Chern characters for equivariant K -theory of proper
G-CW-complexes
by Wolfgang L"uck and Bob Oliver*
In an earlier paper [10], we showed that for any discrete group G, equivaria*
*nt K-theory
for finite proper G-CW-complexes can be defined using equivariant vector bundle*
*s. This was
then used to prove a version of the Atiyah-Segal completion theorem in this sit*
*uation. In
this paper, we continue to restrict attention to actions of discrete groups, an*
*d begin by con-
structing an appropriate classifying space which allows us to define K*G(X) for*
* an arbitrary
proper G-complex X. We then construct rational-valued equivariant Chern charac*
*ters for
such spaces, and use them to prove some more general versions of completion the*
*orems.
In fact, we construct two different types of equivariant Chern character, bo*
*th of which
involve Bredon cohomology with coefficients in the system G=H 7! R(H) . The fi*
*rst,
ch*X: K*G(X) ------! H*G(X; Q R(-));
is defined for arbitrary proper G-complexes. The second, a refinement of the f*
*irst, is a
homomorphism
ech*X: K*G(X) ------! Q H*G(X; R(-));
but defined only for finite dimensional proper G-complexes for which the isotro*
*py subgroups
on X have bounded order. When X is a finite proper G-complex (i.e., X=G is a f*
*inite
CW-complex), then H*G(X; R(-)) is finitely generated, and these two target grou*
*ps are
isomorphic. The second Chern character is important when proving the completio*
*n theo-
rems. The idea for defining equivariant Chern characters with values in Bredon *
*cohomology
H*G(X; Q R(-)) was first due to Slominska [14]. A complex-valued Chern charact*
*er was
constructed earlier by Baum and Connes [5], using very different methods.
The completion theorem of [10] is generalized in two ways. First, we prove i*
*t for real as
well as complex K-theory. In addition, we prove it for families of subgroups i*
*n the sense
of Jackowski [9]. This means that for each finite proper G-complex X and each *
*family F
of subgroups of G, K*G(EF (G) x X) is shown to be isomorphic to a certain compl*
*etion of
K*G(X). In particular, when F = {1}, then EF (G) = EG, and this becomes the us*
*ual
completion theorem.
The classifying spaces for equivariant K-theory are constructed here using S*
*egal's -
spaces. This seems to be the most convenient form of topological group completi*
*on in our
___________________________________
*Partly supported by UMR 7953 of the CNRS
1
situation. However, although -spaces do produce spectra, as described in [12], *
*the spectra
they produce are connective, and hence not what is needed to define equivariant*
* K-theory
directly. So instead, we define K-nG(-) and KO-nG(-) for all n 0 using classif*
*ying spaces
constructed from a -space, then prove Bott periodicity, and use that to define *
*the groups
in positive degrees. One could, of course, construct an equivariant spectrum (o*
*r an Or (G)-
spectrum in the sense of [7]) by combining our classifying space KG with the Bo*
*tt map
2KG ! KG; but the approach we use here seems the simplest way to do it.
By comparison, in [7], equivariant K-homology groups KG*(X) were defined by *
*using
certain covariant functors Ktopfrom the orbit category Or(G) to spectra. This c*
*onstruction
played an important role in [7] in reformulating the Baum-Connes conjecture. I*
*n general,
one expects an equivariant homology theory to be classified by a covariant func*
*tor from the
orbit category to spaces or spectra, and an equivariant cohomology theory to be*
* classified
by a contravariant functor. But in fact, when defining equivariant K-theory her*
*e, it turned
out to be simplest to do so via a classifying G-space, rather than a classifyin*
*g functor from
Or (G) to spaces.
We would like in particular to thank Chuck Weibel for suggesting Segal's pap*
*er and the
use of -spaces, as a way to avoid certain problems we encountered when first tr*
*ying to
define the multiplicative structure on KG(X).
The paper is organized as follows. The classifying spaces for K-nG(-) and KO*
*-nG(-) are
constructed in Section 1; and the connection with G-vector bundles is described*
*. Products
are then constructed in Section 2, and are used to define Bott homomorphisms an*
*d ring
structures on K*G(X); and thus to complete the construction of equivariant K-th*
*eory as
a multiplicative equivariant cohomology theory. Homomorphisms in equivariant K*
*-theory
involving changes of groups are then constructed in Section 3. Finally, the equ*
*ivariant Chern
characters are constructed in Section 5, and the completion theorems are formul*
*ated and
proved in Section 6. Section 4 contains some technical results about rational c*
*haracters.
1. A classifying space for equivariant K-theory
Our classifying space for equivariant K-theory for proper actions of an infi*
*nite discrete
group is constructed using -spaces in the sense of Segal. So we begin by summar*
*izing the
basic definitions in [12].
Let be the category whose objects are finite sets, and where a morphism : *
*S ! T
sends each element s 2 S to a subset (s) T such that s 6= s0 implies (s) \ (s0*
*) = ;.
Equivalently, if P(S) denotes the set of subsets of S, one can regard a morphis*
*m in as a
map P(S) ! P(T ) which sends disjoint unions to disjoint unions. For all n 0, *
*n denotes
the object {1; : :;:n}. (In particular, 0 is the empty set.) There is an obviou*
*s functor from
the simplicial category to , which sends each object [n] = {0; 1; : :;:n} in *
*to n, and
2
where a morphism in _ an order preserving map ' : [m] ! [n] _ is sent to the m*
*orphism
' : m ! n in which sends i to {j | '(i- 1) < j '(i)}.
A -space is a functor A_: op! Spaces which satisfies the following two condi*
*tions:
(i) A_(0) is a point; and
Q n
(ii) for each n > 1, the map A_(n) --! i=1A_(1), induced by the inclusion*
*s i: 1 ! n
(i(1) = {i}), is a homotopy equivalence.
(In fact, Segal only requires that A_(0) be contractible; but for our purposes *
*it is simpler
to assume it is always a point.) Note that each A(S) has a basepoint: the imag*
*e of A(0)
induced by the unique morphism S ! 0. We write A = A_(1), thought of as the "un*
*derlying
space" of the -space A_. A -space A_: op! Spaces can be regarded as a simplicia*
*l space
via restriction to , and |A_| denotes its topological realization (nerve) as a *
*simplicial space.
If A_is a -space, then BA__denotes the -space BA__(S) = |A_(S x -)|; and thi*
*s is iterated
to define BnA__for all n. Thus, BnA = BnA__(1) is the realization of the n-sim*
*plicial space
which sends (S1; : :;:Sn) to A_(S1 x . .x.Sn). Since A_(0) is a point, we can *
*identify A
(= (A_(1))) as a subspace of BA ~=|A_|; and this induces by adjointness a map A*
* ! BA.
Upon iterating this, we get maps (BnA) ! Bn+1A for all n; and these make the se*
*quence
A; BA; B2A; : :i:nto a spectrum. This is "almost" an -spectrum, in that BnA ' B*
*n+1A
for all n 1 [12, Proposition 1.4].
Note that for any -space A_, the underlying space A = A_(1) is an H-space: m*
*ultiplication
is defined to be the composite of a homotopy inverse of the equivalence A_(2) -*
*'-!A_(1)xA_(1)
with the map A_(2) ! A_(1) induced by m2 : 1 ! 2 (m2(1) = {1; 2}). Then A ' BA *
*if
ss0(A) is a group; and BA is the topological group completion of A otherwise. A*
*ll of this
is shown in [12, x1].
We work here with equivariant -spaces; i.e., withQfunctors A_ : op ! G-Space*
*s for
which A_(0) is a point, and for which A_(n) H ! ni=1A_(1) H is a homotopy eq*
*uivalence
for all H G. In other words, restriction to fixed point sets of any H G defi*
*nes a
-space A_H; and the properties of equivariant -spaces follow immediately from t*
*hose of
nonequivariant ones. For example, Segal's [12, Proposition 1.4] implies immedia*
*tely that for
any equivariant -space A_, BnA ! Bn+1A is a weak equivalence for all n 1 in th*
*e sense
that it restricts to an equivalence (BnA)H ' (Bn+1A)H for all H G. This motiva*
*tes the
following definitions.
If F is any family of subgroups of G, then a weak F-equivalence of G-spaces *
*is a G-map
whose restriction to fixed point sets of any subgroup in F is a weak homotopy e*
*quivalence
in the usual sense. The following lemma about maps to weak equivalences is well*
* known; we
note it here for later reference.
Lemma 1.1 Fix a family F of subgroups of G, and let f : Y ! Y 0be any wea*
*k F-
equivalence. Then for any G-complex X all of whose isotropy subgroups are in F,*
* the map
~= 0
f* : [X; Y ]G -----! [X; Y ]G
3
is a bijection. More generally, if A X is any G-invariant subcomplex, and all*
* isotropy
subgroups of Xr A are in F, then for any commutative diagram
A -ff0--!Y
? ?
? ?
y fy
X --ff-!Y 0
of G-maps, there is an extension of ff0 to a G-map eff: X ! Y such that fOeff' *
*ff (equivari-
antly homotopic), and effis unique up to equivariant homotopy.
Proof______: The idea is the following. Fix a G-orbit of cells G=H x Dn ! X *
* in X whose
boundary is in A. Then, since Y H ! (Y 0)H is a weak homotopy equivalence, the *
*map eH x
Dn ! XH ! (Y 0)H can be lifted to Y H (up to homotopy), and this extends equiv*
*ariantly
to a G-map G=H x Dn ! Y . Upon continuing this procedure, we obtain a lifting o*
*f ff to
a G-map eff: X ! Y which extends ff0. This proves the existence of a lifting in*
* the above
square (and the surjectivity of f* in the special case); and the uniqueness of *
*the lifting follows
upon applying the same procedure to the pair Xx I (Xx {0; 1}) [ (Ax I). |
Now fix a discrete group G. Let E(G) be the category whose objects are the e*
*lements of G,
and with exactly one morphism between each pair of objects. Let B(G) be the cat*
*egory with
one object, and one morphism for each element of G. (Note that |E(G)| = EG and *
*|B(G)| =
BG; hence the notation.) When necessary to be precise, ga will denote the morph*
*ism a ! ga
in E(G). We let G act on E(G) via right multiplication: x 2 G acts on objects b*
*y sending
a to ax and on morphisms by sending ga to gax. Thus, for any H G, the orbit ca*
*tegory
E(G)=H is the groupoid whose objects are the cosets in G=H, and with one morphi*
*sm
gaH : aH ! gaH for each g 2 G: a category which is equivalent to B(H). Note in *
*particular
that B(G) ~=E(G)=G.
In order to deal simultaneouslySwith real and complex K-theory, we let F den*
*ote one of
the fields C or R. Set F 1 = 1n=1F n: the space of all infinite sequences in *
*F with finitely
many nonzero terms. Let F -mod be the category whose objects are the finite dim*
*ensional
vector subspaces of F 1, and whose morphisms are F -linear isomorphisms. The se*
*t of objects
of F -mod is given the discrete topology, and the space of morphisms between an*
*y two objects
has the usual topology.
For any finite set S, an S-partitionedLvector space is an object V of F -mo*
*d , together
with a direct sum decomposition V = s2SVs. Let F ~~-mod denote the category*
* of S-
partitioned vector spaces in F -mod , where morphisms are isomorphisms which re*
*spect the
decomposition. In particular, F <0>-mod has just one object 0 F 1 and one mo*
*rphism.
A morphismL : S ! TLinduces a functor FL<> from F -mod to F ~~~~-mod , by se*
*nding
V = t2TVt to W = s2SWs where Ws = t2(s)Vt.
Let Vec_FGbe the -space defined by setting
fi fi
Vec_FG(S) def=fifunc(E(G); F ~~~~-mod )fi
4
for each finite set S. Here, func(C; D) denotes the category of functors from *
*C to D. We
give this the G-action induced by the action on E(G) described above. This is m*
*ade into a
functor on via composition with the functors F <>.
By definition, Vec_FG(0) is a point. To see that Vec_FGisQan equivariant -sp*
*ace, it remains
to show for each n and H that the map Vec_FG(n) H ! ni=1Vec_FG(1) H is a h*
*omotopy
equivalence. The target is the nerve of the category of functors from E(G)=H to*
* n-tuples of
objects in F -mod , while the source can be thought of as the nerve of the full*
* subcategory of
functors from E(G)=H to n-tuples of vector subspaces which are independant in F*
* 1. And
these two categories are equivalent, since every object in the larger one is is*
*omorphic to an
object in the smaller (and the set of objects is discrete).
For all finite H G, VecFGH is the disjoint union, taken over isomorphism *
*classes of
finite dimensional H-representations, of the classifying spaces of their automo*
*rphism groups.
We will see later that VecFGclassifies G-vector bundles over proper G-complexes*
*. So it is
natural to define equivariant K-theory using the its group completion KFG def=B*
*Vec FG,
regarded as a pointed G-space.
In the following definition, [-; -]G and [-; -].Gdenote sets of homotopy cla*
*sses of G-
maps, and of pointed G-maps, respectively.
Definition 1.2 For each proper G-complex X, set
KG(X) = [X; KCG]G and KOG(X) = [X; KRG]G:
For each proper G-CW-pair (X; A) and each n 0, set
K-nG(X; A) = [n(X=A); KCG].G and KO-nG(X; A) = [n(X=A); KRG].G:
The usual cohomological properties of the KFG-n(-) follow directly from the *
*definition.
Homotopy invariance and excision are immediate; and the exact sequence of a pai*
*r and the
Mayer-Vietoris sequence of a pushout square are shown using Puppe sequences to *
*hold in
degrees 0. Note in particular the relations
n
KFG-n(X) ~=Ker KFG(S x X) --! KFG(X)
-n -n (*
*1.3)
KFG-n(X; A) ~=Ker KFG (X [A X) --! KFG (X) ;
for any proper G-CW-pair (X; A) and any n 0.
The following lemma will be needed in the next section. It is a special case*
* of the fact
that Vec_FGand KFG (at least up to homotopy) are independent of our choice of c*
*ategory of
F -vector spaces.
Lemma 1.4 For any monomorphism ff : F 1 ! F 1, the induced map ff* : Vec_FG!*
* Vec_FG,
ff(-)
defined by composition with F -mod ---! F -mod , is G-homotopic to the identit*
*y. In par-
ticular, ff* induces the identity on KG(X).
5
Proof
_________: The functor V 7! ff(V ) is naturally isomorphic to the identity.*
* |
In [10], we defined KG(X), for any proper G-complex X, to be the Grothendiec*
*k group of
the monoid of vector bundles over X. We next construct natural homomorphisms KG*
*(X) !
KG(X), for all proper G-complexes X, which are isomorphisms if X=G is a finite *
*complex
(this is the situation where the K*G(X) form an equivariant cohomology theory).
For each n 0, let F n-mod F -mod be the full subcategory of n-dimensional *
*vector
subspaces in F 1. Let F n-frame denote the category whose objects are the pair*
*s (V; b),
where V is an object of F n-mod and b is an ordered basis of V ; and whose morp*
*hisms are
the isomorphisms which send ordered basis to ordered basis. The set of objects *
*is given the
topology of a disjoint union of copies of GLn(F ) (one for each V in F n-mod )*
*. Note that
there is a unique morphism between any pair of objects in F n-frame. Set
fi fi F;n fi *
* fi
VecF;nG= fifunc(E(G); F n-mod )fi and VgecG = fifunc(E(G); F n-fram*
*e)fi;
with the action of G x GLn(F ) on gVecF;nGinduced by the G-action on E(G) and t*
*he GLn(F )-
action on the set of ordered bases of each n-dimensional V . Let on : gVecF;nG!*
* VecF;nGbe the
G-map induced by the forgetful functor F n-frame ! F n-mod . Then GLn(F ) acts *
*freely
and properly on gVecF;nG. And on induces a G-homeomorphism gVecF;nG=GLn(F ) ~=*
* VecF;nG,
since for any ' : V ! V 0in F -mod , a lifting of V or V 0to F n-frame determin*
*es a unique
lifting of the morphism.
Let H G x GLn(F ) be any subgroup. If H \ (1 x GLn(F )) 6= 1, then (gVecF;n*
*G)H = ;,
since GLn(F ) acts freely on gVecF;nG. So assume H \ (1 x GLn(F )) = 1. Then H *
*is the graph
of some homomorphism ' : H0 ! GLn(F ) (H0 G), and (gVecF;nG)H is the nerve of*
* the
(nonempty) category of '-equivariant functors E(G) ! F n-frame, with a unique m*
*orphism
between any pair of objects (since there is a unique morphism between any pair *
*of objects
in F n-frame). In particular, this shows that (gVecF;nG)H is contractible.
Thus, gVecF;nGis a universal space for those (G x GLn(F ))-complexes upon wh*
*ich GLn(F )
acts freely (cf. [10, x2]). The frame bundle of any n-dimensional G-F -vector b*
*undle over a
G-complex X is such a complex, and hence n-dimensional G-F -vector bundles over*
* X are
classified by maps to VecF;nG= gVecF;nG=GLn(F ). It follows that
EVecF;nG= gVecF;nGxGLn(F)F n-------! VecF;nG
is a universal n-dimensional G-F -vector bundle. And [X; VecF;nG]G ~=VectF;nG(X*
*): the set of
isomorphism classes of n-dimensional G-F -vector bundles over X.
If E is any G-F -vector bundle over X, we let [[E]] 2 KFG(X) = [X; KFG]G be *
*the
composite of the classifying map X ! VecFGfor E with the group completion map V*
*ecFG!
BVec FG= KFG. Any pair E; E0 of vector bundles over X is induced by a G-map
X -----! Vec FGx VecFG= | func(E(G); F -mod x F -mod )| ' | func(E(G); F <2>*
*-mod )|;
6
and upon composing with the forgetful functor F <2>-mod ! F -mod we get the cl*
*assifying
map for E E0. The direct sum operation on VectFG(X) is thus induced by the H-*
*space
structure on VecFG, and [[E E0]] = [[E]] + [[E0]] for all E; E0.
Proposition 1.5 The assignment [E] 7! [[E]] defines a homomorphism
flX : KFG(X) -----! KFG(X);
for any proper G-complex X. This extends to natural homomorphisms fl-nX;A: KFG-*
*n(X; A) !
KFG-n(X; A), for all proper G-CW-pairs (X; A) and all n 0; which are isomorphi*
*sms when
restricted to the category of finite proper G-CW-pairs.
Proof______: By the above remarks, [E] 7! [[E]] defines a homomorphism of mo*
*noids from
VectFG(X) to KFG(X), and hence a homomorphism of groups
flX : KFG(X) ------! KFG(X):
Homomorphisms fl-nX;A(for all proper G-CW-pairs (X; A)) are then constructed vi*
*a the def-
initions
KFG-n(X) def=Ker[KFG(Sn x X) ! KFG(X)]
and KFG-n(X; A) def=Ker[KFG-n(X[A X) ! KFG-n(X)] used in [10], together with th*
*e anal-
ogous relations (1.3) for K*G(-). These homomorphisms clearly commute with bou*
*ndary
maps.
It remains to check that fl-nXis an isomorphism whenever X is a finite prope*
*r G-complex.
Since KFG(-) and KFG(-) are both cohomology theories in this situation, it suff*
*ices, using
the Mayer-Vietoris sequences for pushout squares
G=H x Sm-1 ---! G=H x Dm
? ?
'?y ?y
X ---! (G=H x Dm ) [' X;
to do this when X = G=H x Sm for finite H G and any m 0. Using (1.3) again,*
* it
suffices to show that flX = fl0X is an isomorphism whenever X = G=H x Y for a*
*ny finite
complex Y with trivial G-action. By definition,
H
KFG(G=H x Y ) = G=H x Y; KFG G ~=[Y; (KFG) ];
while KFG(G=H x Y ) is the Grothendieck group of the monoid
F F H
VectFG(G=H x Y ) ~= G=H x Y; VecG G ~= Y; (VecG ) :
Since ss0((VecFG)H ) is a free abelian monoid (the monoid of isomorph*
*ism classes of
H-representations), [12, Proposition 4.1] applies to show that [-; (KFG*
*)H ] is univer-
sal among representable functors from compact spaces to abelian groups which ex*
*tend
VectFG(G=Hx -) ~= VectFH(-). And since KH is representable as a functor on co*
*mpact
spaces with trivial action (H is finite), it is the universal functor, and so [*
*Y; (KFG)H ] ~=
KH (Y ) ~=KG(G=H x Y ). |
7
2. Products and Bott periodicity
We now want to construct Bott periodicity isomorphisms, and define the multi*
*plicative
structures on K*G(X) and KO*G(X). Both of these require defining pairings of c*
*lassifying
spaces; thus pairings of -spaces. A general procedure for doing this was descri*
*bed by Segal
[12, x5], but a simpler construction is possible in our situation.
Fix an isomorphism : F 1 F 1 ! F 1 (F = C or R), induced by some bijection
between the canonical bases. This induces a functor
* : F ~~~~-mod x F -mod -----! F -mod;
and hence (for any discrete groups H and G)
* : Vec_FH(S) ^ Vec_FG(T ) -----! Vec_FHx(GSx T ): (*
*2.1)
This is an (Hx G)-equivariant map of bi--spaces, and after taking their nerves *
*(and loop
spaces) we get maps
F F 2|*| 2 2 F F
BVec FH^ BVec FG ----! 2 BVec H^ BVec G -----! B VecHx G ' BVec Hx G :
= KFH ^ KFG = KFHxG
(*
*2.2)
By Lemma 1.4, these maps are all independent (up to homotopy) of the choice of *
* :
F 1 F 1 ! F 1.
Lemma 2.3 For any discrete groups H and G, any H-space X, and any G-space Y *
*, the
following square commutes:
flXxflY
KFH (X) KFG(Y ) ----! KFH (X) KFG(Y )
? ?
?y * ?y
flXxY
KFHxG (X x Y ) ----! KFHxG (X x Y )
where * is the homomorphism induced by (2.2).
Proof______: The pullback of the universal bundle EVec FHx,Gvia the pairing Ve*
*cFH^ VecFG!
VecFHx Gof (2.1), is isomorphic to the tensor productfofithe universalfbundlesi*
*EVec FHand
EVec FG. This is clear if we identify EVecFG~=fifunc(E(G); F -Bdl)fi(and simila*
*rly for the other
two bundles), where F -Bdlis the category of pairs (V; x) for V in F -mod and x*
* 2 V . |
We now consider case where H = 1, and hence where KFH = Z x BU or Z x BO. The
product map (2.2), after composition with the Bott elements in ss2(BU) or ss8(B*
*O), induces
Bott maps
fiC*: 2KG -----! KG and fiR*: 8KOG -----! KOG: (*
*2.4)
8
Proposition 2.5 For any proper CW-pair (X; A), the Bott homomorphisms
bCX;A: K-nG(X; A) -----! K-n-2G(X; A) and bRX;A: KO-nG(X; A) -----! KO-n-8G(*
*X; A)
are isomorphisms; and commute with the homomorphisms
fl-nX;A: KFG-n(X; A) ! KFG-n(X; A):
Proof______: The last statement follows immediately from Lemma 2.3.
By Lemma 1.1, it suffices to prove that the adjoint maps
KG ----! 2KG and KOG ----! 8KOG
to the pairings in (2.4) are weak homotopy equivalences after restricting to fi*
*xed point sets
of finite subgroups of G. In other words, it suffices to prove that bCX: KG(X) *
*! K-2G(X)
and bRX: KOG(X) ! KO-8G(X) are isomorphisms when X = G=H x Sn for any n 0 and
any finite H G. And this follows since the Bott maps for KG and KO G are isomo*
*rphisms
[10, Theorems 3.12 & 3.15], since KFG-n(X) ~=KFG-n(X) (Proposition 1.5), and si*
*nce these
isomorphisms commute with the Bott maps. |
The K-nG(X) and KO-nG(X) can now be extended to (additive) equivariant cohom*
*ology
theories in the usual way. But before stating this explicitly, we first conside*
*r the ring structure
on KG(X). This is defined to be the composite
*
[X; KFG]G x [X; KFG]G -----! [X; KFGxG ]G -----! [X; KFG]G;
where the first map is induced by the pairing in (2.2), and the second by restr*
*iction to the
diagonal subcategory E(G) E(Gx G).
Before we can prove the ring properties of this multiplication, we must look*
* more closely
at the homotopy equivalence BVec FG'-!2B2VecFGwhich appears in the definition o*
*f the
product. In fact, there is more than one natural map from nBnVec FGto n+1Bn+1V*
*ecFG.
For each n 0 and each k = 0; : :;:n, let kn: nBnVec FG! n+1Bn+1VecFG denote the
map induced as n(f), where f is adjoint to the map BnVec FG! Bn+1VecFG, induced*
* by
identifying BnVec FG(S1; : :;:Sn) with Bn+1VecFG(: :;:Sk-1; 1; Sk; : :):.
By a weak G-equivalence f : X ! Y is meant a map of G-spaces which restrict*
*s to a
weak equivalence fH : XH ! Y H for all H G; i.e., a weak F-equivalence in the*
* notation
of Lemma 1.1 when F is the family of all subgroups of G. Since we are intereste*
*d equivariant
-spaces only as targets of maps from G-complexes, it suffices by Lemma 1.1 to w*
*ork in a
category where weak G-equivalences are inverted.
Lemma 2.6 Let A_ be any G-equivariant -space. Then for any n 1, the maps k*
*n:
nBnA ! n+1Bn+1A (for 0 k n) are all equal in the homotopy category of G-spaces
where weak G-equivalences are inverted.
9
Proof n n n n
_________: For any oe 2 n, let oe* : B A ! B A be the map induced by permut*
*ing the
coordinates of BnA as an n-simplicial set, and by switching the order of loopin*
*g. Then any
two of the kn-1differ by composition with some appropriate oe*, and so it suffi*
*ces to show
that the oe* are all homotopic to the identity.
Consider the following commutative diagram
' n+1 n+1 nn n n
BA - --! B A --- B A
? ? ?
Id?y (1xoe)*?y oe*?y
' n+1 n+1 nn n n
BA - --! B A --- B A;
for any oe 2 n n+1, where ' = 0nO. .O.01is induced by identifying A_(S)*
* with
A(____S; 1; : :;:1). The diagram commutes, and all maps in it are weak G-equiv*
*alences by
[12, Proposition 1.4]. So (1 x oe)* and oe* are both homotopic to the identity *
*after inverting
weak G-equivalences. |
We are now ready to show:
Theorem 2.7 For any discrete group and any proper G-complex X, the pairings X*
* define
a structure of graded ring on K*G(X) and on KO*G(X), which make K*G(-) and KO*G*
*(-)
into multiplicative cohomology theories. The Bott isomorphisms
bCX: K-nG(X) ! K-n-2G(X) and bRX: KO-nG(X) ! KO-n-8G(X)
are KG(X)- or KOG(X)-linear. And the canonical homomorphisms
flCX: K*G(X) ! K*G(X) and flRX: KO *G(X) ! KO*G(X)
are homomorphisms of rings.
Proof______: As usual, set F = C or R. We first check that X makes KFG(X) into *
*a commu-
tative ring _ i.e., that it is associative and commutative and has a unit.
To see that there is a unit, let [F 1] 2 VecFGdenote the vertex for the cons*
*tant functor
E(G) 7! F 12 F <1>-mod , and set [F 1] = 00([F 1]) 2 BVec FG. The following d*
*iagram
commutes:
[F1]^- F F * F
BVec FG ------! VecG ^ BVec G ------! BVec G
? ? ?
Id?y 00^Id?y' 01?y'
[F1] ^- F F * 2 2 F
BVec FG ------! BVec G^ BVec G ------! B VecG ;
and the composite in the top row is homotopic to the identity by Lemma 1.4. So*
* the
element 1 2 KFG(X), represented by the constant map X 7! [F 1] 2 KFG, is an id*
*entity
for multiplication in KFG(X).
10
The commutativity of KF
G(X) follows from Lemma 2.6 (the uniqueness of the map
BA ! 2B2A after inverting weak G-equivalences); together with the fact that the*
* pairing
* : BVec FG^ BVec FG-----! B2VecFG
commutes up to homotopy using Lemma 1.4. And associativity follows since the t*
*riple
products are induced by maps
F ^3 3 F ^3 3|*O(*^Id)| 3 3 F ' F
BVec G -----! (BVec G) ----------!----------! B VecG ----- BVec *
*G;
3|*O(Id^*)|
where the two maps in the middle are homotopic by Lemma 1.4, and the last could*
* be any
of the three possible maps by Lemma 2.6.
The extension of the product to negative gradings is straightforward, via th*
*e identifica-
tions of (1.3). For any n; m 0, the composite
proj* n m n m
KFG(Sn x X) KFG(Sm x X) -----! KFG(S x S x X) KFG(S x S x X)
n m
-----! KFG(S x S x X)
restricts to a product map KFG-n(X) KFG-m(X) ! KFG-n-m(X). To see that the pro*
*duct
has image in KFG-n-m(X), just note that
n+m
KFG-n-m(X) ~= Ker KFG(S x X) --! KFG(X)
n m n m
= Ker KFG(S x S x X) --! KFG(S x X) KFG(S x X) :
This product is clearly associative, and graded commutative (where the change i*
*n sign comes
from the degree of the switching map Sn+m ! Sm+n ).
We next check that this product commutes with the Bott maps in the obvious w*
*ay, so
that it can be extended to KiG(X) for all i. This means showing that the two ma*
*ps
KF (Sn) KFG(X) KFG(X) -------!-------!KFG(Sn x X)
induced by the products constructed above are equal. And this follows from the*
* same
argument as that used to prove associativity of the internal product on KFG(X).
Finally, fl : KFG*(X) ! KFG*(X) is a ring homomorphism by Lemma 2.3. |
3. Induction, restriction, and inflation
In this section we explain how the natural maps defined on KG(X) and KO G (X*
*) by
induction and restriction carry over to KG(X) and KOG(X). Namely, we want to co*
*nstruct
11
for any pair H G of discrete groups, any F = C or R, any G-complex X, and any
H-complex Y , natural induction and restriction maps
~= * G * *
IndGH: KFH*(Y ) ----! KFG(GxH Y ) and ResH : KFG(X) ----! KFH (X|H ):
Furthermore, when H C G is a normal subgroup, we construct an inflation homomor*
*phism
InflGG=H: KFG*=H(X=H) ------! KFG*(X);
which is an isomorphism whenever H acts freely on X. These maps correspond und*
*er
the natural homomorphism KFG*(X) ! KFG*(X) to the obvious homomorphisms induced
by induction, restriction, and pullback of vector bundles. They are all induce*
*d using the
following maps between classifying spaces for equivariant K-theory.
Lemma 3.1 Let f : G0! G be any homomorphism of discrete groups. Then composi*
*tion
with the induced functor E(f) : E(G0) ! E(G) induces an G0-equivariant map f* :*
* Vec_FG!
Vec_____FG0of -spaces, and hence a G0-equivariant map f* : KFG ! KFG0 of classi*
*fying spaces.
And for any subgroup L G0such that L\Ker (f) = 1, f* restricts to a homotopy e*
*quivalence
(KFG)f(L)' (KFG0)L.
Proof______: This is immediate, except for the last statement. And if L *
*G0 is such
that L \ Ker(f) = 1, then L ~= f(L), the categories E(G0)=L and E(G)=f(L) are b*
*oth
equivalent to the category B(L) with one object and endomorphism group L; and t*
*hus
(Vec___FG)f(L)(S) = | func(E(G)=f(L); F ~~~~-mod )| is homotopy equivalent to (V*
*ec_FG0)L(S) =
| func(E(G0)=L; F ~~~~-mod )| for each S in . |
We first consider the restriction and induction homomorphisms.
Proposition 3.2 Fix F = C or R, and let H G be any pair of discrete groups. *
* Let
i* : KFG ! KFH be the map of Lemma 3.1.
(a) For any proper G-CW-pair (X; A), i* induces a homomorphism of rings
ResGH: KFG*(X; A) ------! KFH*(X; A):
(b) For any proper H-CW-pair (Y; B), i* induces an isomorphism
~= *
IndGH: KFH*(Y; B) ------! KFG(GxH Y; GxH B);
which is natural in (Y; B), and also natural with respect to inclusions of*
* subgroups.
The restriction and induction maps both commute with the maps between KFG(-) and
KFH (-) induced by induction and restriction of equivariant vector bundles.
12
Proof *
* *
_________: It suffices to prove this when A = ; = B and * = 0. The fact that*
* i : KFG !
KFH commutes with the Bott homomorphisms and the products follows directly from*
* the
definitions. So part (a) is clear.
The inverse of the homomorphism in (b) is defined to be the composite
*O-
[G xH Y; KFG]G ~=[Y; KFG]H ---i----![Y; KFH ]H :
And since i* restricts to a homotopy equivalence (KFG)L ! (KFH )L for each fini*
*te L H
(Lemma 3.1), this map is an isomorphism by Lemma 1.1.
The last statement is clear from the construction and the definition of fl :*
* KFG(-) !
KFG(-). |
We next consider the inflation homomorphism.
Proposition 3.3 Fix F = C or R. Let G be any discrete group, and let N C G be *
*a normal
subgroup. Then for each proper G-CW-pair (X; A), there is an inflation map
InflGG=N: KFG*=N(X=N; A=N) ------! KFG*(X; A);
which is natural in (X; A), which is a homomorphism of rings (if A = ;), and wh*
*ich com-
mutes with the homomorphism KFG=N(X=N; A=N) ! KFG(X; A) induced considering G=N-
vector bundles as G-vector bundles. And if N acts freely on X, then InflGG=Nis *
*an isomor-
phism.
Proof______: Let f : G ! G=N denote the natural homomorphism, and let f* : KFG=*
*N ! KFG
be the induced map of Lemma 3.1. Define InflGG=Nto be the composite
f*O-
[X=N; KFG=N]G=N ~=[X; KFG=N]G -----! [X; KFG]G:
If N acts freely on X, then for each isotropy subgroup L of X, L \ N = 1, so (f*
**)L :
(KFG=N)L ! (KFG)L is a homotopy equivalence by Lemma 3.1, and the inflation map*
* is an
isomorphism by Lemma 1.1. The other statements are clear. |
Another type of natural map will be needed when constructing the equivariant*
* Chern
character. Fix a discrete group G and a finite normal subgroup N C G, and let I*
*rr(N) be the
set of isomorphism classes of irreducible complex N-representations. Let X be a*
*ny proper
G=N-complex. For any V 2 Irr(N) and any G-vector bundle E ! X, let Hom N (V; *
*E)
denote the vector bundle over X whose fiber over x 2 X is Hom N (V; Ex) (each f*
*iber of E is
an N-representation). If H G is any subgroup which centralizes N, then we can *
*regard
Hom N(V; E) as an H-vector bundle by setting (hf)(x) = h.f(x) for any h 2 H an*
*d any
f 2 Hom N (V; E). We thus get a homomorphism
: KG(X) ------! KH (X) R(N);
P
where ([E]) = V 2Irr(N)[Hom N(V; E)] [V ]. We need a similar homomorphism d*
*efined on
K*G(X).
13
Proposition 3.4 Let G be a discrete group, let N C G be any finite normal subg*
*roup, and
let H G be any subgroup such that [H; N] = 1. Then for any proper G=N-complex*
* X,
there is a homomorphism of rings
= XG;N;H: K*G(X) -------! K*H(X) R(N);
which is natural in X and natural with respect to the degree-shifting maps K*G*
*(X) !
K*+nG(Snx X), and which has the following properties:
(a) For any (complex) G-vector bundle E ! X,
X
([[E]]) = [[Hom N(V; E)]] [V ]:
V 2Irr(N)
(b) For any G0 G, N0 N\ G0, and H0 H\ G0, the following diagram commutes:
XG;N;H
K*G(X) -------! K*H(X) R(N)
? ?
ResGG0?y ResHH0?yResNN0
XG0;N0;H0
K*G0(X) -------! K*H0(X) R(N0):
Proof______: Fix G, H, and N. For any irreducible N-representation V and any*
* surjective
homomorphism p : C[N] -i V , composition with p defines a monomorphism
-Op
Hom N(V; W ) -----! Hom N(C[N]; W ) = W
for any N-representation W ; and thus allows us to identify Hom N (V; W ) as a *
*subspace of
W . In particular, there is a functor
p* : func(Or (G)=N; C~~~~-mod ) -----! func(Or (H); C~~~~-mod )
which sends any ff to the functor h 7! Hom N (V; ff(hN)) ff(hN). If p0 : C[N]*
* -i V 0is
*
* ~=
another surjection of N-representations, where V ~=V 0, then any isomorphism V*
* -! V 0
defines a natural isomorphism between p* and (p0)*. We thus get a map of -spaces
p : Vec_CG------!Vec_CH
which is unique (independant of the projection p) up to H-equivariant homotopy.*
* So this
induces homomorphisms V : K-nG(X) ! K-nH(X), for all proper G=N-complex X (and*
* all
n 0), which depend only on V and not on p. The V clearly commute with the Bot*
*t maps,
and thusPextend to homomorphisms V : K*G(X) ! K*H(X). So we can define by set*
*ting
(x) = V 2Irr(N) V (x) [V ]. Point (a) is immediate; as is naturality in X an*
*d naturality
for restriction to G0 G or H0 H. Naturality with respect to the degree-shifti*
*ng maps
holds by construction.
14
We next show that is natural in N; i.e., that point (b) holds when G0= G an*
*d H0= H.
Let V be the homomorphisms defined above, for each irreducible N-representatio*
*n V ; and
let 0W : K*G(X) ! K*H(X) be the corresponding homomorphism for each irreducibl*
*e N0-
representation W . For each V 2 Irr(N) and each W 2 Irr(N0), set
N
nVW = dimC Hom N0(W; V ) = dimC Hom N(IndN0(W ); V ) :
Thus, nVW is the multiplicity of W in the decomposition of V |N0, as well as th*
*e multiplicity
of V in the decomposition of IndNN0(W ). So for any x 2 K*G(X),
X X X
(Id ResNN0)(G;N;H(x)) = V (x) [V |N0] = nVW. V (x) [*
*W ];
V 2Irr(N) W2Irr(N0)V 2Irr(N)
P
and we will be done upon showing that 0W = V nVW. V for each W 2 Irr(N0). F*
*ix a
P k
surjection p0 : C[N0]P-i W , and a decomposition IndNN0(W ) = i=1Vi (where t*
*he Vi are
irreducible and k = V nVW). For each 1 i k, let pi : C[N] -i Vi be the com*
*posite of
IndNN0(p0) followed by projection to Vi. Then
Mk
N C
p0= pi: Vec_CG ------! Vec_H
i=1
P k
as maps of -spaces, and so W ' i=1 Vias maps K*G(X) ! K*H(X).
It remains to show that is a homomorphism of rings. Since it is natural in*
* N, and
since R(N) is detected by characters, it suffices to prove this when N is cycli*
*c. For any
x; y 2 KG(X),
X X
(x).(y) = V (x). W (y) [V W ] and (xy) = U(xy) [U]:
V;W2Irr(N) U2Irr(N)
And thus (x).(y) = (xy) since
M N N
UO* = *O( V ^ W ) : Vec_CG ^ Vec_CG ------! Vec_CH;
V;W2Irr(G)
V W~=U
as maps of -spaces, for each U 2 Irr(N). |
4. Characters and class functions
Throughout this section, G will be a finite group. We prove here some result*
*s showing
that certain class functions are characters; results which will be needed in th*
*e next two
sections.
15
For any field K of characteristic zero, a K-character of G means a class fun*
*ction G ! K
which is the character of some (virtual) K-representation of G. Two elements g*
*; h 2 G
are called K-conjugate if g is conjugate to ha for some a prime to n = |g| = |h*
*| such that
(i 7! ia) 2 Gal(Ki=K), where i = exp(2ssi=n). For example, g and h are Q-conjug*
*ate if
and are conjugate as subgroups, and are R-conjugate if g is conjugate t*
*o h or h-1.
Proposition 4.1 Fix a finite extension K of Q, and let A K be its ring of int*
*egers. Let
f : G ! A be any function which is constant on K-conjugacy classes. Then |G|.f*
* is an
A-linear combination of K-characters of G.
Proof______: Set n = |G|, for short. Let V1; : :;:Vk be the distinct irr*
*educible K[G]-
representations, let Oi be the character of Vi, set Di = End K[G](Vi) (a divisi*
*on algebra
over K), and set di= dimK (Di). Then by [13, Theorem 25, Cor. 2],
Xk
1 X -1
|G|.f = riOi where ri= __ f(g)Oi(g );
i=1 dig2G
and we must show that ri2 A for all i. This means showing, for each i = 1; : :;*
*:k, and each
g 2 G with K-conjugacy class conjK(g), that |conjK(g)|.Oi(g) 2 diA.
Fix i and g; and set C = , m = |g| = |C|, and i = exp(2ssi=m). Then Gal(K*
*(i)=K)
acts freelyfonithe set conjK (g): the element (i 7! ia) acts by sending h to*
* ha. So
[K(i):K]fi|conjK(g)|.
Let Vi|C = W1a1 . . .Wtatbe the decomposition as a sum of irreducible K[C]-m*
*odules.
For each j, Kj def=EndK[C](Wj) is the field generatedfbyiK and the r-th rootsfo*
*fiunity for
some r|m (m = |C|), and dimKj (Wj) = 1.fiSo dimK (Wj)fi[K(i):K]. Also, difidim*
*K(Wjaj),
since Wjajis a Di-module; and thus difiaj.|conjK(g)|. So if we set j = OWj(g) 2*
* A, then
Xt
|conjK(g)|.Oi(g) = |conjK(g)|. ajj 2 diA;
j=1
and this finishes the proof. |
For each prime p and each element g 2 G, there are unique elements gr of ord*
*er prime to
p and gu of p-power order, such that g = grgu = gugr. As in [13, x10.1], we ref*
*er to gr as the
p0-component of g. We say that a class function f : G ! C is p-constant if f(g)*
* = f(gr) for
each g 2 G. Equivalently, f is p-constant if and only if f(g) = f(g0) for all g*
*; g0 2 G such
that [g; g0] = 1 and g-1g0has p-power order.
Lemma 4.2 Fix a finite group G, a prime p, and a field K of characteristic z*
*ero. Then a p-
constant class function ' : G ! K is a K-character of G if and only if '|H is a*
* K-character
of H for all subgroups H G of order prime to p.
16
Proof
_________: Recall first that G is called K-elementary if for some prime q, G *
*= Cm o Q, where
Cm is cyclic of order m, q|-|m, Q is a q-group, and the conjugation action of Q*
* on K[Cm ]
leaves invariant each of its field components. By [13, x12.6, Prop. 36], a K-*
*valued class
function of G is a K-character if and only if its restriction to any K-elementa*
*ry subgroup of
G is a K-character. Thus, it suffices to prove the lemma when G is K-elementary.
Assume first that G is q-K-elementary for some prime q 6= p. Fix a subgroup *
*H G of
p-power index and order prime to p, and let ff : G i H be the surjection with f*
*f|H = Id.
Set pa = | Ker(ff)|. Then
Aut(Ker (ff)) ~=(Z=pa)* ~=(1 + pZ=pa) x (Z=p)*;
where the first factor is a p-group. Hence for any g 2 H and x 2 Ker(ff), eithe*
*r [g; x] = 1
and hence g = (gx)r; or gxg-1 = xi for some i 6 1 (mod p) and hence g is conjug*
*ate to gx.
In either case, '(gx) = '(g). Thus, ' = ('|H )Off, and this is a K-character of*
* G since '|H
is by assumption a K-character of H.
Now assume G is p-K-elementary. Write G = Cm oP , where p|-|m and P is a p-g*
*roup. Let
S be the set of primes which divide m. For each I S, let CI Cm be the product*
* of the
Sylow p-subgroups for p 2 I, set GI = CIo P , and let ffI : G i GI be the homom*
*orphism
which is the identity on GI.
For each I S, we can consider K[CI] as a G-representation via the conjugati*
*on action
of P ; and each CI-irreducible summand of K[CI] is P -invariant and hence G-inv*
*ariant. Thus,
each irreducible K[CI]-representation can be extended to a K[GI]-representation*
* upon which
P \ CG(CI) acts trivially. Hence, since '|CI is a K-character of CI; there is a*
* K-character
OI of GI such that OI(gx) = OI(x) = '(x) for all x 2 CI and g 2 P such that [g;*
* CI] = 1.
Now set X
O = (-1)|Ir J|(OIOffJ);
JIS
a K-character of G. We claim that ' = O. Since both are class functions, it suf*
*fices to show
that '(gx) = O(gx) for all commutingfgi2 P and x 2 Cm = CS. Fix such g and x, a*
*nd let
X S be the set of all primes pfi|x|. Then [g; CX ] = 1, and so
X X
O(gx)= (-1)|Ir J|OI(ffJ(gx)) = (-1)|Ir J|OI(g.ffJ(x))
JIS JIS
X X
= (-1)|Ir J|'(ffJ(x)) + (-1)|Ir J|OI(g.ffJ(x))
JIX JI6X
= '(x) = '(gx):
Note, in the second line, that all terms in the second sum cancel since ffJ(x) *
*= ffJ0(x) if
J = J0\ X, and all terms in the first sum cancel except that where J = I = X. *
* |
When A = Z and K = Q, Proposition 4.1 and Lemma 4.2 combine to give:
17
Corollary 4.3 Fix a finite group G and a prime p. Let f : G ! Z be any functio*
*n which
is p-constant, and constant on Q-conjugacy classes in G. Set |G| = m.pr where p*
*|-|m. Then
m.f is a Q-character of G.
5. The equivariant Chern character
We construct here two different equivariant Chern characters, both defined o*
*n the equiv-
ariant complex K-theory of proper G-complexes. The first is defined for arbitra*
*ry X (with
proper G-action), and sends K*G(X) to the Bredon cohomology group H*G(X; QZ R(*
*-)).
The second is defined only when X is finite dimensional and has bounded isotrop*
*y, and
takes values in QZ H*G(X; R(-)).
We first fix our notation for dealing with Bredon cohomology [6]. Let Or (G*
*) denote
the orbit category: the category whose objects are the orbits G=H for H G, and*
* where
Mor Or(G)(G=H; G=K) is the set of G-maps. A coefficient system for Bredon cohom*
*ology is
a functor F : Or (G)op ! Ab . For any such functor F and any G-complex X, the B*
*redon
cohomology H*G(X; F ) is the cohomology of a certain cochain complex C*G(X; F )*
*, where
CnG(X; F ) is the direct product over all orbits of n-cells of type G=H of the *
*groups F (G=H).
This can be expressed functorially as a group of morphisms of functors on Or (G*
*):
CnG(X; F ) = Hom Or(G)C_n(X); F ;
where C_n(X) : Or(G)op! Ab is the functor C_n(X)(G=H) = Cn(XH ).
Clearly, the coefficient system F need only be defined on the subcategory of*
* orbit types
which occur in the G-complex X. In particular, since we work here only with pro*
*per actions,
we restrict attention to the full subcategory Or f(G) of orbits G=H for finite *
*H G. Let
R(-) denote the functor on Or f(G) which sends G=H to R(H): a functor on the o*
*rbit
category via the identification R(H) ~=K0G(G=H). More precisely, a morphism G=H*
* ! G=K
in Orf(G), where gH 7! gaK for some a 2 G with a-1Ha K, is sent to the homomor*
*phism
R(K) ! R(H) induced by restriction and conjugation by a.
Since R(-) is a functor from the orbit category to rings, there is a pairing
C*G(X; R(-)) C*G(X; R(-)) ------! C*G(X x X; R(-))
for any proper X, and hence a similar pairing in cohomology. Via restriction to*
* the diagonal
subspace X X x X this defines a ring structure on H*G(X; R(-)).
The equivariant Chern character will be constructed here by first *
*reinterpreting
H*G(X; Q R(-)) as a certain group of homomorphisms of functors, and then direc*
*tly con-
structing a map from KG(X) to such homomorphisms. This will be done with the he*
*lp of
18
another category, Sub
f(G), which is closely related to Orf(G). The objects of *
*Sub f(G) are
the finite subgroups of G, and
Mor Subf(G)(H; K) Hom (H; K)= Inn(K)
is the subset consisting of those monomorphisms induced by conjugation and incl*
*usion in G.
There is a functor Or f(G) ! Sub f(G) which sends an orbit G=H to the subgroup *
*H, and
which sends a morphism xH 7! xaK in Orf(G) to the homomorphism x 7! a-1xa f*
*rom
H to K. Via this functor, we can think of Sub f(G) as a quotient category of Or*
* f(G).
Let C_qt*(X); H_qt*(X) : Sub f(G)op--! Ab be the functors
C_qt*(X)(H) = C*(XH =CG(H)) and H_qt*(X)(H) = H*(XH =CG(H)): (*
*5.1)
For any functor F : Sub f(G)op! Ab , regarded also as a functor on Or f(G)op,
Hom Orf(G)(C_*(X); F ) ~=Hom Subf(G)(C_qt*(X); F ); (*
*5.2)
since
Hom CG(H)(C*(XH ); F (H)) ~=Hom (C*(XH =CG(H)); F (H))
for each H (and CG(H) is the group of automorphisms of G=H in Orf(G) sent to th*
*e identity
in Sub f(G)). In particular, (5.2) will be applied when F = R(-), regarded as a*
* functor on
Sub f(G) as well as on Or f(G).
As noted above, for any coefficient system F , the cochain complex C*G(X; F *
*) can be
identified as a group of homomorphisms of functors on Or (G). The following le*
*mma says
that the Bredon cohomology groups H*G(X; Q R(-)) have a similar description, b*
*ut using
functors on Sub f(G)op.
Lemma 5.3 Fix a discrete group G and a proper G-complex X. Then (5.2) induc*
*es an
isomorphism of rings
~= qt
X : H*G(X; Q R(-)) -----! Hom Subf(G)H_*(X); Q R(-) :
Proof______: Since
C*G(X; Q R(-)) ~=Hom Orf(G)(C_*(X); Q R(-)) ~=Hom Subf(G)(C_qt*(X); Q R(*
*-));
this will follow immediately once we show that Q R(-) is injective *
*as a functor
Sub f(G)op ! Ab. It suffices to prove this after tensoring with C; i.e., it *
*suffices to
prove that Cl(-) (complex valued class functions) is injective. And this holds*
* since for
any F : Sub f(G)op! Ab ,
Y Y
Hom Subf(G)(F; Cl(-)) ~= Hom Subf(G)(F; Clg(-)) ~= Hom (F (); C);
g g
19
where both products are taken over any set of conjugacy class representatives f*
*or elements of
finite order in G, and where Clg(H) denotes the space of class functions on H w*
*hich vanish
on all elements not G-conjugate to g. |
We are now ready to define the Chern character
ch*X: K*G(X) -------! H*G(X; Q R(-))
for any proper G-complex X. Here and in the following theorem, we regard K*G(-*
*) as
being Z=2-graded; so that ch*Xsends K0G(X) to HevG(X; Q R(-)) and sends K1G(X)*
* to
HoddG(X; Q R(-)). By Lemma 5.3, it suffices to define homomorphisms
H
chHX: K*G(X) ------! Hom H*(X =CG(H)) ; Q R(H) ;
for each finite subgroup H G, which are natural in H in the obvious way. We de*
*fine chHX
to be the following composite:
(proj)** H
K*G(X) -Res--!K*NG(H)(XH ) ---! K*CG(H)(XH ) R(H) -----! KCG(H)(EGx X ) R(*
*H)
--Infl-1---!* H chId * H
~= K (EGxCG(H) X ) R(H) ----! H EGxCG(H) X ; Q R(H)
(proj)* * H H
-----~ H X =CG(H); Q R(H) ~=Hom H*(X =CG(H)); Q R(H) : (5.4)
=
Here, is the homomorphism defined in Proposition 3.4, ch denotes the ordinary *
*Chern
character, and (proj)* in the bottom line is an isomorphism since all fibers of*
* the projec-
tion from EG xCG(H)XH to XH =CG(H) are Q-acyclic (classifyingQspaces of finite*
* groups).
By the naturality properties of shown in Proposition 3.4, HchHX takes value*
*s in
Hom Subf(G)H_qt*(X); Q R(-) , and hence (via Lemma 5.3) defines an equivarian*
*t Chern
character
qt
ch*X: K*G(X) ------! H*G(X; Q R(-)) ~=Hom Subf(G)H_*(X); Q R(-) :
All of the maps in (5.4) are homomorphisms of rings, and hence ch*Xis also a ho*
*momorphism
of rings. Also, the ch*Xcommute with degree-changing maps K*G(X) ! K*+m (Sm xX)*
* (i.e.,
product with the fundamental class of Sm ) and similarly in cohomology, since a*
*ll maps
in (5.4) do so. They are thus natural with respect to boundary maps in Mayer-V*
*ietoris
sequences.
Theorem 5.5 For any finite proper G-complex X, the Chern character ch*Xextend*
*s to an
isomorphism of rings
~= *
Q ch*X: Q K*G(X) ------! HG(X; Q R(-)):
Proof______: For any finite subgroup H G,
K0G(G=H) ~=R(H) ~=H0G(G=H; R(-)); and K1G(G=H) = 0 = H6=0G(G=H; R(-)):
20
From the definition in (5.4) (and since the non-equivariant Chern character K(p*
*t) ! H0(pt)
is the identity map), we see that Qch*G=His the identity map under the above id*
*entifications.
The Chern characters for G=Hx Dn and G=Hx Sn-1) are thus isomorphisms for all n*
*. The
theorem now follows by induction on the number of orbits of cells in X, togethe*
*r with the
Mayer-Vietoris sequences for pushouts X = X0[' (G=Hx Dn) (and the 5-lemma). *
* |
Theorem 5.5 means that the Q-localization of the classifying space KG splits*
* as a prod-
uct of equivariant Eilenberg-Maclane spaces. Hence for any proper G-complex X, *
*there is
~=
an isomorphism K*G(X; Q) ---! H*G(X; Q R(-)), where the first group is defined*
* via the
localized spectrum (and is not in general isomorphic to Q K*G(X; A)).
The coefficient system Q R(-), and hence its cohomology, splits in a natura*
*l way as a
product indexed over cyclic subgroups of G of finite order. For any cyclic grou*
*p S of order
n < 1, we let Z[iS] Q(iS) denote the cyclotomic ring and field generated by th*
*e n-th roots
of unity; but regarded as quotient rings of the group rings Z[S*] Q[S*] (S* = *
*Hom (S; C*)).
In other words, we fix an identification of the n-th roots of unity in Q(iS) wi*
*th the irreducible
characters of S. The kernel of the homomorphism R(S) ~=Z[S*] i Z[iS] is precis*
*ely the
ideal of elements whose characters vanish on all generators of S.
Lemma 5.6 Fix a discrete group G, and let S(G) be a set of conjugacy class r*
*epresentatives
for the cyclic subgroups S G of finite order. Then for any proper G-complex X,*
* there is
an isomorphism of rings
Y N(S)
H*G(X; Q R(-)) ~= H*(XS=CG(S); Q(iS)) ;
S2S(G)
where N(S) acts via the conjugation action on Q(iS) and translation on XS=CG(S)*
*. If,
furthermore, the isotropy subgroups on X have bounded order, then the homomorph*
*ism of
rings
Y i N(S)j
H*G(X; R(-)) -----! H C*(XS=CG(S); Z[iS])
S2S(G)
Y N(S)
-----! H*(XS=CG(S); Z[iS]) ; *
* (1)
S2S(G)
induced by restriction to cyclic subgroups and by the projections R(S) -i Z[iS*
*], has kernel
and cokernel of finite exponent.
Proof______: By (5.2),
C*G(X; R(-)) ~=Hom Orf(G)(C_*(X); R(-)) ~=Hom Subf(G)(C_qt*(X); R(-)):
For each S 2 S(G), let OS 2 Cl(G) be the idempotent class function: OS(g) = *
*1 if is
conjugate to S, and OS(g) = 0 otherwise. By Proposition 4.1, for each finite su*
*bgroup H
21
G, (O H H
S)|H is the character of an idempotent eS 2 QR(H). Set QRS(H) = eS .(QR(H)*
*),
and let RS(H)Q QRS(H) be the image of R(H) under the projection. This defines a*
* splitting
Q R(-) = S2S(G)QRS(-) of the coefficient system. For each S and H,
i j
QRS(S) = Q(iS) and so QRS(H) ~=map N(S) MorSubf(G)(S; H) ; Q(iS) :
It follows that
S
C*G(X; QRS(-)) ~=Hom Subf(G)(C_qt*(X); QRS(-)) ~=Hom Q[N(S)]C*(X =CG(S)); Q(iS*
*) ;
N(S)
and hence H*G(X; QRS(-)) ~= H*(XS=CG(S)); Q(iS) .
Now assume there is a bound on the orders of isotropy subgroups on X, and le*
*t m be
the least common multiple of the |Gx|. By Proposition 4.1 again, meHS 2 R(H) f*
*or each
S 2 S(G) and each isotropy subgroup H. So there are homomorphisms of functors
i Y
R(-) ------!------ RS(-);
j S2S(G)
where i is induced by the projections R(H) i RS(H) and j by the homomorphisms
meHS.
RS(H) ---! R(H) (regarding RS(H) as a quotient of R(H)); and iOj and jOi are bo*
*th
multiplication by m. For each S, the monomorphism
S * S
C*G(X; RS(-)) ~=Hom Z[N(S)]C*(X =CG(S)); Z[iS] ----! C (X =CG(S); Z[iS])
is split by the norm map for the action of N(S)=CG(S), and hence the kernel and*
* cokernel
of the induced homomorphism
* S N(S)
H*G(X; RS(-)) -----! H (X =CG(S); Z[iS])
fi fi
have exponent dividing '(m) (since |N(S)=CG(S)|fi| Aut(S)|fi'(m)). The composi*
*te in (1)
thus has kernel and cokernel of exponent m.'(m). |
By the first part of Proposition 5.6, the equivariant Chern character can be*
* regarded as
a homomorphism
Y N(S)
ch*X: K*G(X) ------! H*(XS=CG(S); Q(iS)) ;
S2S(G)
where S(G) is as above. This is by construction a product of ring homomorphisms.
We now apply the splitting of Lemma 5.6 to construct a second version of the*
* equivari-
ant rational Chern character: one which takes values in Q H*G(X; R(-)) rather *
*than in
H*G(X; Q R(-)). The following lemma handles the nonequivariant case.
22
Lemma 5.7 There is a homomorphism n!ch : K*(X) ! H2n (X; Z), natural on the *
*cat-
egory of CW-complexes, whose composite to H*(X; Q) is n! times the usual Chern *
*char-
acter truncated in degrees greater than 2n. Furthermore, n!ch is natural with *
*respect to
suspension isomorphisms K*(X) ~=Ke*+m(m (X+)), and is multiplicative in the sen*
*se that
n!ch(x) . n!ch(y) = n!. n!ch(xy) for all x; y 2 K(X) (in both cases after re*
*stricting to
the appropriate degrees).
Proof______: Define n!ch: K0(X) ! Hev;2n(X; Z) to be the following polynomial i*
*n the Chern
classes:
Xn i 2 nj
xi xi
n!. 1 + xi+ __ + . .+.___ 2 Z[c1; : :;:cn] = Z[x1; x2; : :;:xn]n :
i=1 2! n!
Here, as usual, ck is the k-th elementary symmetric polynomial in the xi. This *
*is extended to
K-1(X) ~=Ke((X+)) in the obvious way. The relations all follow from the usual r*
*elations
between Chern classes in the rings H*(BU(m)). |
We are now ready to construct the integral Chern character. What this really*
* means is
that under certain restrictions on X, some multiple of the rational Chern chara*
*cter ch*Xof
Theorem 5.5 can be lifted to the integral Bredon cohomology group H*G(X; R(-)).
Proposition 5.8 Let G be a discrete group, and let X be a finite dimensional p*
*roper G-
complex whose isotropy subgroups have bounded order. Then there is a homomorphi*
*sm
ech*X: K*G(X) -----! Q H*G(X; R(-));
natural in such X, whose composite to H*G(X; Q R(-)) is the map ch*Xof Theorem*
* 5.5.
~=
Furthermore, ceh*Xinduces an isomorphism of rings Q K*G(X) -! Q H*G(X; R(-)*
*).
And for any finite subgroup K G, ceh0G=Kis the identity map under the identifi*
*cations
KG(G=K) ~=R(G=K) ~=H0G(G=K; R(-)).
fi
Proof______: Fix X, and choose any integer n dim(X)=2. Set m = lcm |Gx| fix *
*2 X and
N = n!.m4n. For each S 2 G of finite order, let echSXbe the following composite:
(proj)** S
K*G(X) -Res--!K*NG(S)(XS) ---! K*CG(S)(XS) R(S) -----! KCG(S)(EG x X ) R(S)
-Infl-1----!* S n!ch 2n S
~= K (EGxCG(S)X ) R(S) ----! H EGxCG(S)X R(S)
m4n(proj*)-1* S * S
----------! H X =CG(S) R(S) -----! H (X =CG(S); Z[iS]):
Here, is the homomorphism of Lemma 3.4, and Inflis the inflation isomorphism o*
*f Propo-
*
* proj*
sition 3.3. The first map in the bottom row is well defined since H*(XS=C(S); Z*
*[iS]) ---!
H*(EGxC(S)XS; Z[iS]) has kernel and cokernel of exponent m2n (this follows from*
* the spec-
tral sequence for the projection, all of whose fibers are of the form BGx for x*
* 2 X). The last
23
map is induced by the projection R(S) -i Z[i
S]. All of these maps are homomorp*
*hisms of
rings (up to the obvious integer multiples).
Now let S(G) be any set of conjugacy class representatives for cyclic subgro*
*ups S G
of finite order. Define ech*Xto be the composite
_1_QechS i Y j
ceh*X: K*G(X) -N----X--!Q H*(XS=CG(S); Z[iS]) N(S) ~=Q H*G(X; R(-));
S2S(G)
where the isomorphism is that of Lemma 5.6. The naturality of ech*X, its indepe*
*ndence of the
choice of n, and its relation with ch*X, are immediate from the construction. *
*Also, ceh*Xis
natural with respect to the degree-changing maps K*(X) ! K*+m (Sm xX) (and simi*
*larly
in cohomology). In particular, this means that it commutes with all maps in May*
*er-Vietoris
sequences.
It remains to prove that ech*Xinduces an isomorphism on Q K*G(X). This is d*
*one by
induction on dim (X), using the obvious Mayer-Vietoris`sequences. So it suffic*
*es to show
it for (possibly infinite) disjoint unions i2IG=Hi of orbits. Both groups ar*
*e zero in odd
degrees. And in even degrees,
` ech0X ev `
Q KG i2IG=Hi ------! Q HG i2IG=Hi; R(-)
~=Q Q iR(Hi) ~=Q Q iR(Hi)
is the identity map under these identifications. |
6. Completion theorems
Throughout this section, G is a discrete group. We want to prove completion *
*theorems for
finite proper G-complexes: theorems which show that K*G(E x X), when E is a "un*
*iversal
space" of a certain type, is isomorphic to a certain completion of K*G(X). The*
* key step
will be to construct elements of K*G(X) whose restrictions to orbits in X are s*
*ufficiently
"interesting". And this requires a better understanding of the "edge homomorph*
*ism" for
K*G(X).
For any finite dimensional proper G-complex X, the skeletal filtration of K**
*G(X) induces
a spectral sequence
Ep;2*2~=HpG(X; R(-)) =) K*G(X):
If X also has bounded isotropy, the Chern character echXof Proposition 5.8 is a*
*n isomorphism
(after tensoring with Q) from the limit of this spectral sequence to its E2-ter*
*m. It follows
that the spectral sequence collapses rationally; i.e., that the images of all d*
*ifferentials in the
spectral sequence consist of torsion elements.
24
Of particular interest is the edge homomorphism of the spectral sequence. T*
*his is a
homomorphism
fflX : K*G(X) ------! H0G(X; R(-));
which is induced by restriction to the 0-skeleton of X under the identification
(0) 1 (1) (0) (1) (0)
H0G(X; R(-)) = Ker KG(X ) ---! KG(X ; X ) = Im KG(X ) ---! KG(X ) :
Alternatively, H0G(X; R(-)) can be thought of as the inverse limit, taken over *
*all isotropy
subgroups H of X and all connected components of XH , of the representation rin*
*gs R(H);
and the edge homomorphism sends an element of K*G(X) to the collection of its r*
*estrictions
to elements of K*G(Gx) ~=R(Gx) at all points x 2 X.
As an application of the integral Chern character of Proposition 5.8, we get:
Proposition 6.1 Let X be any finite dimensional proper G-complex whose isotrop*
*y sub-
groups have bounded order. Then for any 2 H0G(X; R(-)), there is k > 0 such t*
*hat k.
and k lie in the image of the edge homomorphism
fflX : KG(X) ------! H0G(X; R(-)):
Similarly, for any 2 H0G(X; RO(-)), there is k > 0 such that k. and k lie in t*
*he image
of the edge homomorphism
fflX : KOG(X) ------! H0G(X; RO(-)):
Proof______: The usual homomorphisms between R(-) and RO(-), and between K*G(-*
*) and
KO*G(-), induced by (CR) and by forgetting the complex structure, show that up *
*to 2-
torsion, KO*G(X) and H0G(X; RO(-)) are the fixed point sets under complex conju*
*gation
of the groups K*G(X) and H0G(X; R(-)), respectively. So the edge homomorphism *
*in the
orthogonal case is also surjective modulo torsion. The rest of the argument is *
*identical in
the real and complex cases; we restrict to the complex case for simplicity.
By Proposition 5.8, the integral Chern character for X(0)is the identity und*
*er the usual
identifications KG(G=K) ~=R(K) ~=H0G(G=K; R(-)) for an orbit G=K (K finite). S*
*o by
the naturality of echX, the composite
ech0X
Q KG(X) ----!~ Q HevG(X; R(-)) --- i Q H0G(X; R(-)) Q KG(X(0)) *
* (1)
=
is just the map induced by restriction to X(0). So rationally, the edge homomor*
*phism is just
the projection of the integral Chern character cehonto H0G(X; R(-)), and is in *
*particular
surjective. And hence, for any 2 H0G(X; R(-)), there is some k > 0 such that *
*k. 2
fflX (KG(X)).
It remains to show that k 2 Im(fflX ) for some k. If we knew that the Atiyah*
*-Hirzebruch
spectral sequence
Ep;2*2~=HpG(X; R(-)) =) K*G(X)
25
were multiplicative (i.e., that the differentials were derivations), then the r*
*esult would follow
directly. As we have seen, all differentials in the spectral sequence have fini*
*te order. Hence,
for each r 2 and each j 2 E0;2*r, there is some k > 0 such that
k.dr(j) = 0; and hence dr(jk) = k.dr(j)jk-1 = 0:
Upon iteration, this shows that for any 2 H0G(X; R(-)) = E0;02, there exists k*
* > 0 such
that k. and k both survive to E0;01; and hence lie in the image of the edge hom*
*omorphism.
Rather than prove the multiplicativity of the spectral sequence, we give the*
* following
more direct argument. Identify
(1) (0)
2 H0G(X; R(-)) = Im KG(X ) --! KG(X ) :
Assume, for some r 2, that lies in the image of KG(X(r-1)); we prove that som*
*e power
of lies in the image of KG(X(r)).
Fix e 2 KG(X(r-1)) such that resX(0)(e) = . Since r 2,
0 (r-1) 0 (0) 0 (r) 0 (0)
Im HG(X ; R(-)) --! HG(X ; R(-)) = Im HG(X ; R(-)) --! HG(X ; R(-)) :
Hence, since the Chern character is rationally an isomorphism, there exists k s*
*uch that k.
lies in the image of KG(X(r)), or equivalently such that
h d i
k.e 2 Ker KG(X(r-1)) ---! KG(X(0)) ---! K1G(X(r); X(0))
h d i
= Ker KG(X(r-1)) ---! K1G(X(r); X(r-1)) ---! K1G(X(r); X(0)) : *
* (1)
In Lemma 6.2 below, we will show that there is a KG(X(r-1))-module structure on*
* the relative
group K1G(X(r); X(r-1)) which makes the boundary map d : KG(X(r-1)) ! K1G(X(r);*
* X(r-1))
into a derivation. Then d(ek) = k.ek-1.d(e), so ek lies in the kernel in (1), *
*and hence
k = resX(0)(ek) lies in the image of KG(X(r)). |
It remains to prove:
Lemma 6.2 Let X be any proper G-complex. Then, for any r 2, one can p*
*ut a
KG(X(r-1))-module structure on K1G(X(r); X(r-1)) in such a way that fo*
*r any ff; fi 2
KG(X(r-1)),
d(fffi) = ff.dfi + fi.dff 2 K1G(X(r); X(r-1)):
Proof______: We can assume X = X(r). Write Y = X(r-1), for short. Fix a map*
* : X !
Z def=X x Y [ Y x X which is homotopic to the diagonal, and such that |Y is equ*
*al to the
diagonal map. Since Z contains the r + 1-skeleton of X x X, is unique up to ho*
*motopy
(rel Y ). In particular, if T : Z ! Z is the map which switches coordinates, th*
*en T O '
(rel Y ).
26
Now, for ff 2 K 1 1
G(Y ) and x 2 KG(X; Y ), let ff.x 2 KG(X; Y ) be the image of*
* ffxx under
the following composite
* 1 * 1
ff x x 2 K1G(Y x X; Y x Y ) ~=K1G(Z; X x Y ) -incl---!KG(Z; Y x Y ) ----! KG*
*(X; Y ):
Here, the external product ff x x is induced by the pairing KG ^ KG ! KGxG ! K*
*G of
(2.2); or equivalently is defined to be the internal product of proj*1(ff) 2 KG*
*(Y x X) and
proj*2(x) 2 K1G(Y x X; Y x Y ). We can thus consider KG(X; Y ) as a KG(Y )-mod*
*ule. In
particular, the relation (fffi).x = ff.(fi.x) follows since the two composites *
*( x IdX)O and
(IdX x)O are homotopic as maps from X to
(Xx Y xY ) [ (Y xXx Y ) [ (Y xY xX):
Now consider the following commutative diagram:
~=
KG(Y x Y ) - -d-! K1G(Z; Y x Y )---! K1G(X; Y ) x Y K1GY x (X; Y )
? ? x
* ?y * ?y ~=??
KG(Y ) - -d-! K1G(X; Y ) --- K1GZ; Y x X K1GZ; X x Y
where the isomorphisms hold by excision. For any ff; fi 2 KG(Y ), the external*
* product
ff x fi 2 KG(Y x Y ) is sent, by the maps in the top row, to the pair (dff x fi*
*; ff x dfi). This
follows from the linearity of the differential (which holds in any multiplicati*
*ve cohomology
theory). And since T O ' , as noted above, we have
d(fffi) = *(d(ff x fi)) = fi.dff + ff.dfi: |
As an immediate consequence of Proposition 6.1, we now get:
Corollary 6.3 Assume that G is discrete. Fix any family F of finite subgroups*
* of G of
bounded order, and let
0 0
V = VH 2 lim-R(H) or V = VH 2 lim-RO(H)
H2F H2F
be any system of compatible (virtual) representations. Then for any finite dime*
*nsional proper
G-complex X all of whose isotropy subgroups lie in F, there is an integer k > 0*
*, and
elements ff; fi 2 KG(X) (or ff0; fi02 KOG(X)), such that ff|x = k.VGx and fi|x *
*= (VGx)k (or
ff0|x = k.VG0xand fi0|x = (VG0x)k) for all x 2 X.
Proof______: Let be the image of V under the ring homomorphism
lim-R(H) ----! H0G(X; R(-))
H2F
27
(or similarly in the orthogonal case); and apply Proposition 6.1. |
Corollary 6.3 can be thought of as a generalization of [10, Theorem 2.7]. I*
*t was that
result which was the key to proving the completion theorem in [10], and Corolla*
*ry 6.3 plays
a similar role in proving the more general completion theorems here.
In what follows, a family of subgroups of a discrete group G will always mea*
*n a set of
subgroups closed under conjugation and closed under taking subgroups.
Lemma 6.4 Let X be a proper n-dimensional G-complex. Set
I = Ker[K*G(X) -res--!K*G(X(0))]:
Then In+1 = 0.
Proof______: Fix any elements x 2 In and y 2 I. By induction, we can assume t*
*hat x van-
ishes in K*G(Xn-1), and hence that it lifts to an element x0 2 K*G(X; X(n-1)). *
*Recall that
K*G(X; X(n-1)) is a K*G(X)-module, and the map K*G(X; X(n-1)) ! K*G(X) is K*G(X*
*)-linear.
But I.K*G(X; X(n-1)) = 0, since I vanishes on orbits; so yx0= 0, and hence yx =*
* 0 in K*G(X).
|||
As in earlier sections, in order to handle the complex and real cases simult*
*aneously, we set
F = C or R, and write KFG*(-) and RF (-) for the equivariant K-theory and repre*
*sentation
rings over F .
Fix any finite proper G-complex X, and let f : X ! L be any map to a finite *
*dimensional
proper G-complex L whose isotropy subgroups have bounded order. Let F be any fa*
*mily of
finite subgroups of G. Regard KFG*(X) as a module over the ring KFG(L). Set
h res Y i
I = IF;L = Ker KFG(L) ----! KFH (L(0)) :
H2F
For any n 0, the composite
proj* * res * (n-1)
In.KFG*(X) KFG*(X) ----! KFG(EF (G) x X) ----! KFG((EF (G) x X) )
is zero, since the image is contained in IKFG*((EF (G)xG X)(n-1))n = 0 which va*
*nishes by
Lemma 6.4. This thus defines a homomorphism of pro-groups
* ffin * * (n-1)
X;fF: KFG(X) I .KG(X) n1 ------! KFG (EF (G) x X) n1 :
Theorem 6.5 Fix F = C or R. Let G be a discrete group, and let F be a family *
*of subgroups
of G closed under conjugation and under subgroups. Fix a finite proper G-comple*
*x X, a finite
28
dimensional proper G-complex Z whose isotropy subgroups have bounded order, and*
* a G-map
f : X ! Z. Regard KFG*(X) as a module over KFG(Z), and set
h res Y i
I = IFF;Z= Ker KFG(Z) ----! KFH (Z(0)) :
H2F
Then
* ffin * * (n-1)
X;fF: KFG(X) I .KFG(X) n1 ------! KFG (EF (G) x X) n1
is an isomorphism of pro-groups. Also, the inverse system KFG*(EF (G) x X)(n)*
* n1
satisfies the Mittag-Leffler condition. In particular,
(n)
lim-1KFG*(EF (G) x X) = 0;
and X;fFinduces an isomorphism
~= * * (n)
KFG*(X)bI------! KFG(EF (G) x X) ~=lim-KFG (EF (G) x X) :
Proof______: Assume that X;fFis an isomorphism. Then the system KFG*(EF (G)x X*
*)(n) n1
satisfies the Mittag-Leffler condition because KFG*(X)=In does. In particular,
(n) * * (n)
lim-1KFG*(EF (G)x X) = 0; and so KFG(EF (G)x X) ~=lim-KFG (EF (G)x X)
(cf. [4, Proposition 4.1]).
It remains to show that X;fFis an isomorphism.
Step 1 Assume first that X = G=H, for some finite subgroup H G. Let F|H be *
*the
family of subgroups of H contained in F, and consider the following commutative*
* diagram:
f* * pr1 *
KFG(Z) - --! KFG(G=H) ---! KFG(EF (G)x G=H)
? ?
eveH?y~= q?y~=
pr2 *
RF (H) - --!~ KFH*(pt) ---! KFH (EF|H(H)):
=
Here, pr2induces an isomorphism of pro-groups
* n * * (n-1)
KFH (*)=IF (H) .KFH (*) n1 -----! KF (BH) n1
by the theorem of Jackowski [9, Theorem 5.1], where
h Y i def
IF (H) = Ker RF (H) ---! RF (L) I0 = evf(eH)(I):
L2F|H
29
(The theorem in [9] is stated only for complex K-theory, but as noted afterward*
*s, the proof
applies equally well to the real case.) We want to show that pr1induces an isom*
*orphism of
pro-groups
* n * * (n-1)
KFG(G=H)=I .KFG(G=H) n1 -----! KFG (EF (G) x G=H) n1 :
So we must show that for some k, IF (H)k I0.
This means showing that the ideal IF (H)=I0 is nilpotent; or equivalently (s*
*ince R(H) is
noetherian) that it is contained in all prime ideals of R(H)=I0 (cf. [3, Propos*
*ition 1.8]). In
other words, we must show that every prime ideal of R(H) which contains I0 also*
* contains
IF (H). Fix any prime ideal P R(H) which does not contain IF (H). Set i = exp(*
*2ssi=|H|)
and A = Z[i]. By a theorem of Atiyah [1, Lemma 6.2], there is a prime ideal p *
*A and an
element s 2 H such that
P = {v 2 R(G) | Ov(s) 2 p}:
(This is stated in [1] only in the complex case, but the same arguement applies*
* to prime ideals
in the real representation ring.) Also, s is not an element of any L 2 F, since*
* P 6 IF (H).
Set p = char(A=p) (possibly p = 0).
For any g 2 G of finite order, we let gr represent its p-regular component: *
*the unique
gr 2 such that p|-||gr| and |(gr)-1g| is a power of p (gr = g if p = 0). By*
* [1, Lemma 6.3],
we can replace s by sr without changing the ideal P; and can thus assume that p*
*|-||s|.
Let m0 be the least common multiple of the orders of isotropy subgroups in Z*
*, and let
m be the largest divisor of m0 prime to p (m = m0 if p = 0). Define ' : tors(G*
*) ! Z by
setting '(g) = 0 if gr 2 L for some L 2 F, and '(g) = m otherwise. By Corollary*
* 4.3, '|L
is a rational character of L for each L 2 Isotr(Z). So by Corollary 6.3, there *
*is k > 0 and
an element 2 KG(Z) whose restriction to any orbit has character the restrictio*
*n of 'k. In
other words, 2 I = IFF;Z, and so 'k|H is the character of an element v 2 I0. *
* But then
Ov(s) = '(s)k 62 p, so v 62 P, and thus P 6 I0.
Step 2 By Step 1, the theorem holds when dim(X) = 0. So we now assume that dim(*
*X) =
m > 0. Assume X = Y [' G=Hx Dm , for some attaching map ' : G=Hx Sm-1 ! Y . We
can assume inductively that the theorem holds for Y , G=Hx Sm-1 , and G=Hx Dm '*
* G=H.
All terms in the Mayer-Vietoris sequence
----! KFG*(X) ----! KFG*(Y ) KFG*(G=Hx Dm ) ----! KFG*(G=Hx Sm-1 ) ----!
are KFG(X)-modules and all homomorphisms are KFG(X)-linear; and the KFG(Z)-modu*
*le
structure on each term is induced from the KFG(X)-module structure. So if we l*
*et I0
KFG(X) be the ideal generated by the image of I; then dividing out by (I0)n is *
*the same
as dividing out by In for all terms. In addition, KFG(X) is noetherian (in fact*
*, a finitely
generated abelian group), and so this Mayer-Vietoris sequence induces an exact *
*sequence of
30
pro-groups
* n * n * m n
-----! KFG(X)=I n1 -----! KFG(Y )=I KFG(G=Hx D )=I n1
* m-1 n
-----! KFG(G=Hx S )=I n1 -----!
by [10, Lemma 4.1]. There is a similar Mayer-Vietoris exact sequence of the pr*
*o-groups
KFG*(EF (G) x -)(n-1) n1 ; and the theorem now follows from the 5-lemma for pr*
*o-
groups together with the induction assumptions. |
References
[1]M. Atiyah, Characters and cohomology of finite groups, Publ. Math. I.H.E.S.*
* 9_(1961),
23-64
[2]M. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. *
*Math. Oxford
19_(1968), 113-140
[3]M. Atiyah & I. Macdonald, Introduction to commutative algebra, Addison-Wesl*
*ey
(1969)
[4]M. Atiyah & G. Segal, Equivariant K-theory and completion, J. Diff. Geometr*
*y 3_
(1969), 1-18
[5]P. Baum and A. Connes: Chern character for discrete groups, in: Matsumoto, *
*Miyu-
tami, and Morita (eds.), A f^ete of topology; dedicated to Tamura, 163-232,*
* Academic
Press (1988)
[6]G. Bredon, Equivariant cohomology theories, Lecture notes in mathem*
*atics 34_,
Springer-Verlag (1967)
[7]J. Davis & W. L"uck, Spaces over a category and assembly maps in isomorphism
conjectures in K-and L-Theory, MPI-preprint (1996), to appear in K-theory
[8]D. Grayson, Higher algebraic K-theory: II, Algebraic K-theory, Evanston, 1*
*976,
Springer Lecture Notes in Math. 551 (1976).
[9]S. Jackowski, Families of subgroups and completion, J. Pure Appl. Algebra 3*
*7_(1985),
167-179
[10]W. L"uck & B. Oliver: The completion theorem in K-theory for proper actions*
* of a
discrete group, preprint (1997)
[11]S. Mac Lane, Categories for the working mathematician, Springer-Verlag (197*
*1)
[12]G. Segal, Categories and cohomology theories, Topology 13_(1974), 293-312
31
[13]J.-P. Serre, Linear representations of finite groups, Springer-Verlag (1977)
[14]J. Slominska, On the equivariant Chern homomorphism, Bull. Acad. Pol. Sci. *
*24_
(1976), 909-913
Addresses:
Wolfgang L"uck Bob Ol*
*iver
Institut f"ur Mathematik und Informatik Laboratoire de Mathem*
*atiques
Westf"alische Wilhelms-Universtit"at Universite P*
*aris Nord
Einsteinstr. 62 Avenue J.-B. Cl*
*ement
48149 M"unster 93430 Villeta*
*neuse
Germany Fran*
*ce
lueck@math.uni-muenster.de bob@math.univ-paris1*
*3.fr
http://wwwmath.uni-muenster.de/math/u/lueck http://zeus.math.univ-paris13.fr/*
*~bob
32
~~