*Corresponding author. Tel.:#33 1 49 40 3594; fax:#33 1 49 40 3568.
E-mail addresses:lueck@math.uni.muenster.de (W. LuKck), bob@math.univ-paris13.fr (B. Oliver)
The completion theorem in K-theory for proper actions of a discrete group
Wolfgang Lu K ck , Bob Oliver *
Institut f u(r Mathematik und Informatik, Westfa(lische Wilhelms-Universita(t, Einsteinstr. 62, 48149 Mu(nster, Germany LAGA - UMR 7539 du CNRS, Universite& Paris Nord, Avenue, J.-B. Cle&ment, 93430 Villetaneuse, France
Received 22 May 1998; accepted 20 August 1999
Abstract
We prove a version of the Atiyah}Segal completion theorem for proper actions of an in"nite discrete group G. More precisely, for any"nite properG-CW-complexX,KH(EG;%X) is the completion ofKH
%(X) with respect to a certain ideal. We also show, for suchGandX, thatK
%(X) can be de"ned as the Grothendieck group of the monoid of G-vector bundles overX. 2001 Elsevier Science Ltd. All rights reserved.
MSC: Primary 55N91; secondary 19L47
Keywords: K-theory; Proper actions; Vector bundles
LetGbe any discrete group. For suchG, aG-CW-complexis a CW-complex withG-action which permutes the cells such that an element g3G sends a cell to itself only by the identity map.
AG-CW-complexXisproperif all of its isotropy subgroups have"nite order, and isxniteif it is made up of "nitely many orbits of cells. A G-CW-pairis a pair of G-spaces (X,A), where X is a G-CW-complex andAis a G-invariant subcomplex.
The main results of this paper are Theorems 3.2 and 4.3 below. The"rst says that equivariant K-theory KH
%(!) can be de"ned on the category of "nite proper G-CW-pairs using ("nite dimensional) G-vector bundles, in the sense that this does de"ne an equivariant cohomology theory. In particular, for anyX,K%(X) is just the Grothendieck group of the monoid ofG-vector bundles overX.
0040-9383/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 4 0 - 9 3 8 3 ( 9 9 ) 0 0 0 7 7 - 4
The second theorem is an extension of the Atiyah}Segal completion theorem to this situation. It says that for any"nite proper G-CW-complex X,KH(EG;%X) is the completion ofKH
%(X) with respect to a certain ideal. In particular, when the universal properG-spaceEFIN(G) ("E
M Gin the notation of Baum and Connes [7]) has the homotopy type of a"nite G-CW-complex, then this completion is taken with respect to the augmentation ideal of K
%(EFIN(G)). For example, when X"EFIN(G), Theorem 4.3 implies thatKH(BG) is the completion ofKH
%(EFIN(G)) with respect to the augmentation ideal in K%(EFIN(G)).
There are two ways in which the proofs of these theorems, whenGis in"nite and discrete, diverge from the usual proofs for "nite group actions. First, since the category of spaces with proper G-action does not contain cones or suspensions ("xed points are not allowed), we need to"nd other ways to de"neK%(X,A) andK\L% (X). This is easily handled. A more crucial di!erence is that special constructions are needed, carried out in Section 2, to get around the lack of `su$ciently manya product bundles. This second di$culty is illustrated by the fact that both of these theorems fail in general whenGis a positive dimensional noncompact Lie group. This is discussed in detail, with examples, in Section 5. Examples which show thatK%(!) de"ned usingG-vector bundles is not an equivariant cohomology theory in this situation were originally due to Phillips [15], who instead de"ned K
%(!) using in"nite dimensional G-vector bundles with Hilbert space "bers (see also [17]).
In a separate paper, we will construct an equivariant cohomology theoryKH
%(!) for arbitrary (not necessarily proper) G-CW-complexes using spectra. More precisely, this will be done using Or(G)-spectra: contravariant functors from the orbit category of G to spectra. We will also construct an equivariant Chern character for proper G-C=-complexes which takes values in equivariant Bredon cohomology, and which is rationally an isomorphism for"nite properG-C=- complexes.
Let*H
%(X) be the Grothendieck group of ("nite-dimensional)G-vector bundles overX. There is a natural transformationu%:*H
%(X)PKH
%(X), which is an isomorphism for "nite properG-CW- complexes. In the nonequivariant case, this is well known to be an isomorphism for any "nite dimensional CW-complexX(since any mapXPB;factors through some B;(n)). But even for
"niteGO1, Example 3.11 below shows that*H
% isnota cohomology theory on the category of all
"nite dimensional proper G-CW-pairs.
The paper is organized as follows:
1. G-vector bundles over properG-CW-complexes, 2. Constructions ofG-vector bundles,
3. EquivariantK-theory for"nite proper G-CW-complexes, 4. The completion theorem,
5. Proper actions of Lie groups, References.
1. G-vector bundles over properG-CW-complexes
Throughout this section G is a Lie group. We collect here some basic facts about G-vector bundles over proper G-CW-complexes.
A G-CW-complex X is a space with G-action, which is "ltered by its `skeletaa XL, such that Xhas the weak topology as the union of theXL, and such that eachXLis obtained fromXL\by attaching orbits of cells G/HG;DL via attaching maps G/HG;SL\PXL\. (Here X\".) WhenG is discrete, aG-CW-complex can be thought of as a CW-complex withG-action which permutes the cells, such that an elementg3Gsends a cell to itself only by the identity map. Note that the orbit space of a G-CW-complex inherits the structure of an (ordinary) CW-complex.
For more details aboutG-CW-complexes, see, e.g., [9, Sections II.1 and II.2] or [13, Sections I.1 and I.2].
A G-CW-complex X is xnite if it is made up of "nitely many orbits of cells G/H;DL, or equivalently if X/Gis a"nite CW-complex. A G-CW-complexXwill be calledproperif all of its isotropy subgroups are compact. (ForG-CW-complexes, this is equivalent to the various de"ni- tions of proper actions which have been given in more general situations.)
A G-vector bundle over a G-CW-complex X consists of a (complex) vector bundle p:EPX, together with aG-action on E such thatp is G-equivariant and each g3Gacts on E and Xvia a bundle isomorphism. We letE"V denote the"ber over a pointx3X. AmapofG-vector bundles fromp:EPXtop:EPXis just a map (fM,f) of vector bundles, such thatfM:EPEandf:XPX areG-equivariant. Here, we assume only thatfM restricts to a linear mapE"VPE"DVfor eachx3X.
We call (fM,f) astrong mapiffM restricts to a linear isomorphismE"VP" E"DVfor eachx3X. This is clearly equivalent to the condition thatp:EPXis isomorphic to the pullback ofp:EPXoverf.
Most of the properties of G-vector bundles over G-CW-complexes we need will be easy consequences of the following elementary lemma.
Lemma 1.1. (a)AnyG-vector bundle over an orbit of cellsG/H;DLis isomorphic toG;&(<;DL)for someH-representation <.
(b)For anyG-CW-complexX,both XandX/Gare paracompact.
(c)Fix aG-vector bundlep:EPXover aG-CW-complexX.LetXLbe then-skeleton ofX,and set EL"p\(XL).Then the squares
are pushout squares for eachn.Also,XandEhave the weak topology with respect to the subspaces XLandE
L,respectively. More generally,if +X
G,GZ' is any set of subcomplexes which coverX,then XandE have the weak topology with respect to the subspacesXG andp\(XG),respectively.
(d)For any G-CW-pair (X,A), there is a neighborhood=of Ain X,which can be chosen to be closed or open,such that Ais an equivariant strong deformation retract of=.
Proof. (a) Note that for any G-map p:XPG/H, the canonical map G;&p\(eH)PX is a G- homeomorphism. (This will be used frequently throughout the paper.) It thus su$ces to show that anyH-vector bundle overDL is isomorphic to the product bundle<;DL for someH-representa- tion <, and this follows from [3, Proposition 1.6.2].
(c) The pushout square forXL, and the fact thatXhas the weak topology with respect to its skeleta, follow from the de"nition of a G-CW-complex. In particular, a functionXP>(for any space >) is continuous if and only if its composite with each equivariant cell G/H;DLPX is continuous; and from this one sees immediately thatXhas the weak topology with respect to any covering set of subcomplexes. TheG-pushout property forE
L follows the pushout property forXL, together with (a) and [13, Lemma 1.26].
We now claim foranyX, any vector bundlep:EPX, and any coveringX"GZ'XG by closed subspaces, that E has the weak topology with respect to its subsets p\(XG) if X has the weak topology with respect to the XG. Upon restricting to a neighborhood of any given x3X, this is reduced to the case whereEis a product bundle; and the result then follows easily since the"bers are locally compact.
(b) Given an open coveringU of Xor of X/G, a partition of unity subordinate to Ucan be constructed by applying Zorn's lemma to the set of such partitions of unity over subcomplexes of X(and using (c) above). For more details (in the case of a nonequivariant CW-complex), see [14, Theorem II.4.2].
(d) For eachn, one easily constructs a collar neighborhood<
L ofXLinXL>(open or closed), together with an equivariant deformation retractionoL:<
LPXL, which restricts to a deformation retraction ofo\L (B) toBfor any B-XL. Now set
=\"A, =
L"A6o\L (=
L\5XL) (alln*0), and ="8
L
=L, let rL:=
LP=
L\ be the identity on Aand oL on=
L!A, and let r:=PA be the composite of therL. 䊐
The next three results, which list some of the standard properties ofG-vector bundles, are easy consequences of Lemma 1.1. We begin with homotopy invariance. As usual, I denotes the unit interval [0,1].
Theorem 1.2. Let X be a proper G-CW-complex, let p:EPX;I be a G-vector bundle, and set E"E"6",regarded as a G-vector bundle overX.Then there is an isomorphismo:EP" E;I of G-vector bundles,which is the identity onE and covers the identity onX;I.If,in addition,A-Xis any G-invariant subcomplex, then o can be chosen to extend any given isomorphism o:E""'P" E""'.
Proof. Using Lemma 1.1(c), this is quickly reduced to the case where (XQ ,A)"
(G/H;DL,G/H;SL\). By Lemma 1.1(a),E"G;&(<;DL;I). Thus,o is equivalent to a map o:SL\;IPAut&(<) which sendsSL\;0 to the identity, and this can be extended toDL;Isince SL\PDL is a co"bration. 䊐
The proof of the next two lemmas is similiar to that of Theorem 1.2.
Lemma 1.3. Let(X,A)be a properG-CW-pair,and letEandEbeG-vector bundles overX.Then any mapf:E"PE" ofG-vector bundles overAextends to a mapfM:EPEofG-vector bundles overX.
Proof. Via Lemma 1.1, it su$ces to prove this when (X,A)"(G/H;DL,G/H;SL\), E"G;&(<;DL), andE"G;&(<;DL). A mapEPEofG-vector bundles thus corresponds to anH-mapDLPHom&(<,<), and any map overAextends to a map overXsince Hom&(<,<) is contractible. 䊐
Lemma 1.4. Let (X,A) be a proper G-CW-pair,and let E be a G-vector bundle over X.Then any G-invariant Hermitian metric ofE" extends to a G-invariant Hermitian metric onE.
Proof. Again, it su$ces to prove this when (X,A)"(G/H;DL,G/H;SL\), and E"G;&(<;DL). A Hermitian metric overX then corresponds to a map DLPHerm&(<) (the space ofH-invariant Hermitian metrics over<); and any such map onSL\can be extended to one on DLsince Herm&(<) is convex (and hence contractible). 䊐
To"nish the section, we check that a pushout ofG-vector bundles is aG-vector bundle over the
pushout of the base spaces. This will, of course, be used to prove excision in Section 3.
Lemma 1.5. Let u: (X,X)P(X,X) be a map of G-CW-pairs, set u"u"6, and assume that XXPX. Let p:EPX and p:EPX be G-vector bundles, let u:E"6PE be a strong map coveringu,and setE"EPE.Thenp"p6p:EPXis aG-vector bundle overX.
Proof. The only problem is to show that p:EPXis locally trivial (in a non-equivariant sense).
SinceE is locally trivial, so isE"6!6E"6!6. So it remains to"nd a neighborhood ofX over whichEis locally trivial. Choose a closed neighborhood=
ofXinXfor which there is a strong deformation retraction r:=
PX (Lemma 1.1(d)). By the homotopy invariance for nonequivariant vector bundles over paracompact spaces (cf. [10, Corollary 3.4.5]),ris covered by a strong map of vector bundles r :E"5PE which extends iM. Set ="XP=
. Then r extends, via the pushout, to a strong map of vector bundlesE"5PEwhich extendsiM, and hence E"5 is locally trivial. 䊐
Let p:EPB be a G-vector bundle over a proper G-CW-complex. Each orbit Gx-X has aG-invariant neighborhood;
Vsuch thatGxis an equivariant retract of;
V, and the neighborhood can in fact be chosen such that the retraction is covered by a strong mapE"3VPE"%V. There is thus aG-coveringUofXsuch that eachE"3 (for;3U) is`trivialain the sense that it is the pullback of a bundle over a (proper) orbit. Also, sinceX/Gis paracompact by Lemma 1.1(b),UisG-numerable in the sense that there is a locally "nite partition of unity +t
3";3U, by G-invariant functions t3 with supp(t3)-;. Hence our notion ofG-vector bundles agrees with that of tom Dieck in [9, Section I.9]. For these same reasons, the results of this section can easily be extended to proper G-spaces which have paracompact quotients (any such space has tubes, i.e., equivariant neighbor- hood retracts of orbits).
2. Constructions ofG-vector bundles
The main result in this section is Theorem 2.6. Given a discrete groupG, aG-CW-complexX, and a family+<
&,of representations of the isotropy subgroups inX, we would like to be able to
construct a G-vector bundle EPXwhose "ber over anyx3X is isomorphic to<
%V. This is in general not possible, even for"niteG, for reasons discussed at the end of the section. What we show here is that we can do this, assuming certain conditions onXand the<
&, but only after replacing the <
& by some iterated direct sum (<
&)I, or by some iterated tensor product (<
&)BI. These bundles are the crucial ingredients in the proof that G-vector bundles de"ne an equivariant cohomology theory (Theorem 3.2), and the proof of the completion theorem (Theorem 4.3).
Throughout the"rst part of the section,GandCwill denote arbitrary Lie groups. AfamilyFof subgroups ofGis a set of (closed) subgroups ofGwhich is closed under conjugation. We will need to work with some classifying spaces and universal spaces:"rst for (proper)G-actions and then for bundles.
De5nition 2.1. For any familyF of subgroups of G, let EF(G) denote the topological category whose objects are the pairs (G/H,gH) forH3Fandg3G, and where Mor((G/H,gH),(G/K,gK)) is the set ofG-mapsG/HPG/Kwhich sendgHtogK(a set of cardinality at most one). LetEF(G) be the realization of the nerve of EF(G), considered as a G-CW-complex:
EF(G)"
LZ Z%&2%&L
G/H;DL
&,where the identi"cations are those induced by the obvious face and degeneracy maps.
As usual, we are assuming thatEF(G) has the weak topology with respect to its cellular structure.
Lemma 2.2. Fix a Lie groupGand a family Fof subgroups of G.
(a)For any K3F, (EF(G))) is contractible.
(b) Let(X,A) be any G-CW-pair such that GV3Ffor all x3X.Then any G-map f:APEF(G), extends to a G-map f6:XPEF(G),and any two such extensions areG-homotopic relative A.
Proof. (a) For anyK-G, (EF(G)))is the nerve of the full subcategory ofEF(G) with objects those (G/H,gH) such thatK-gHg\. And ifK3F, then this category has the initial object (G/K,eK).
(b) This follows immediately from point (a) (see [13, Proposition 2.3 p. 35]). 䊐
AG-equivariantC-bundle(or (G,C)-bundle for short) consists of aC-principal bundlep:EPX, together with left G-actions on E and X, such that p is G-equivariant, and such that the left G-action and the right C-action on E commute. We let Bdl%CX denote the set of isomorphism classes of (G,C)-bundles over theG-spaceX.
One natural example of this is the caseC";(n). A (G,;(n))-bundleEPXis just the principal bundle associated with the G-vector bundle E;3L"LPX. Similarly, a (G,&L)-bundle is the principal bundle associated with a G-equivariant n-sheeted covering space. In the constructions below, we will have to consider (G,C)-bundles for certain"nite subgroupsC-;(n).
Now "x a family F of compact subgroups of G. For each H3F, set RepC(H)"
Hom(H,C)/Inn(C); i.e., the set of conjugacy classes of homomorphisms fromHtoC. For example, Rep3L(H) is the set of isomorphism classes of n-dimensional complex representations ofH, and
RepL(H) is the set of isomorphism classes ofH-sets of ordern. Note that for anyHandC, there are natural bijections
RepC(H)Bdl&C(pt)Bdl%C(G/H). (2.3)
We need a way to specify the isomorphism types of the"bers of a (G,C)-bundle. Suppose we are given an element
A"(a&)3lim &
&ZF
RepC(H)-
&ZF
RepC(H),
where the limit is taken with respect to all homomorphisms induced by inclusions and conjugation inG. This is equivalent to an element in lim
QOrF%Bdl
%C(!), where Bdl
%C(!) is considered as a contravariant functor (via pullback) on the orbit categoryOrF(G). IfXis aG-space all of whose isotropy subgroups lie inF, then we de"ne a (G,A)-bundleoverXto be a (G,C)-bundle such that
the"ber over any pointx3Xis isomorphic to (C,a%V), regarded as a (GV,C)-bundle over a point
(see (2.3)). WhenC";(n), this corresponds to thoseG-vector bundles whose"bers are isomorphic to certain given representations of the isotropy subgroups.
We want to de"ne classifying spaces for (G,C)-bundles and for (G,A)-bundles. In fact, these are just the universal (G;C)-CW-complexes with respect to appropriate families.
De5nition 2.3. LetFbe a family of compact subgroups of G. De"ne EF(G,C)"EFC(G;C) and BF(G,C)"EF(G,C)/C,
where
FC"+H-G;C"pr(H)3F,H5(1;C)"1,. For any element
A"(a&)3lim &
&ZF
RepC(H)-
&ZF
RepC(H), de"ne
FA"+H-G;C"H"graph (a:KPC), some K3F, somea conjugate to a),, and set
EF(G,A)"EFA(G;C) and BF(G,A)"EF(G,A)/C.
In the above situation, ifEPN Xis any (G,C)-bundle, whereXis a properG-CW-complex all of whose isotropy subgroups lie inF, thenEis a proper (G;C)-CW-complex all of whose isotropy subgroups lie in FC. Conversely, ifE is any proper (G;C)-CW-complex all of whose isotropy subgroups lie inFC, thenE/Cis a properG-CW-complex all of whose isotropy subgroups lie inF, and the projection EPE/C is a (G,C)-bundle. Similarly, for any A, there is a correspondence
between (G,A)-bundles and (G;C)-CW-complexes all of whose isotropy subgroups lie inFA. This leads to the following:
Lemma 2.4. Fix a familyFof compact subgroups ofG,and an element A"(a&)3lim
&
&ZF
RepC(H),
where the limit is taken with respect to inclusions and conjugation in G.Then the following hold:
(a) For eachH3F,letCC(a&)denote the centralizer of the image of a&:HPC(well dexned up to conjugacy). Then there is a homotopy equivalence
(BF(G,A))&KBCC(a&),
which is natural with respect to maps induced by homomorphismsCPC. (b) The(G,A)-bundle
EF(G,A)PBF(G,A)
is the universal(G,A)-bundle in that it dexnes,via pullbacks,a bijection [X,BF(G,A)]%PBdl%A(X),
for any properG-CW-complex X all of whose isotropy subgroups are in F. Proof. (a) Fix H, and write C"CC(a&) for short. Consider the (G,A)-bundle
G;&(EC;!C)PG/H;BC,
whereHacts on EC;!C viah(x,c)"(x,a&(h)c). The classifying map for this bundle restricts to a map
BCP(BF(G,A))&.
Similarly, the restriction of the universal (G,A)-bundle over (BF(G,A))&is an (H,a&)-bundle over a space with trivial H-action, and hence has structure group C"CC(a&). It is thus classi"ed by a map
(BF(G,A))&PBC,
and the above two maps are homotopy inverses by the universal properties of the spaces.
(b) This follows immediately from Lemma 2.2(b). 䊐
We now assume, throughout the rest of the section, thatGis discrete. We need to construct maps to the classifying spaces de"ned in De"nition 2.3. The obstructions to doing so lie in certain Bredon cohomology groups.
Let (X,A) be anyG-CW-pair such that the isotropy group of each point inX!Alies inF. For eachn*0, let
CM L(X,A) :OrF(G)PAb
denote the contravariant functor which sends G/H to CL(X&,A&). Here, CL(X&,A&) is the free abelian group with one generator for each n-cell in X&!A&. For any contravariant functor M:OrF(G)PAb, HomOrF%(C
ML(X,A),M) is the direct sum of one copy ofM(G/H) for each orbit G/H;DL of n-cells in X!A. In particular, C
M L(X,A) is projective in the category OrF(G)-mod of contravariant functors OrF(G)PAb. The Bredon cohomology groups HH
%(X,A;M) are thus the homology groups of the cochain complex
0PHomOrF%(C
M (X,A),M) BPHomOrF%(C
M(X,A),M) BPHomOrF%(C
M (X,A),M) BP 2.
Lemma 2.5. Assume thatGis discrete.Fix a familyFofxnite subgroups ofG,axnite groupC,and a system of representations
A"(a&)&ZF3lim &
&ZF
RepC(H).
Set B"BF(G,A),and let bA:BPEF(G)
be any G-map. (This exists and is unique up to G-homotopy by Lemma 2.2(b), since all isotropy subgroups for B lie in F.) Let Z denote the mapping cylinder of bA. Let M:OrF(G)PAb be any contravariant functor.Then for eachn*0,
"C"L)HL%(Z,B;M)"0.
Proof. There is a cohomology spectral sequence ENO "ExtNOrF%(H
O(Z,B),M) NHN>O% (Z,B;M), where H
N(Z,B) denotes the functor OrF(G)PAb which assigns to G/H the abelian group HN(Z&,B&). It is induced by the double complex HomOrF%(C
M O(Z,B),I
N), where+I
N,is any injective resolution in OrF(G)-mod of M. We have just seen that the C
M O(Z,B) are all projective in OrF(G)-mod. This category does have enough injectives by, e.g., [19, Example 2.3.13].
SinceZ&K(EF(G))&is contractible by Lemma 2.2(a), we conclude from Lemma 2.4(a) that HO(Z&,B&)HI O\(B&)HI O\(CC(a&)).
In particular, since CC(a&)-C, this shows that
"C")HH(Z&,B&)"0.
So "C" annihilates all terms in the above spectral sequence, and hence (since EN "0) "C"L annihilates HN%(Z,B;M). 䊐
Given anyA"(a&)3lim &
&ZF
RepC(H), and any homomorphismo:CP;(n), there is a natural map oH:BF(G,A)PBF(G,oA) whereoA"(oa&)3lim
&
&ZF
Rep3L(H);
and oH commutes with the mapsbA and bMA to EF(G).
Theorem 2.6. Assume that G is discrete. Fix any familyFofxnite subgroups of G,and let V"(<
&)3lim &
&ZF
Rep3L(H)
be any system of compatible n-dimensional representations. Assume that there is a xnite group C, a system
A"(a&)3lim &
&ZF
RepC(H),
and a homomorphism o:CP;(n) such that V"oA. Then for any d'0 there is an integer k"k(d)'0,such that for any d-dimensional G-CW-complex X all of whose isotropy subgroups lie in F,there are G-vector bundlesE,EPXsuch that thexbersE"V andE"V over each point x3Xare isomorphic as GV-representations to(<
%V)I and(<
%V)BI,respectively.
Proof. We only treat the case of direct sums here; the tensor product case is analogous. By the universal property of EF(G) (Lemma 2.2(b)), it su$ces to prove this when X"EF(G)B (the d-skeleton).
WriteB"BF(G,A) andBI"BF(G,VI) for short (anyk*1), and letZbe the mapping cylinder ofbA:BPEF(G). We must construct, for somek, a mapZBPBI; and we will do so by extending the map oIH:BPBI. By Lemma 2.4(a), (BI)&KBAut&(<I&) for each H3F, and is in particular a product of B;(m)'s and hence simply connected. So there is no obstruction to extending oH:BPB to a map f:B6ZPB.
Assume inductively that fB\:B6ZB\PBP has been constructed, (where r"k(d!1)). We now apply standard equivariant obstruction theory. For eachH3F, let
cB(fB\)(G/H) :CB(Z&,B&)PnB\((BP)&)
be the map which sends each generator, corresponding to a d-cell p in Z&!B&, to the element fB\(*p). This is well de"ned independently of the basepoint, since (BP)& is simply connected. By naturality, this de"nes an element
cB(fB\)3CB%(Z,B;n
B\(BP))"HomOrF%(C
M B(Z,B),n
B\(BP)),
andfB\can be extended to a mapB6ZBPBPif and only ifcB(fB\)"0. Furthermore,cB(fB\) is a cocycle by [20, Theorem V.5.6], and hence de"nes an element
oB(fB\)3HB%(Z,B;n
B\(BP)).
Finally, for anyc3CB\% (Z,B;n
B\(BP)), there is a mapf:B6ZB\PBP such thatfagrees with fB\ on B6ZB\, and such that cB(f)!cB(fB\)"d(c) (as in [20, Theorem V.5.6]). So if oB(fB\)"0, then fB\"B6ZB\ can be extended to B6ZB. For more details, see [20, Section V.5], and also [8, Section II.1] (where equivariant obstruction theory is developed for actions of
a "nite group).
By Lemma 2.5, HB%(Z,B;n
B\(BP)) has exponent "C"B. Furthermore, by Lemma 2.4(a) again, nB\(BP) is the functor G/HCnB\(BAut&(<P&)). Write m""C"B for short, and consider the homomorphisms
nB\(BAut&(<P&))G&&2GK
(nB\(BAut&(<P&)))K &=H nB\(BAut&(<KP&)).
These are all homomorphisms of functors onOrF(G). Also,BHi
Q"BHi
for alls, since the corresponding maps between spaces di!er by conjugation by an element of Aut&(<KP&) (and the automorphism group is connected). Since diag" KQi
Q, it follows that (BH)diag factors through multiplication by m, and hence that the induced map
HB%(Z,B;n
B\(BP))DKHPHB%(Z,B;n
B\(BKP))
is zero. Here, DK:;(rn)P;(mrn) denotes the diagonal inclusion. We can thus extend DKHfB\"B6ZB\to a map
fB:B6ZBPBKP"BF(G,VKP).
Setk"k(d)"mr; the pullback toEF(G)B-ZBof the (G,VI)-vector bundle EF(G,VI);3LI"LIPBF(G,VI)
now has the desired properties. 䊐
As a"rst consequence of Theorem 2.6, we show the following result, which will be needed when
proving excision for equivariantK-theory de"ned viaG-vector bundles.
Corollary 2.7. Assume that G is discrete,and let X be any xnite dimensional proper G-CW-complex whose isotropy subgroups have bounded order. Then there is a G-vector bundleEPXsuch that for each x3X,the xberE"V is a multiple of the regular representation ofGV.
Proof. LetFbe the family of isotropy subgroups inX, and letnbe the least common multiple of their orders. For eachH3F, let<
& be the free complexH-representation of dimensionn, and let a&:HPRL be a homomorphism corresponding to a free H-set of order n. Then
V"(<
&)3lim &
&ZF
Rep3L(H) and A"(a&)3lim &
&ZF RepR
L(H),
and these satisfy the hypotheses of Theorem 2.6 (withC"RL). So by the theorem, there is somek, and a G-vector bundle EPX, such that for each x3X, E"V<I%V is a multiple of the regular representation of GV. 䊐
AnH-representation<will be calledp-freeif for any subgroupK-Hof order prime top,<"K is a multiple of the regular representation of K. This is equivalent to the condition that the character of any elementh3Hnot ofp-power order is zero. The next result, a second consequence of Theorem 2.6, is the main technical ingredient in our extension of the completion theorem of
Atiyah and Segal from "nite groups and compact spaces to arbitrary discrete groups and "nite proper G-CW-complexes.
Corollary 2.8. Assume that G is discrete,and let X be any xnite dimensional proper G-CW-complex whose isotropy subgroups have bounded order. Then for any prime p,there is a G-vector bundleEPX of dimension prime to p,such that for eachx3X,E"V isp-free as a GV-representation.
Proof. LetFbe the family of isotropy subgroups inX, and letmbe the least common multiple of the"H"forH3F. For each H3F, leta&:HPRK be the homomorphism corresponding to any free action of Hon+1,2,m,. Thea& clearly form an element
A"(a&)3lim &
&ZF RepR
K(H).
Setn""RK/SylN(RK)", let o:RKP;(n) be the corresponding permutation representation, and (for eachH) let<
& be then-dimensional representation de"ned byoa&. By Theorem 2.6, there is k'0 and aG-bundleEPX, such that the"berE"Vover any pointx3Xis isomorphic to (<
%V)BI. This bundle has dimension nI, which is prime to p. Furthermore, for each H3F, and each subgroup K-Hof order prime to p, (<
&)") is a free "[K]-module by construction, and so the same holds for (<&BI)"). In other words,E"V isp-free as aGV-representation for eachx, and soEhas all of the required properties. This "nishes the proof of Corollary 2.8. 䊐
In view of Theorem 2.6, the following question arises. Let X be a G-CW-complex. Given a compatible family +<
&, of representations of the isotropy subgroups ofX, is there a G-vector bundleEPXsuch thatE"V<
%V asGV-representations for eachx3X? Here,`compatibleameans that if a(K)-H, wherea is an inner automorphism ofG, then (aH<
&)")<
). This question can also be posed more generally, requiring di!erent representations on di!erent components of"xed point sets.
It is in fact easy to"nd counterexamples to this question, even in the case whereGis"nite. Fix
a"nite groupGand a normal subgroupH¢G, and setI&"Ker[R(G) PR(H)]. By a theorem of
Jackowski [11, Theorem 5.1 and Example 5.5], the pro-rings +K%(E(G/H)L),LV and +R(G)/(I&)L,LV are isomorphic. In particular, for n su$ciently large, the "bers of any G-vector bundle over E(G/H)L, considered as H-representations, can always be extended to virtual G- representations. On the other hand, anyG/H-invariantH-representation<
& de"nes a`compatible familyaof representations of the isotropy subgroups ofE(G/H). It is not hard to"nd examples of GandHwhere Im[R(G)PR(H)]TR(H)%&, and hence of a compatible family which cannot be the
"bers of aG-vector bundle over the"nite G-CW-complexE(G/H)L.
What we really would like to "nd is an example of an in"nite discrete group G, such that EFIN(G) has the homotopy type of a"niteG-CW-complex, and for which not every compatible family +<
&, of representations of the"nite subgroups can be realized as aG-vector bundle over EFIN(G) (not even stably). Presumably such examples exist, but we have so far been unable to"nd any.
3. EquivariantK-theory for5nite proper G-CW-complexes
The main result in this section is that whenGis discrete,G-vector bundles de"ne a9/2-graded multiplicative cohomology theoryKH
%(!) on the category of"nite properG-CW-complexes. This is summarized in Theorem 3.2 below.
The assumption here that G is discrete is essential, even in the case of "nite proper CW- complexes. So this will be assumed throughout most of the section. The problems arising in the case of positive dimensional Lie groups will be discussed in Section 5 below.
The usual way to de"neK%(X,A), whenGis"nite, is as the reducedK-theory of the mapping cone of the inclusion of A in X (cf. [3] or [18]). That approach is not possible here, since the mapping cone of a map of properG-CW-complexes has aG-"xed point, and hence is not proper if Gis not compact. For the same reason, we are unable to use suspensions in this situation to de"ne the groups K\L% (X,A). Instead, we make the following de"nitions:
De5nition 3.1. For any Lie groupGand any properG-CW-complexX, let*%(X)"*%(X) be the Grothendieck group of the monoid of isomorphism classes of G-vector bundles over X. De"ne
*\L% (X), for alln'0, by setting
*\L% (X)"Ker[*%(X;SL)
H
P*%(X)].
For any properG-CW-pair (X,A), set
*\L% (X,A)"Ker[*\L% (X6X)PGH *\L% (X)].
WhenG is discrete and (X,A) is axnite properG-CW-pair, write K%(X,A)"*%(X,A) and K\L% (X,A)"*\L% (X,A).
The pullback construction makes *\L% (!) and K\L% (!) into contravariant functors on the categories of proper, or"nite proper, G-CW-pairs.
Note that we get a natural isomorphism prH
6i:K%(X)K\L% (X)P" K%(X;SL),
whereiis the inclusion and pr6 the projection. We can now state the main theorem in this section.
Theorem 3.2. For any discrete group G,the groupsK\L% (X,A) extend to a9/2-graded multiplicative equivariant cohomology theory on the category ofxnite proper G-CW-pairs. In particular,KH
%(!)is
a homotopy invariant contravariant functor,satisxes excision,and there is an exact sequence
of KH
%(X)-modules for any xnite proper G-CW-pair (X,A). For any pushout X"X6X where
(X,A) is a xnite proper G-CW-pair,all maps in the induced Mayer}Vietoris sequence are KH
%(X)- linear. For xnite subgroups H-G, there are natural isomorphisms K%(G/H)R(H), and K%(G/H)"0.If G isxnite and X is compact,this construction agrees with the classicial dexnition.
The proof of Theorem 3.2 will occupy most of the rest of the section. We"rst show some of the properties of*H
%(!) which hold for any Lie groupGand any properG-CW-complexX, beginning with homotopy invariance.
Lemma 3.3 (Homotopy invariance). LetGbe a Lie group. Iff,f:(X,A)P(>,B)areG-homotopic G-maps between properG-CW-pairs,then
fH "fH
:*\L% (>,B)P*\L% (X,A) for all n*0.
Proof. Whenn"0 andA"B", this follows immediately from Theorem 1.2. The general case then follows from the de"nition of*\L% (X,A). 䊐
We next note the following relation between equivariantK-theory for di!erent groups.
Lemma 3.4 (Induction). Let H-G be an inclusion of Lie groups and let (X,A) be a proper H-CW-pair. ThenG;&(X,A) is a properG-CW-pair,and there are isomorphisms
i%&:*\L& (X,A)P" *\L% (G;&(X,A))
(for alln*0)dexned by sending [E] to[G;&E].
Proof. This is clear whenA". WhenAO, it follows since G;&(X6X)(G;&X)6%"&(G;&X). 䊐
The next two lemmas are also very elementary.
Lemma 3.5 (Free quotients). LetGbe a Lie group. Let(X,A)be a properG-CW-pair for which the normal subgroupH¢Gacts freely onX.Then the projectionpr: XPX/Hinduces an isomorphism
prH:*\L%&(X/H,A/H)P" *\L% (X,A).
Proof. This is quickly reduced to the casen"0 andA", for which the inverse of the above map is de"ned by sending [E]3*%(X) to [E/H]3*%&(X/H). 䊐
Lemma 3.6. LetGbe a Lie group,and let(X,A)be a properG-CW-pair. Suppose thatX"ZGZ'XG, the disjoint union of openG-invariant subspacesXG (for any index setI), and setAG"A5XG.Then there is a natural isomorphism
*\L% (X,A)P"
GZ'*\L% (XG,AG)
induced by the inclusions of the components.
We now assume, throughout (most of) the rest of the section, thatGis a discrete group. When proving excision and constructing the exact sequences for equivariantK-theory, we need to know when a G-vector bundleE
over aG-subspace A-Xcan be embedded as a summand of some bundleEoverX. Suppose for simplicity thatX/Gis compact. IfGis a"nite group (or a compact Lie group), then it is easy to"ndE, since anyG-vector bundle over a compactG-CW-complex is a summand of some product bundle. This is no longer the case whenGis not compact, and instead of product bundles we will constructE using the bundles constructed in Corollary 2.7.
Note that the following lemma doesnot hold whenG is a noncompact Lie group of positive dimension, even in the special case whereXis"nite. In fact, Phillips [15, Section 9] has shown that in this situation, equivariantK-theory de"ned via ("nite-dimensional)G-vector bundles need not be an equivariant cohomology theory. We will discuss this in more detail in Section 5.
Lemma 3.7. Assume G is discrete, let u:XP> be an equivariant map between xnite proper G-CW-complexes,and letEPXbe aG-vector bundle. Then there is aG-vector bundleEP>such thatEis a summand of uHE.
Proof. Letmbe the maximum dimension of any"ber ofE. By Corollary 2.7, there is aG-vector bundleFP>such that each"berF"W is a multiple of the regularGW-representation. After possibly replacing F by some iterated direct sum with itself, we can assume that for each x3X, (uHF)"VF"PVcontains at leastmcopies of the regular representation ofGV; and hence that there is a GV-linear injection of E"V into (uHF)"V. This extends to a monomorphism of G-vector bundles fromE"%V into (uHF)"%V, which by Lemma 1.3 extends to a bundle mapf
V:EPuHFcovering the identity on X. In particular, fV is a monomorphism over some open G-invariant neighborhood
;V ofGxinX.
Since X/G is compact, we can choose x,2,xL3X such that X is covered by the sets
;V,2,;
VL. The sum of thefVG is then a monomorphismf:EPuH(FL) of bundles covering the identity onX. The image offis aG-invariant subbundle ofuH(FL) (cf. [3, Lemma 1.3.1]). And via a Hermitian metric on F, it is seen to be aG-vector bundle summand. 䊐
The Mayer}Vietoris exact sequence follows as a consequence of Lemma 3.7.
Lemma 3.8 (Mayer}Vietoris sequence). AssumeGis discrete. Let
be a pushout square ofxnite properG-CW-complexes,wherei andj are inclusions of subcomplexes.
Then there is a natural exact sequence,inxnite to the left
2&B\L\K\L% (X)H&H=HH K\L% (X)K\L% (X)G&H\GH K\L% (A)&B\L
2PK\% (A)&B\K%(X)H&H=HH K%(X)K%(X)G&H\GH K%(A). (1)
Proof. We"rst show that the sequence
K%(X)H&H=HH K%(X)K%(X)G&H\GH K%(A) (2)
is exact, and hence that sequence (1) is exact atK\L% (X)K\L% (X) for alln. Clearly the composite in (2) is zero. So"x an element (a,a)3Ker (iH
!iH
). By Lemma 3.7, we can add an element of the form ([jH
E], [jH
E]) for someG-vector bundleEPX, and arrange thata"[E] anda"[E] for some pair ofG-vector bundlesEIPXI. TheniH
E and iH
E are stably isomorphic, and after adding the restrictions of another bundle over X (Lemma 3.7 again), we can arrange that iH
EiH
E. Lemma 1.5 now applies to show that there is aG-vector bundleEoverXsuch that jH
IEE
I (k"1,2), and hence that ([E ],[E
])3Im (jH jH
).
Assume now thatA is a retract ofX. We claim that in this case, Ker [K
%(X)PHH K
%(X )]PHH
"
Ker [K
%(X
)PGH K
%(A)] (3)
is an isomorphism. It is surjective by the exactness of (2). So "x an element [E]! [E]3Ker (jH
jH
). To simplify the notation, we writeE"6"jH
E, E""iH jH
E, etc. (But we arenot assuming thatj and i are injective.) Let p:XPA be a retraction, and let p:XPX be its extension to X. Using Lemma 3.7, we can arrange that E"6IE"6I for k"1,2. Upon applying Lemma 3.7 to the retraction p:XPX, we obtain a G-vector bundle FPX such that E is a summand ofpHF. Upon stabilizing again, we can assume thatEpHF, and hence thatFE"6 and E"6pH
(F")pH
(E"). Fix isomorphisms tI:E"6IPE"6I covering Id6I. The automor- phism (t")(t")\ of E" pulls back, under p, to an automorphism u of E"6; and by replacingtbyutwe can arrange thatt""t". Thent6tis an isomorphism fromEto E, and this proves (3).
We now return to the general case. For eachn*1, K\L% (A)"Ker [K
%(A;SL)PK
%(A)]
Ker [K%(X6"䢇(A;SL))
H
&K%(X)] (by (3))
Ker [K
%((X
;DL)6"1L\(X
;DL))\&EHK
%(X)], (hty. invar.)
the last step since ((X;E)6(A;DL)) is a strong deformation retract of X;DL. De"ne d\L: K\L% (A)PK\L>% (X) to be the homomorphism which makes the following diagram commute:
We have already shown that sequence (1) is exact at K\L% (X)K\L% (X) for all n. To see its exactness atK%\L>(X) andK\L% (A) (for anyn*1), apply the exactness of (2) to the following split inclusion of pushout squares:
The upper pair of squares induces a split surjection of exact sequences whose kernel yields the exactness of (1) at K\L>% (X). And since
Ker [K%((X;SL)6"䢇(X;SL))PK%(X)]
Ker [K%((X;SL)P(X;SL))PK%(XPX)]K\L% (X)K\L% (X)
by (3), the lower pair of squares induces a split surjection of exact sequences whose kernel yields the exactness of (1) at K\L% (A). 䊐
Excision, and the long exact sequence for a pair, follow as immediate consequences of the Mayer}Vietoris sequence.
Lemma 3.9 (Excision). AssumeGis discrete. Let u: (X,A)P(>,B)
be a map of xnite properG-CW-pairs,such that>B6PX.Then uH: K\L% (>,B)PK\L% (X,A)
is an isomorphism for alln*0.
Proof. For eachn, the square
is a pushout, andXis a retract ofX6X. So its Mayer}Vietoris sequence splits into short exact sequences
0PK\L% (>6 >)PK\L% (X6X)K%\L(>)PK\L% (X)P0.
And hence K\L% (>,B)K\L% (X,A). 䊐
Lemma 3.10 (Exactness). AssumeGis discrete,and let(X,A)be axnite properG-CW-pair. Then the following sequence,extending inxnitely far to the left,is natural and exact:
2BP\L\K\L% (X,A)PGH K\L% (X)PHH K\L% (A)PB\L K\L>% (X,A)PGH
2PB\ K%(X,A)PGH K%(X)PHH K%(A).
Proof. This follows immediately from the Mayer}Vietoris sequence for the square
In the nonequivariant case,*(X)K(X) for any "nite dimensional CW-complexX: since any mapXPB;factors through someB;(n). The following example shows that this is no longer true in the equivariant case, even for actions of"nite groups: the Mayer}Vietoris sequence need not be exact in this situation.
Example 3.11. Fix any"nite groupGO1. De"neX"(G;1)/&, where (g,n)&(1,n) for anyg3G and any n39. For each n39, set A
L"(G;[n!,n#])/(G;+n,). Set X "
ZLZ9AL, X"ZLZ9AL>, and X"X5X. Let iI:XIPX and jI:XPXI (k"1,2)
denote the inclusions. Then the sequence
*%(X)GPHGH*%(X)*%(X)H&H\HH *%(X) is not exact.
Proof. For each n,K%(AL)R(G) (each AL is equivariantly contractible); and the kernel of the restriction mapK%(AL)PK%(AL5X) is (under this identi"cation) the augmentation idealIR(G).
Choose representations<
L,=
L (alln39) such that dim (<
L)"dim (=
L), Hom (<
L,=
L)"0, and +dim (<
L),is unbounded. Then the element (+[<
L;A
L]![= L;A
L],LZ9,+[<
L>;A
L>]![=
L>;A
L>],LZ9) lies in Ker (jH
!jH
), but not in Im (iH ,iH
). 䊐 We now consider products on KH
%(X) and on KH
%(X,A). For any proper G-CW-complex X, tensor product ofG-vector bundles makes*%(X) into a commutative ring, and all induced maps fH:*%(>)P*%(X) are ring homomorphisms. For eachn,m*0,
K\L\K% (X)Ker [K\K% (X;SL)PK\K% (X)]
"Ker [K%(X;SL;SK)PK%(X;SL)K%(X;SK)],
where the"rst isomorphism follows from the usual Mayer}Vietoris sequences. Hence K%(X;SL)K%(X;SK)&&&&NHBNHK%(X;SL;SK)
restricts to a homomorphism
K\L% (X)K%\K(X)PK\K\L% (X).
By applying the above de"nition withn"0 or m"0, the multiplicative identity forK%(X) is seen to be an identity for KH
%(X). Associativity of the graded product is clear, and graded commutativity follows upon showing (using a Mayer}Vietoris sequence) that composition with a degree !1 map SLPSL induces multiplication by !1 on K\L(X). This product thus makes KH
%(X) into a ring. Clearly,fH:KH
%(>)PKH
%(X) is a ring homomorphism for anyG-mapf:XP>. For a "nite proper G-CW-pair (X,A), KH
%(X6X)PKH
%(X) is a split surjection and ring homomorphism (and split by a ring homomorphism), and so its kernel is aKH
%(X)-module. For any X"X6X, where (X,A) is a"nite properG-CW-pair, the boundary map in the correspond- ing Mayer}Vietoris sequence isKH
%(X)-linear, since it is de"ned via a certain map between spaces which commutes with their (split) projections ontoX. And hence the boundary maps in the long exact sequence for a pair (X,A) areKH
%(X)-linear, since they are de"ned to be the boundary maps of a certain Mayer}Vietoris sequence all of whose spaces map to X.
It remains to prove Bott periodicity in this situation. Recall that KI (S)"
Ker [K(S)PK(pt)]9, and is generated by the Bott element B3KI (S): the element [S;"]![H]3KI (S), whereH is the canonical complex line bundle over S"" /. For any
"nite properG-CW-complex X, there is an obvious pairing
K\L% (X)KI (S)PB Ker [K\L% (X;S)PK\L% (X;pt)]K\L\% (X),
induced by (external) tensor product of bundles. Evaluation at the Bott element now de"nes a homomorphism
b"b(X) :K\L% (X)PK%\L\(X),
which by construction is natural in X. And this is then extends to a homomorphism b"bH(X,A) :K\L% (X,A)PK\L\% (X,A)
de"ned for any"nite properG-CW-pair (X,A) and alln*0.
Theorem 3.12 (Equivariant Bott periodicity). AssumeG is discrete. Then the Bott homomorphism b"b(X,A) :K\L% (X,A)PK\L\(X,A)
is an isomorphism for any discrete groupGand any xnite properG-CW-pair(X,A) (and alln*0).
Proof. Assume "rst thatX">6P(G/H;DK), where H-G is "nite and u:G/H;SK\P> is a G-map; and assume inductively thatb(>) is an isomorphism. Since
K\L% (G/H;SK\)K\L& (SK\) and K\L% (G/H;DK)K\L& (DK),
the Bott homomorphisms b(G/H;SK\) and b(G/H;DK) are isomorphisms by the equivariant Bott periodicity theorem for actions of"nite groups [4, Theorem 4.3]. The Bott map is natural, and compatible with the various boundary operators in the Mayer}Vietoris sequence (in nonpositive degrees) for>,X,G/H;SK\, andG/H;DK; and sob(X) is an isomorphism by the 5-lemma. The proof thatb(X,A) is an isomorphism for an arbitrary proper"niteG-CW-pair (X,A) now follows immediately from the de"nitions of the relative groups. 䊐
We are now ready to prove the main theorem. De"ne, for alln39, KL%(X,A)"
KK%\%(X,(X,A)A) ifif nn is even,is odd.For any"nite properG-CW-pair (X,A), de"ne the boundary operatordL:KL%(A)PKL>% (X,A) to be d:K\% (A)PK%(X,A) if nis odd, and to be the composite
K%(A)P@
" K\% (A)PB\K\% (X,A)
if nis even.
Proof of Theorem 3.2. We have already proven excision (Lemma 3.9) and homotopy invariance (Lemma 3.3). The long exact sequence of a pair follows from that in negative degrees (Lemma 3.10), and the fact that the Bott map is natural and commutes with the boundary operators d\L. The
same holds for the product structure which comes from that onK\L% (X,A). For anyX"X6X, the boundary map in the corresponding Mayer}Vietoris sequence is KH
%(X)-linear, since it is de"ned via a certain map between spaces which commutes with their (split) projections ontoX.
And hence the boundary maps in the long exact sequence for a pair (X,A) areKH
%(X)-linear, since they are de"ned to be the boundary maps of a certain Mayer}Vietoris sequence all of whose spaces map toX. The other claims are immediate. 䊐
We next consider the Thom isomorphism theorem for proper actions of in"nite discrete groups.
This "rst requires a slight detour. The Thom class of a G-vector bundle E is an element in
K%(D(E),S(E)), where S(E)-D(E) denote the unit sphere and disk bundles in E(with respect to someG-invariant metric). This is most easily de"ned in terms of a chain complex of vector bundles overD(E), and we must"rst explain how such a chain complex determines an element inK-theory.
AG-vector bundle chain complex over a properG-CW-pair (X,A) is a"nite dimensional chain complex (CH,cH) ofG-vector bundles overXwhose restriction toAis acyclic. In other words, for some N'0,
0PC,PA, C,\AP 2,\ PA CPA CPA CP0
is a sequence ofG-vector bundles and bundle maps, such thatcL\cL"0 for alln, and such that restriction to the"bers over anyx3Ais exact. WhenGis compact, the monoid ofG-vector bundle chain complexes over (X,A), modulo an appropriate submonoid, is isomorphic to K%(X,A) by a theorem of Segal [18, Proposition 3.1]. In a later paper, we will prove this in our present setting, for proper actions of in"nite discrete groups. But for now, all we need to know is that any such complex de"nes an element ofK
%(X,A) in a natural (functorial) way.
Fix a G-vector bundle chain complex (CH,cH) over (X,A). For each n, set CL"
Im(cL>")"Ker(cL"). Each CL-CL is a G-invariant subbundle: this follows by induction on n, since the kernel of the surjection CL"PAL CL\ is a subbundle (cf. [12, Theorems 5.13 and 6.3]). LetCL-CL" be anyG-invariant complementary bundle toCL; de"ned, for example, using aG-invariant Hermitian metric onCL. Thus, for eachn,cL sends CLisomorphically toCL\. Set C"LZ9CL> and C"LZ9CL, let f!:C"PC" be the sum of the isomorphisms
CL>A&&L>\
"
CL> and CL>&&AL>
"
CL. Finally, de"ne
[CH,cH]"[C6D!C]![C6'C]3Ker[K%(X6X)PGH K%(X)]"K%(X,A).
This is independent of the choice of CL, since there is an a$ne structure on the space of all complementary bundles (and hence a homotopy between any two of them).
Now letp:EPXbe ann-dimensionalG-vector bundle over a properG-CW-complexX, and set p""p"D(E). Consider the cochain complex ofG-vector bundles (KIpH
"E,d) over (D(E),S(E)), which
over anyv3D(E) takes the form
0PKENTP;T KENTP;T KENTP 2;T P;T KLENTP0.
Here, v denotes the exterior product with the element v3ENT. One easily checks that this sequence is exact for all v not in the zero section ofE.
There is a technical problem here: D(E) and S(E) do not have natural structures as G-CW- complexes, and so KH
%(D(E),S(E)) is not de"ned in De"nition 3.1. It is not di$cult, however, to modify the de"nitions (and the proof of Theorem 3.2) to include this case: either by showing that (D(E),S(E)) has theG-homotopy type of a"nite properG-CW-pair, or via a more general de"nition of equivariant cellular complexes, or by constructingKH
%(!) as an equivariant cohomology theory for all properG-spaces with compact quotient. This last approach will be taken by the authors in a later, more technical, paper. For now, we just assume that equivariantK-theory has been de"ned, in some way or other, for disk and sphere bundles of G-vector bundles over "nite proper G-CW-complexes.
De5nition 3.13. For any G-vector bundleEover X, theThom class ofEis the element j#3K%(D(E),S(E)),
de"ned to be the class of the cochain complex (KH(pH
"E),d) over (D(E),S(E)) as de"ned above. The Thom homomorphismis the composite
¹#:KH
%(X)PNH"
"
KH
%(D(E))PH# KH
%(D(E),S(E)),
where the second map is multiplication with the Thom class.
Theorem 3.14 (Thom isomorphism theorem). AssumeG is discrete. Then for any G-vector bundle p:EPXover axnite proper G-C=-complex X,the Thom homomorphism
¹#:KH
%(X)P" KH
%(D(E),S(E)) is an isomorphism.
Proof. Assume "rst that X"G/H;>, where >"SL\ or DL, and whereE"7<;>for some H-representation<. Then
KL%(X)KL%(G/H;>)KL&(>);
and
KL%(D(E),S(E))KL%(G;&(D(<);>),G;&(S(<);>)) KL&(D(<);>,S(<);>)
(the last step by Lemma 3.4). So in this case,¹
# is an isomorphism by the Thom isomorphism theorem for actions of "nite groups [4, Theorem 4.3].
