An equivariant Poincar\'e duality for proper cocompact actions by matrix groups

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 holds between $G$-equivariant $K$-theory of $M$, defined using finite-dimensional $G$-vector bundles, and $G$-equivariant $K$-homology of $M$, defined through the geometric model of Baum and Douglas.


Introduction
Poincaré 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 with the fundamental class in K-homology.This class can be represented geometrically by the spin c -Dirac operator.More generally, if a compact Lie group acts on the manifold preserving the spin c structure, the analogous map implements Poincaré duality between the equivariant versions of K-theory and K-homology.In the case when 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 proper actions.The observation underlying Poincaré duality in all of these cases is that there exist enough equivariant vector bundles with which to pair the fundamental class.
In contrast, Phillips [14] showed through a counter-example with a non-linear group that, for proper actions by a general Lie group G on a manifold X with compact quotient space, finite-dimensional vector bundles do not exhaust the Gequivariant K-theory of X, and that it is necessary to introduce infinite-dimensional vector bundles into the description of K-theory (see also [10]).
One case in which finite-dimensional bundles are sufficient is when the group G is linear (see [13]), owing to the key fact that in this case every G-equivariant vector bundle over X is a direct summand of a G-equivariantly trivial bundle, ie.one that is isomorphic to X × V for some finite-dimensional representation of G on V .
G-equivariant vector bundle over M , and f : M → X is a G-equivariant continuous map.The Baum-Douglas map takes where ϕ is the * -representation of C 0 (M ) on B(L 2 (E)) given by pointwise multiplication, and the right-hand side is the pushforward under f of a class in KK G * (C 0 (M ), C).Corollary 1.2 then implies: Corollary 1.3.Suppose a linear Lie group G acts properly and cocompactly on a G-equivariantly spin c manifold X.Then for any e ∈ KK G * (C 0 (X), C), there exist a G-equivariantly cocompact spin c manifold M and a G-equivariant continuous map f : M → X such that e is the pushforward under f of the class of a Dirac-type operator on M in KK G * (C 0 (M ), C).More precisely, there exists a vector bundle Acknowledgements.The authors would like to thank Peter Hochs and Hang Wang for their useful feedback on this paper.
Hao Guo was partially supported by funding from the Australian Research Council through the Discovery Project DP200100729, and partially by the National Science Foundation through the NSF DMS-2000082.Varghese Mathai was partially supported by funding from the Australian Research Council, through the Australian Laureate Fellowship FL170100020.

Preliminaries
We begin by recalling the definitions and facts we will need.Unless specified otherwise, G will always denote a closed subgroup of GL(n, R) for some n.All vector bundles will be complex.For this section, let X be a locally compact proper G-space.
2.1.Equivariant K-theory.In [13], Phillips showed that the G-equivariant Ktheory of the space X with G-cocompact supports can be defined in a such a way that 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 the complement 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 x ⊕ id)a x = t x ⊕ id for all x in the complement of a G-cocompact subset of X.The set of equivalence classes [E, F, t] of K-cocycles forms a semigroup under the direct sum operation, and we define the group K 0 G (X) is the Grothendieck completion 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 will omit the map t from the cycle, and simply denote a class in , where G acts trivially on R i .By [13, Lemma 2.2], K i G satisfies Bott periodicity, so that we have a natural isomorphism K i G (X) ∼ = K i+2 G (X) for each i.We will use the notation In addition, K i G are contravariant functors from the category of proper G-spaces and proper G-equivariant maps to the category of abelian groups, and form an equivariant extraordinary cohomology theory with a continuity property.In particular, Bott periodicity implies that for any G-invariant open subset U ⊆ X, there is a six-term exact sequence of abelian groups is induced by the extension-by-zero homomorphism -see Remark 2.5 below, and the boundary maps ∂ are defined as in equivariant Ktheory for compact group actions [15].

Remark 2.5 (Extension-by-zero). Any inclusion of G-invariant open subsets U 1 ֒→ U 2 induces in the obvious way an extension-by-zero
The induced map on operator K-theory, together with the identification To prove Poincaré duality, we will make use of the following Thom isomorphism theorem for G-spin c bundles: Theorem 2.6 ([14, Theorem 8.11]).Let E be a finite-dimensional G-equivariant spin c vector bundle over X.Then there is a natural isomorphism 2.2.The Gysin homomorphism.Theorem 2.6 can be used to give an explicit geometric description of the Gysin (pushforward) homomorphism in G-equivariant K-theory, which we will need later.
Let Y 1 and Y 2 be two G-cocompact G-spin c manifolds and f : Y 1 → Y 2 a Gequivariant continuous map.By cocompactness, both Y 1 and Y 2 contain only finitely many orbit 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 denote the zero section, and define the G-equivariant embedding Let ν 1 be the normal bundle of i Y 1 , which we identify with a G-invariant tubular neighbourhood U 1 of its image.Note that 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 assumption that Y 1 and Y 2 are G-spin c , that ν 1 has a G-spin c structure, and so Theorem 2.6 applies.Identifying the normal bundle with U 1 , we have the Thom isomorphism for i = 0, 1.The Gysin homomorphism where λ is induced by the extension-by-zero map associated to the inclusion of U 1 into Y 2 × R 2n (see Remark 2.5 below), and the right horizontal isomorphism is due to Bott periodicity.
Remark 2.7.It can be seen from the above that f !depends only on the G-homotopy class of f and that the Gysin map is functorial under compositions.
, where For i = 0 or 1, the G-equivariant geometric K-homology group K G i (X) is the abelian group generated by geometric K-cycles (M, E, f ) where dim M = i mod 2, subject to the equivalence relation generated by the following three elementary relations: (i) (Direct sum -disjoint union) For two G-equivariant Hermitian vector bundles E 1 and E 2 over M and a G-equivariant continuous map f : M → X, (ii) (Bordism) Suppose two cycles (M 1 , E 1 , f 1 ) and (M 2 , E 2 , f 2 ) are bordant, so that 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 , where −M 2 denotes M 2 with the opposite G-spin c structure.Then (iii) (Vector bundle modification) Let V be a G-spin c vector bundle of real rank 2k over M .Upon choosing a G-invariant metric on V , let M be the unit sphere bundle of (M × R) ⊕ V , where the bundle M × R is equipped with the trivial G-action.Let F be the Bott bundle over M , which is fibrewise the non-trivial generator of K 0 (S 2k ).(See [4, Section 3] for a more detailed description.)Then where π : M → M is the canonical projection.
the additive inverse of [M, E, f ] is its opposite [−M, E, f ], while the additive identity is given by the empty cycle where M = ∅.
Remark 2.9.The above definition of classes [M, E, f ] continues to make sense if we replace the bundle E by a K-theory class.Indeed, if , then there exists a G-vector bundle F over X such that Adding the inverse of [M, F, f ] to both sides and rearranging shows that the class Finally, we can describe vector bundle modification using the Gysin homomorphism 2.3.To do this, let M be the manifold underlying the vector bundle modification of a cycle (M, E, f ) by a bundle V , as in Definition 2.8 (iii).Then M is the unit sphere bundle of (M × R) ⊕ V .We will refer to the G-equivariant embedding as the north pole section.Lemma 2.10.Let (M, E, f ) be a geometric cycle for X.Let ( M , F ⊗π * (E), f •π) be its modification by a G-spin c vector bundle V of even real rank, and let π : M → M be the projection.Let s : M → M be the north pole section.Then Proof.The proof we give is similar to that of [5, Lemma 3.5] concerning the case of compact 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 a G-invariant tubular neighbourhood U of the embedding s : M → M .The Gysin map s ! is then the composition where T G is the Thom isomorphism in the form (2.2), while λ is the homomorphism induced by the extension-by-zero map C 0 (U ) → C 0 ( M ).Note that T G is essentially given via tensor product with a "Bott element", and (2.4) admits the following geometric description.Let F be the Bott bundle over M , and let F 0 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 0 ].On the other hand, since M is the boundary of the unit sphere bundle of (M × R) ⊕ W , and the bundle F 0 is pulled back from M , the cycle whence the right-hand side is equal to ( M , s![E], f • π) by the description of the composition (2.4) given above.

Poincaré duality for geometric K-homology
In this section we prove Theorem 1.1.For the rest of this section, let X be a proper G-spin c manifold with X/G is compact.We can define the following natural map between K * G (X) and K G * (X), which can be thought of as cap product with the fundamental K-homology class on X.For this, let S 1 be the unit circle in C, and define the map c : X → X × S 1 x → (x, 1). (3.1) where pr 1 : X × S 1 is the projection onto the first factor, and we have used the notation from Remark 2.9.
We now show that φ is an isomorphism by defining explicitly a map ψ that will turn out to be its inverse.
so the degrees make sense.
We first need to show that ψ is well-defined.For this, we will use the following: 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, and let i : ∂W ֒→ W be the natural inclusion.Then the composition is the zero map.
Proof.We give the proof when dim ∂W = dim X mod 2 and show that is the zero map; the proofs for the other cases are similar.
Let us consider the composition (3.2) upon applying the Thom isomorphism, Theorem 2.6, 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 2n for some n, where G is realized as a subgroup of GL(2n, R); note that this is possible because G is assumed to be linear.Let j ∂W denote the restriction of j W to ∂W .Let i X : X → X × R 2n be the zero section.Define the embedding and let i ∂W be the restriction of i W to ∂W .Let ν W and ν ∂W denote the respective normal bundles of the embeddings i W and i ∂W .We may identify these normal bundles with G-invariant tubular neighbourhoods U W and U ∂W in X × R 2n , noting that in general U W has boundary.Since the normal bundle of ∂W in W is trivial and one-dimensional, there is a natural G-equivariant identification for some ε > 0. It follows from the two-out-of-three lemma for G-equivariant spin cstructures (see [12,Section 3.1] and [8, Remark 2.6]), together with the fact that W and X are G-spin c , that ν W and ν ∂W are G-spin c vector bundles, and hence Theorem 2.6 applies.The resulting Thom isomorphisms for W , ∂W , and X (in the notation of (2.2)) are shown as vertical arrows in the following commutative diagram: where the map T G i * is determined uniquely by commutativity, and the homomorphism λ is induced by the extension-by-zero map where j * is induced by the inclusion j : ∂U W ֒→ U W and the maps λ are again induced by the extension-by-zero map Using this, the bottom row of (3.4) fits into the following commutative diagram: It follows from exactness of (3.5) that λ • T G i * = 0, and hence (h| ∂W ) ! • i * = 0.
Proof.That ψ respects disjoint union/direct sum is clear, since for any element of the form [M, hence ψ is well-defined with respect to the bordism relation.To see that ψ is welldefined with respect to vector bundle modification, let ( 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).By Lemma 2.10, we have Functoriality of the Gysin map with respect to composition, together with the fact that π • s = id, now implies Proposition 3.6.The map φ is injective.
Proof.For any e ∈ K i G (X), we have which are both equal to e, where we have used functoriality of the Gysin map.Hence ψ • φ = id, so φ is injective.
For surjectivity, we will use the Gysin homomorphism from subsection 2.2, together with the following result, which is a special case of [5, Theorem 4.1] but applied to linear instead of compact G. Lemma 3.7.Let M, N, X be three G-cocompact G-spin c manifolds, g : N → X a G-equivariant continuous map, and f : M → N a G-equivariant embedding with even codimension.Then for any G-vector bundle E → M , we have Proof.The proof of Theorem [5, Theorem 4.1], which was stated for compact Lie groups, goes through with no changes to our setting.Proposition 3.8.The map φ is surjective.
Proof.Examining Definitions 3.1 and 3.2, one sees that φ • ψ is given, at the level of geometric cycles, by where the map c was defined in (3.1).Let i : M → R 2n be a G-equivariant embedding for some n, and let j : X → R 2n × X be the zero section.Upon compactifying R 2n , f factors as where pr 2 is the projection onto the second factor, and j becomes an embedding X → S 2n × X.
Suppose first that dim M = dim X mod 2. Then it suffices to prove that any geometric cycle of the form (M, E, f ) is equivalent to (X, f ![E], id).By Lemma 3.7 applied to the embedding i × f , we have Propositions 3.8 and equation (3.6) together imply that φ : K i G (X) → K G i+dim X (X) is an isomorphism for i = 0, 1, which establishes Theorem 1.1.
as in Remark 2.5.It thus suffices to show that the composition λ • T G i * vanishes.By [13, Lemma 2.2] or [14, Chapter 5], we have a six-term exact sequence