H for some finite
group H ⊂ G such that |H| is invertible in Λ. Since Hk(BH;Hq({•};Λ)) = 0
vanishes for all k by [13, Corollary 10.2 in Chapter III on page 84], this follows
from Lemma 3.1 (ii). e (cid:3)
Lemma 3.4. Let H be a finite group. Let P(H) be the set of primes dividing |H|.
The canonical map of pro-Z-modules
{Hk(BHn−1)}−∼=→ {Hk(BHn−1;Z )}
p
p∈P(H)
Y
is a pro-isomorphismefor k ∈Z. The pro-moduele b
{Hk(BHn−1;Z )}
p
pprime
Y
p6∈P(H) e b
is pro-trivial.
Proof. We have the exact sequence of abelian groups
0→Z−→i Z →coker(i)→0
p
p∈YP(H)
whereiistheproductofthecanonicalembbeddingsZ→Z . Itinducesalongexact
p
sequence
b
...→Hk−1(BHn−1;coker(i))→Hk(BHn−1)→Hk(BHn−1; Z )
p
p∈YP(H)
e e →eHk(BHn−1;coker(i)b)→...
and thus an exact sequence of pro-Z-modules
e
{Hk−1(BHn−1;coker(i))}→{Hk(BHn−1)}→{Hk(BHn−1; Z )}
p
p∈YP(H)
e e e →{Hk(BHn−1;cobker(i))}
e
10 MICHAELJOACHIMANDWOLFGANGLU¨CK
Multiplication with the order of |H| induces an isomorphism |H|·id: coker(i) −∼=→
coker(i). HenceHk(BH;Hl({•};coker(i)))vanishesforallk,l∈Zby[13,Corollary
10.2 in Chapter III on page 84]. We conclude from Lemma 3.1 (i) that the pro-
Z-module {Hk(BeHn−1;coker(i)} is trivial. This shows that the obvious map of
pro-Z-modules
e {Hk(BHn−1)}−∼=→{Hk(BHn−1; Z )}
p
p∈YP(H)
is bijective. The canoenical map e b
Hk(Y; Z )→ Hk(Y;Z )
p p
p∈YP(H) p∈YP(H)
is a natural transformeation of cohombology theoriees satisfbying the disjoint union
axiom and is an isomorphism for Y ={•} since the set P(H) is finite. Hence it is
an isomorphism for every finite-dimensional Y and in particular for Y = BHn−1.
We conclude that the obvious map of pro-Z-modules
{Hk(BHn−1)}−∼=→ {Hk(BHn−1;Z )}
p
p∈YP(H)
is bijective. e e b
If p does not divide |H|, then Hk(BH;Hq({•};Z )) = 0 by [13, Corollary 10.2
p
inChapterIII onpage84]. We concludefromLemma3.1(i)thatthe mapinduced
by the inclusion e b
{Hk(BHn;Z )}→{Hk(BHn−1;Z )}
p p
is trivial for all k ∈Z and n≥2. This implies that
e b e b
{Hk(BHn−1;Z )}
p
p6∈YP(H)
is pro-trivial. e b (cid:3)
Lemma 3.5. Let X be a proper G-CW-complex. Let P be a set of primes con-
taining P(X). Then the canonical map
Hk(q(X): EG× X →G\X)−∼=→ Hk(q(X): EG× X →G\X;Z )
G G p
p∈P
Y
is an isomorphism. b
Proof. WeconcludefromtheMilnor’sexactsequence[38,Theorem1.3inXIII.1on
page 605] and the Five-Lemma that it suffices to treat the case, where X is finite
dimensional. Using Mayer-Vietorissequences the claim can be reduced to the case
X = G/H for some finite group H such that P contains the set P(H) of primes
dividing the order of H. So we must show the canonical map
Hk(BH)−∼=→ Hk(BH;Z )
p
p∈P
Y
is bijective for any finite greoup H with (H)e⊆ P. BybMilnor’s exact sequence [38,
Theorem 1.3 in XIII.1 on page 605] and the Five-Lemma, it remains to show the
bijectivity of
lim Hk(BHn−1) → lim Hk(BHn−1;Z );
←−n→∞ ←−n→∞ p
p∈P
Y
lim1 Hek(BHn−1) → lim1 Hek(BHn−1;Zb).
←−n→∞ ←−n→∞ p
p∈P
Y
This follows from Lemema 1.1 and Lemma 3.4. e b (cid:3)