An equivariant Poincaré duality for proper cocompact actions by matrix groups
aa r X i v : . [ m a t h . K T ] O c t AN EQUIVARIANT POINCAR´E DUALITY FOR PROPERCOCOMPACT ACTIONS BY MATRIX GROUPS
HAO GUO AND VARGHESE MATHAI
Abstract.
Let G be a linear Lie group acting properly and isometrically on a G -spin c manifold M with compact quotient. We show that Poincar´e duality holdsbetween G -equivariant K -theory of M , defined using finite-dimensional G -vectorbundles, and G -equivariant K -homology of M , defined through the geometricmodel of Baum and Douglas. Introduction
Poincar´e duality in K -theory asserts that the K -theory group of a closed spin c manifold is naturally isomorphic to its K -homology group via cap product withthe fundamental class in K -homology. This class can be represented geometricallyby the spin c -Dirac operator. More generally, if a compact Lie group acts on themanifold preserving the spin c structure, the analogous map implements Poincar´eduality between the equivariant versions of K -theory and K -homology. In the casewhen the Lie group is non-compact but has finite component group, induction on K -theory and K -homology allow one to establish the analogous result [7] for properactions. The observation underlying Poincar´e duality in all of these cases is thatthere exist enough equivariant vector bundles with which to pair the fundamentalclass.In contrast, Phillips [14] showed through a counter-example with a non-lineargroup that, for proper actions by a general Lie group G on a manifold X withcompact quotient space, finite-dimensional vector bundles do not exhaust the G -equivariant K -theory of X , and that it is necessary to introduce infinite-dimensionalvector bundles into the description of K -theory (see also [10]).One case in which finite-dimensional bundles are sufficient is when the group G islinear (see [13]), relying on the key fact that in this case every G -equivariant vectorbundle over X is a direct summand of a G -equivariantly trivial bundle, ie. one thatis isomorphic to X × V for some finite-dimensional representation of G on V .Motivated by this fact, we give a short proof that Poincar´e duality in this set-ting. That is, we show that the natural map from G -equivariant K -theory to Mathematics Subject Classification. G -equivariant K -homology – which for us means Baum-Douglas’ geometric K -homology – given by pairing with the fundamental class for proper cocompact G -spin c manifolds when G is linear. Theorem 1.1.
Suppose a linear Lie group G acts properly, isometrically, and co-compactly on a G -equivariantly spin c manifold X . Then there is a natural isomor-phism ψ : K ∗ G ( X ) → K G ∗ ( X ) , (1.1) where the left and right-hand sides denote G -equivariant K -theory and geometric K -homology respectively. For compact Lie group actions, Theorem 1.1 is implied by the work of [5] on theisomorphism between the equivariant geometric and analytic models of K -homology.Meanwhile, by the Peter-Weyl theorem, such groups form a subclass of linear Liegroups. Consequently, Theorem 1.1 provides another approach to some of the resultsin [5]. Remark 1.2.
While Theorem 1.1 makes no reference to the analytic model of K -homology [1, 9], we note that (1.1) still holds when the right-hand side is replaced bythe analytic K -homology group KK G ∗ ( C ( X ) , C ). Indeed, the second duality resultof [6, Section 6] specialised to our setting, together with the Thom isomorphismfrom [14, Theorem 8.11], imply that K ∗ G ( X ) ∼ = KK G ∗ ( C ( X ) , C ).In particular, this means that for linear Lie groups acting properly, isometrically,and cocompactly on a G -equivariantly spin c manifold X , the G -equivariant analyticand geometric K -homology groups of X are isomorphic. Acknowledgements.
The authors would like to thank Peter Hochs and Hang Wangfor their useful feedback on this paper.Hao Guo was partially supported by funding from the Australian Research Coun-cil through the Discovery Project DP200100729, and partially by the National Sci-ence Foundation through the NSF DMS-2000082. Varghese Mathai was partiallysupported by funding from the Australian Research Council, through the AustralianLaureate Fellowship FL170100020.2.
Preliminaries
We begin by recalling the definitions and facts we will need. Unless specifiedotherwise, G will always denote a closed subgroup of GL ( n, R ) for some n . For thissection, let X be a locally compact proper G -space. OINCAR´E DUALITY FOR ACTIONS BY MATRIX GROUPS 3
Equivariant K -theory. In [13], Phillips showed that the G -equivariant K -theory of the space X with G -cocompact supports can be defined in a such a waythat is directly analogous to non-equivariant, compactly supported K -theory. Definition 2.1 ([13, Definition 1.1]) . A G -equivariant K -cocycle for X is a triple( E, F, t ) consisting of two finite-dimensional complex G -vector bundles E and F over X and a G -equivariant bundle bundle map t : E → F whose restriction to thecomplement of some G -cocompact subset of X is an isomorphism. Two K -cocycles( E, F, t ) and ( E ′ , F ′ , t ′ ) are said to be equivalent if there exist finite-dimensional G -vector bundles H and H ′ and G -equivariant isomorphisms a : E ⊕ H → E ′ ⊕ H, b : F ⊕ H → F ′ ⊕ H ′ such that b − x ( t ′ x ⊕ id) a x = t x ⊕ id for all x in the complement of a G -cocompactsubset of X . The set of equivalence classes [ E, F, t ] of K -cocycles forms a semigroupunder the direct sum operation, and we define the group K G ( X ) is the Grothendieckcompletion of the semigroup of finite-dimensional complex G -vector bundles over X . Remark 2.2.
When it is clear from context, or when X is G -cocompact, we willomit the map t from the cycle, and simply denote a class in K G ( X ) by [ E ] − [ F ]. Remark 2.3.
For general locally compact groups, Definition 2.1 needs to be modi-fied to include infinite-dimensional bundles [14, Chapter 3]. For G linear, this is notnecessary [13, Theorem 2.3]. Definition 2.4.
For each non-negative integer i , let K iG ( X ) = K G ( X × R i ) , where G acts trivially on R i .By [13, Lemma 2.2], K iG satisfies Bott periodicity, so that we have a naturalisomorphism K iG ( X ) ∼ = K i +2 G ( X ) for each i . We will use the notation K ∗ G ( X ) = K G ( X ) ⊕ K G ( X ) . In addition, K iG are contravariant functors from the category of proper G -spacesand proper G -equivariant maps to the category of abelian groups, and form an equi-variant extraordinary cohomology theory with a continuity property. In particular,Bott periodicity implies that for any G -invariant open subset U ⊆ X , there is asix-term exact sequence of abelian groups K G ( U ) K G ( X ) K G ( X \ U ) K G ( X \ U ) K G ( X ) K G ( U ) , ∂∂ (2.1)where the maps K iG ( U ) → K iG ( X ) are induced by the natural inclusion of U into X , and the boundary maps ∂ are defined as in equivariant K -theory for compactgroup actions [15]. HAO GUO AND VARGHESE MATHAI
To prove Poincar´e duality, we will make use of the following Thom isomorphismtheorem for G -spin c bundles: Theorem 2.5 ([14, Theorem 8.11]) . Let E be a finite-dimensional G -equivariantspin c vector bundle over X . Then there is a natural isomorphism K iG ( X ) ∼ = K i +dim EG ( E ) for i = 0 , , where dim E is the real dimension of E and i + dim E is taken mod . The Gysin homomorphism.
Theorem 2.5 can be used to give an explicitgeometric description of the Gysin (pushforward) homomorphism in G -equivariant K -theory, which we will need later.Let Y and Y be two G -cocompact G -spin c manifolds and f : Y → Y a G -equivariant continuous map. By cocompactness, both Y contains only finitely manyorbit types. Together with the fact that G is a linear group, this implies, by [11,Theorem 4.4.3], that there exists a G -equivariant embedding j Y : Y → R n for some n , where G is considered as a subgroup of GL (2 n, R ). Let i Y : Y → Y × R n denote the zero section, and define the G -equivariant embedding i Y : Y → Y × R n ,y ( f ( y ) , j Y ( y )) . Let ν be the normal bundle of i Y , which we identify with a G -invariant tubu-lar neighbourhood U of its image. Note that it follows from the two-out-of-threelemma for G -equivariant spin c -structures (see [12, Section 3.1] and [8, Remark 2.6]),together with the assumption that Y and Y are G -spin c , that ν has a G -spin c structure, and so Theorem 2.5 applies. Hence we have a natural isomorphism T : K iG ( Y ) ∼ = K i +dim ν G ( U ) , (2.2)for i = 0 ,
1. The Gysin homomorphism f ! : K iG ( Y ) → K i +dim Y − dim Y G ( Y )associated to f is then the composition K iG ( Y ) K i +dim Y +2 n − dim Y G ( U ) K i +dim Y +2 n − dim Y G ( Y × R n ) K i +dim Y − dim Y G ( Y ) , T λ ∼ = (2.3) OINCAR´E DUALITY FOR ACTIONS BY MATRIX GROUPS 5 where λ is induced by the extension-by-zero map associated to the inclusion of U into Y × R n (see Remark 2.7 below), and the right horizontal isomorphism is dueto Bott periodicity. Remark 2.6.
It can be seen from the above that f ! depends only on the G -homotopyclass of f and that the Gysin map is functorial under compositions. Remark 2.7.
For any inclusion of G -invariant open subsets U ֒ → U , the extension-by-zero map C ( U ) → C ( X ) naturally induces a map C ( U ) ⋊ G → C ( U ) ⋊ G )between crossed products. The vertical map λ in (2.3) is the induced map onoperator K -theory, upon applying the identification K iG ( U j ) ∼ = K i ( C ( U j ) ⋊ G )from [14, Theorem 6.7].2.3. Equivariant geometric K -homology. We briefly review the equivariant ver-sion of Baum and Douglas’ geometric definition of K -homology [2]; see [3], [4], [5],or [7] for more details. As before, X is a locally compact proper G -space. Definition 2.8. A G -equivariant geometric K -cycle for X is a triple ( M, E, f ),where • M is a proper G -cocompact manifold with a G -equivariant spin c -structure; • E is a smooth G -equivariant Hermitian vector bundle over M ; • f : M → X is a G -equivariant continuous map.For i = 0 or 1, the G -equivariant geometric K -homology group K Gi ( X ) is the abeliangroup generated by geometric K -cycles ( M, E, f ) where dim M = i mod 2, subjectto an equivalence relation generated by the following three elementary relations:(i) (Direct sum – disjoint union) For two G -equivariant Hermitian vector bun-dles E and E over M and a G -equivariant continuous map f : M → X ,( M ⊔ M, E ⊔ E , f ⊔ f ) ∼ ( M, E ⊕ E , f );(ii) (Bordism) Suppose two cycles ( M , E , f ) and ( M , E , f ) are bordant , sothat there exists a G -cocompact proper G -spin c manifold W with boundary,a smooth G -equivariant Hermitian vector bundle E → W and a continuous G -equivariant map f : W → X such that ( ∂W, E | ∂W , f | ∂W ) is isomorphicto ( M , E , f ) ⊔ ( − M , E , f ) , where − M denotes M with the opposite G -spin c structure. Then( M , E , f ) ∼ ( M , E , f );(iii) (Vector bundle modification) Let V be a G -spin c vector bundle of real rank2 k over M . Let c M be the unit sphere bundle of ( M × R ) ⊕ V , where thebundle M × R is equipped with the trivial G -action. Let F be the Bott
HAO GUO AND VARGHESE MATHAI bundle over c M , which is fibrewise the non-trivial generator of K ( S k ). (See[4, Section 3] for a more detailed description.) Then( M, E, f ) ∼ ( c M , F ⊗ π ∗ ( E ) , f ◦ π ) , where π : c M → M is the canonical projection.Addition in K geo ,Gi ( X ) is given by[ M , E , f ] + [ M , E , f ] = [ M ⊔ M , E ⊔ E , f ⊔ f ] , the additive inverse of [ M, E, f ] is its opposite [ − M, E, f ], while the additive identityis given by the empty cycle where M = ∅ . Remark 2.9.
The above definition of classes [
M, E, f ] continues to make sense ifwe replace the bundle E by a K -theory class. Indeed, if x = [ E ] − [ E ] = [ E ′ ] − [ E ′ ] ∈ K G ( M ) , then there exists a G -vector bundle F over X such that E ⊕ E ′ ⊕ F ∼ = E ′ ⊕ E ⊕ F. By Definition 2.8 (i), this means[
M, E , f ] + [ M, E ′ , f ] + [ M, F, f ] = [
M, E ′ , f ] + [ M, E , f ] + [ M, F, f ] . Adding the inverse of [
M, F, f ] to both sides and rearranging shows that the class[
M, x, f ] := [
M, E , f ] − [ M, E , id] = [ M, E ′ , id] − [ M, E ′ , id]is well-defined.Finally, we can describe vector bundle modification using the Gysin homomor-phism 2.3. To do this, let c M be the manifold underlying the vector bundle modifi-cation of a cycle ( M, E, f ) by a bundle V , as in Definition 2.8 (iii). Then c M is theunit sphere bundle of ( M × R ) ⊕ V . We will refer to the G -equivariant embedding s : M → c M ⊆ ( M × R ) ⊕ V,m ( m, , north pole section . Lemma 2.10.
Let ( M, E, f ) be a geometric cycle for X . Let ( c M , F ⊗ π ∗ ( E ) , f ◦ π ) beits modification by a G -spin c vector bundle V of even real rank, and let π : c M → M be the projection. Let s : M → c M be the north pole section. Then ( c M , F ⊗ π ∗ ( E ) , f ◦ π ) ∼ ( c M , s ![ E ] , f ◦ π ) . Proof.
The proof we give is similar to that of [5, Lemma 3.5] concerning the case ofcompact Lie group actions; compare also the discussion following [4, Definition 6.9].To begin, observe that the total space of V can be identified G -equivariantly with OINCAR´E DUALITY FOR ACTIONS BY MATRIX GROUPS 7 a G -invariant tubular neighbourhood U of the embedding s : M → c M . The Gysinmap s ! is then simply the composition K ∗ G ( M ) T −→ K ∗ G ( U ) λ −→ K ∗ G ( c M ) , (2.4)where T is the Thom isomorphism in the form (2.2), while λ is the homomorphisminduced by the extension-by-zero map C ( U ) → C ( c M ). Let F be the Bott bundleover c M , and let F be the bundle defined by pulling back the restriction F | M along π . The composition (2.4) is then given by pulling back a vector bundle over M along π and tensoring with the class [ F ] − [ F ]. On the other hand, since c M is theboundary of the unit sphere bundle of ( M × R ) ⊕ W , and the bundle F is pulledback from M , the cycle ( c M , F ⊗ π ∗ ( E ) , f ◦ π ) is bordant to the empty cycle. Thuswe have a bordism of cycles( c M , F ⊗ π ∗ ( E ) , f ◦ π ) ∼ ( c M , ([ F ] − [ F ]) ⊗ π ∗ ( E ) , f ◦ π ) , whence the right-hand side is equal to ( c M , s ![ E ] , f ◦ π ) by the description of thecomposition (2.4) given above. (cid:3) Poincar´e duality
In this section we prove Theorem 1.1. For the rest of this section, let X be aproper G -spin c manifold with X/G is compact.Recall that an element K G ( X ) = K G ( X × R ) is equivalent to a difference oftwo vector bundles over X × S that are G -equivariantly isomorphic over X × { } ,where S is equipped with the trivial G -action. With this in mind, we can definethe following natural map between K ∗ G ( X ) and K G ∗ ( X ), which can be thought of ascap product with the fundamental K -homology class. Definition 3.1.
Define the map φ : K ∗ G ( X ) → K G ∗ ( X ) by φ : K iG ( X ) → K Gi +dim X ( X ) ,x ( [ X, x, id] , if i = 0 , [ X × S , x, pr ] , if i = 1 , where pr : X × S is the projection onto the first factor, and we have used thenotation from Remark 2.9.We now show that φ is an isomorphism by defining explicitly a map ψ that willturn out to be its inverse. HAO GUO AND VARGHESE MATHAI
Definition 3.2.
Define the map ψ : K G ∗ ( X ) → K ∗ G ( X ) by ψ : K Gi +dim X ( X ) → K iG ( X ) , [ M, E, f ] f ! [ E ] , for i = 0 ,
1, where f ! is the Gysin homomorphism from (2.3). Remark 3.3.
Note that f ! [ E ] ∈ f ! ( K G ( M )) ⊆ K dim X − dim MG ( X ) = K dim X − ( i +dim X ) G ( X ) = K iG ( X ) , so the dimensions make sense.We first need to show that the map ψ is well-defined. For this, we will use thefollowing: Lemma 3.4.
Let W be a G -cocompact, G -spin c manifold-with-boundary, and let X be a G -cocompact G -spin c manifold. Let h : W → X be a G -equivariant map, andlet i : ∂W ֒ → W be the natural inclusion. Then the composition K ∗ G ( W ) i ∗ −→ K ∗ G ( ∂W ) ( h | ∂W ) ! −−−−−→ K ∗ G ( X ) is the zero map.Proof. We give the proof in the case dim ∂W = dim X mod 2, showing that K G ( W ) i ∗ −→ K G ( ∂W ) ( h | ∂W ) ! −−−−−→ K G ( X ) (3.1)is the zero map; the proofs for the other cases are similar.Let us consider the composition (3.1) upon applying the Thom isomorphism,Theorem 2.5, in the form of (2.2), and use the description of the Gysin map from(2.3).Let j W be a G -equivariant embedding of W into R n for some n , where G isrealized as a subgroup of GL (2 n, R ); note that this is possible because G is assumedto be linear. Let j ∂W denote the restriction of j W to ∂W . Let i X : X → X × R n be the zero section. Define the embedding i W : W → X × R n ,w ( h ( w ) , j W ( w )) , and let i ∂W be the restriction of i W to ∂W . Let ν W and ν ∂W denote the respectivenormal bundles of the embeddings i W and i ∂W . We may identify these normalbundles with G -invariant tubular neighbourhoods U W and U ∂W in X × R n , notingthat in general U W has boundary. Since the normal bundle of ∂W in W is trivialand one-dimensional, there is a natural G -equivariant identification U ∂W ∼ = ∂U W × ( − ε, ε ) (3.2)for some ε >
0. It follows from the two-out-of-three lemma for G -equivariant spin c -structures (see [12, Section 3.1] and [8, Remark 2.6]), together with the fact that OINCAR´E DUALITY FOR ACTIONS BY MATRIX GROUPS 9 W and X are G -spin c , that ν W and ν ∂W are G -spin c vector bundles, and henceTheorem 2.5 applies. The resulting Thom isomorphisms for W , ∂W , and X (inthe notation of (2.2)) are shown as vertical arrows in the following commutativediagram: K G ( W ) K G ( ∂W ) K G ( X ) K G ( U W ) K G ( U ∂W ) K G ( X × R n ) , i ∗ T ( h | ∂W ) ! T TT i ∗ λ (3.3)where the map T i ∗ is determined by uniquely commutativity, and the homomor-phism λ is induced by the extension-by-zero map C ( U ∂W ) → C ( X × R n ) as inRemark 2.7. It thus suffices to show that the composition λ ◦ T i ∗ vanishes.By [13, Lemma 2.2] or [14, Chapter 5], we have a six-term exact sequence K G ( U W \ ∂U W ) K G ( U W ) K G ( ∂U W ) K G ( ∂U W ) K G ( U W ) K G ( U W \ ∂U W ) , λ j ∗ δδ j ∗ λ (3.4)where j ∗ is induced by the inclusion j : ∂U W ֒ → U W and the maps λ are againinduced by the extension-by-zero map C ( U W \ ∂U W ) → C ( U W ). The identification(3.2) gives a natural isomorphism K G ( ∂U W ) ∼ = K G ( U ∂W ). Using this, the bottomrow of (3.3) fits into the following commutative diagram: K G ( U W ) K G ( U ∂W ) K G ( U X ) K G ( ∂U W ) K G ( U W \ U ∂W ) . T i ∗ j ∗ λδ ∼ = λ (3.5)It follows from exactness of (3.4) that λ ◦ T i ∗ = 0, and hence ( h | ∂W ) ! ◦ i ∗ = 0. (cid:3) Proposition 3.5.
The map ψ is well-defined.Proof. That ψ respects disjoint union/direct sum is clear, since for any element ofthe form [ M, E ⊕ E , f ] ∈ K G ∗ ( M ), we have ψ [ M, E ⊕ E , f ] = f ! [ E ⊕ E ] = f ! [ E ] + f ! [ E ] ∈ K ∗ G ( M ) . Next, let [
W, E, h ] be an equivariant bordism between two elements [ M , E , h ] and[ − M , E , h ]. Then Lemma 3.4 applied to ∂W = M ⊔ − M implies that( h ) ! [ E ] = ( h ) ! [ E ] , hence ψ is well-defined with respect to the bordism relation. To see that ψ is well-defined with respect to vector bundle modification, let ( c M , F ⊕ π ∗ ( E ) , f ◦ π ) be the modification of a cycle ( M, E, f ) for X by a bundle V , as in Definition 2.8 (iii). ByLemma 2.10, we have[ c M , F ⊕ π ∗ ( E ) , f ◦ π ] = [ c M , s ! [ E ] , f ◦ π ] . Functoriality of the Gysin map with respect to composition, together with the factthat π ◦ s = id, now implies ψ [ c M , s ! [ E ] , f ◦ π ] = ( f ◦ π ) ! s ! [ E ]= f ! ◦ ( π ◦ s ) ! [ E ]= f ! [ E ]= ψ [ M, E, f ] . (cid:3) It is clear that ψ ◦ φ = id, and hence the map φ is injective. Indeed, for any x ∈ K G ( X ), we have ψ ◦ φ ( x ) = ψ ([ X, x, id]) = id ! ( x ) = x, (3.6)where we have used the notation from Remark 2.9.For surjectivity, we will use the following result, which is a special case of [5,Theorem 4.1] but applied to linear instead of compact G . Lemma 3.6.
Let
M, N, X be three G -cocompact G -spin c manifolds, f : N → X a G -equivariant continuous map, and g : M → N a G -equivariant embedding. Then [ M, E, g ◦ f ] = [ N, f ![ E ] , g ] ∈ K G ∗ ( X ) . Proof.
The proof of Theorem [5, Theorem 4.1], which was stated for compact Liegroups, goes through with no changes to our setting. (cid:3)
Proposition 3.7.
The map φ is surjective.Proof. At the level of geometric cycles, the composition φ ◦ ψ is given by[ M, E, f ] f ! [ E ] [ X, f ! [ E ] , id] . Thus it suffices too prove that any geometric cycle of the form (
M, E, f ) is equivalentto (
X, f ! [ E ] , id). Let i : M → R n be a G -equivariant embedding for some n , and let j : X → R n × X be the zero section. Upon compactifying, the map f factors as S n × XM X, pr i × f f (3.7)where pr is the projection onto the second factor, and j becomes an embedding X → S n × X . By Lemma 3.6 applied to the embedding i × f , we have[ M, E, f ] = [ S n × X, ( i × f ) ! [ E ] , pr ] ∈ K G ∗ ( X ) . (3.8) OINCAR´E DUALITY FOR ACTIONS BY MATRIX GROUPS 11
Meanwhile, by Lemma 3.6 to j , together with functoriality of the Gysin map, yields[ X, f ! [ E ] , id] = [ X, f ! [ E ] , pr ◦ i ]= [ S n × X, i ! ( f ! [ E ]) , pr ]= [ S n × X, ( i ◦ f ) ! [ E ] , pr ] . (3.9)Finally, since the maps i × f and i ◦ f are G -homotopic through F : M × I → S n × X, ( m, t ) ((1 − t ) i ( m ) , f ( m )) , invariance of the Gysin homomorphism under G -homotopy implies that (3.8) and(3.9) are equal, and we conclude. (cid:3) Propositions 3.7 and equation (3.6) together imply that φ : K iG ( X ) → K Gi +dim X ( X )is an isomorphism for i = 0 ,
1, which proves Theorem 1.1.
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Department of Mathematics, Texas A&M University
Email address : [email protected] (Varghese Mathai) School of Mathematical Sciences, University of Adelaide
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