Twisted K-theory and Poincare duality

Using methods of KK-theory, we generalize Poincare duality to the framework of twisted K-theory.


Introduction
In [4], Connes and Skandalis showed, using Kasparov's KK-theory, that given a compact manifold M , the K-theory of M is isomorphic to the K-homology of T M and vice-versa. It is well-known to experts that a similar result holds in twisted K-theory, although this is apparently written nowhere in the literature. In this paper, using Kasparov's more direct approach, we show that given any (graded) locally trivial bundle A of elementary C * -algebras over M , the C * -algebras of continuous sections C(M, A) and C(M, A op⊗ Cliff(T M ⊗ C)) are K-dual to each other. When A = M is the trivial bundle, we recover Poincaré duality between C(M ) and C τ (M ) := C(M, Cliff(T M ⊗C)) [8], which is equivalent to Poincaré duality between C(M ) and C 0 (T M ) since C τ (M ) and C 0 (T M ) are KK-equivalent to each other.

Preliminaries
In this paper, we will assume that the reader is familiar with the language of groupoids (although this is not crucial in the proof of the main theorem concerning Poincaré duality).
We just recall the definition of a generalized morphism (see e.g. [6]), since it is used at several places. Suppose that G ⇒ G (0) and Γ ⇒ Γ (0) are two Lie groupoids. Then a generalized morphism from G to Γ is given by a space P , two maps G (0) τ ← P σ → Γ (0) , a left action of G on P with respect to τ , a right action of Γ on P with respect to σ, such that the two actions commute, and P → G (0) and is a right Γ-principal bundle. The set of isomorphism classes of generalized morphisms from G to Γ is denoted by H 1 (G, Γ). There is a category whose objects are Lie groupoids and arrows are isomorphism classes of generalized morphisms; isomorphisms in this category are called Morita equivalences.
Finally, we recall that any element of H 1 (G, Γ) is given by the composition of a Morita equivalence with a strict morphism.

Graded twists and twisted K-theory
In this section, we review the basic theory of twisted K-theory in the graded setting, sometimes in more detail than some other references like [1,5,3]. This is a probably well-known and straightforward generalization of the ungraded case as developed e.g. in [1,10], hence we will omit most proofs.

Graded Dixmier-Douady bundles
Let M ⇒ M (0) be a Lie groupoid (more generally, most of the theory below is still valid for locally compact groupoids having a Haar system). The reader who is not interested in equivariant K-theory may assume that M = M (0) = M is just a compact manifold.
A graded Dixmier-Douady bundle of parity 0 (resp. of parity 1) A over M ⇒ M (0) is a locally trivial bundle of Z/2Z-graded C * -algebras over M (0) , endowed with a continuous action of M, such that for all x ∈ M (0) , the fiber A x is isomorphic to the Z/2Z-graded algebra K(Ĥ x ) of compact operators over a Z/2Z-graded Hilbert spaceĤ x (resp. to the where H x is some Hilbert space and Cℓ 1 is the first complex Clifford algebra). Beware that H x orĤ x does not necessarily depend continuously on x. Of course, the action of M is required to preserve the degree. The usual theory of graded twists [1] corresponds to even graded D-D bundles (i.e. D-D bundles of parity 0), but our slightly more general definition allows to cover Clifford bundles as well: if E → M is a Euclidean vector bundle of dimension d, then Cliff(E⊗C) → M is a graded D-D bundle of parity (d mod 2).
Denote byĤ the graded Hilbert space is the M-equivariant M (0) -Hilbert module obtained from C c (M) by completion with respect to the scalar product Two graded D-D bundles A and A ′ are said to be Morita equivalent if (they have the same parity and) A⊗K(Ĥ M ) ∼ = A ′⊗ K(Ĥ M ). The set of Morita equivalence classes of graded D-D bundles forms a group Br * (M) = Br 0 (M) ⊕ Br 1 (M), the graded Brauer group of M. The sum of A and A ′ is A⊗A ′ (note that the parities do add up), and the opposite A op of A is the bundle whose fibre at x ∈ M (0) is the conjugate algebra of A x . In other words, ) in the even (resp. odd) case.
Note that H 1 (M,Û (Ĥ)) is not a monoid, since given two morphisms f 1 , f 2 : Γ →Û (Ĥ) (with Γ Morita equivalent to M), the map f : g → f 1 (g)⊗f 2 (g) is not a morphism since On the other hand, if we restrict to degree 0 operators, i.e. if we considerÛ ( The sequence where the first map is the quotient map and the second is P → P × PÛ(Ĥ) K(Ĥ), is canonically split-exact (the proof is analogue to [10]), and the splitting identifies There is a split exact sequence [1,5] Indeed, from the exact

Furthermore, in the decomposition
. This can be seen by direct checking using (1)

Graded S 1 -central extensions
One defines the sum of two graded central extensions ( Γ 1 , δ 1 ) and ( Γ 2 , δ 2 ) as ( Γ, δ), where δ(g) = δ 1 (g) + δ 2 (g) and Γ = ( The multiplication for the groupoid Γ is ( Note that the set of isomorphism classes of graded S 1 -central extensions of Γ forms an abelian group. To see that the product is commutative, To see that ( Γ, δ) has an inverse, let Γ op be equal to Γ as a set, but the S 1 -principal bundle structure is replaced by the conjugate one, and the product * op in Γ op is is an isomorphism ( g ∈ Γ is any lift of g ∈ Γ).
Let us define the group Ext(M, S 1 ). Consider the collection of triples ( , the K-theory of the reduced crossed-product of the graded C * -algebra A by the action of M. If A corresponds to the graded central extension ( The C * -algebra C * r ( Γ) S 1 is considered as a Z/2Z-graded C * -algebra, using the grading automorphism Note that it suffices to study K 0

Example of manifolds
Let M be a manifold. Elements of Ext(M, S 1 ) are given by an open cover (U i ) i∈I , smooth maps c ijk : Define a product on Γ by (x ij , λ)(x jk , µ) = (x ik , λµc ijk ). Then Γ is a groupoid, and there is a central extension S 1 → Γ → Γ.
The sum of (c 1 , δ 1 ) and (c 2 , Let us consider the particular case when c = 1 is the trivial cocycle. In that case, C * r ( Γ) S 1 ∼ = C * r (Γ). Let us compare this Z/2Z-graded C * -algebra to the Z/2Z-graded C * -algebra C 0 ( M ), where M → M is the double cover determined by the cocycle δ. Let P = (∐U i ) × M M . Then P is a Z/2Z-equivariant Morita equivalence from Γ to M ⋊ Z/2Z, [5,Remark A.13].

Twistings by Euclidean vector bundles
Suppose that E is a Euclidean vector bundle over M ⇒ M (0) . Then E is given by an O(n)-principal bundle over M ⇒ M (0) , hence by a morphism f : Γ → O(n) together with a Morita equivalence from Γ to M.
Let E be the graded S 1 -central extension We first need two lemmas. Proof. Since G is compact, every S 1 -central extension is of finite order. Let us recall the argument: given a central extension Then the representation of G in W is a map G → U (W ) ∼ = S 1 which is a splitting of n E, hence E is of order at most n.
Therefore, the extension E comes from a central extension 0 → Z/nZ → G → G → 1. Since G is simply connected, the central extension must be trivial as Z/nZ-principal bundle, i.e. G = G × Z/nZ, and the product on G is given by (g, λ)(h, µ) = (gh, λ + µ + c(g, h)) where c : G × G → Z/nZ is a 2-cocycle. Using connectedness of G 0 , c must factor through G/G 0 × G/G 0 , i.e. the central extension is pulled back from a central extension of G/G 0 , which must be trivial by assumption. If E and E ′ are S 1 -central extensions whose restriction to G 0 are isomorphic, then E and E ′ are isomorphic.
Proof. After taking the difference of E and E ′ , we may assume that E ′ is the trivial extension. Denote by S 1 → G → G the extension E. Let g → g be a splitting G 0 → G. Choose a family (s i ) such that G = ∐ i s i G 0 , and for each i, choose a lift s i of s i . Define then s i g by s i g. By construction, γh = γ h for all (γ, h) ∈ G × G 0 .
Next, define the 2-cocycle c : G × G → S 1 by g h = c(g, h) gh. Let c ij = c(s i , s j ).
For all j, let ϕ j : G 0 → S 1 such that s −1 j g s j = ϕ j (g) s −1 j gs j . It is immediate to check that ϕ j is a group morphism, hence ϕ j is trivial by assumption, i.e. s −1 j g s j = s −1 j gs j . Multiplying on the right by h and on the left by s i s j , we get s i g s j h = s i s j s −1 j gs j h = c ij s i s j s −1 j gs j h = c ij s i gs j h, hence s i g s j h = c ij s i gs j h. It follows that c(s i g, s j h) = c ij , i.e. that c factors through G/G 0 × G/G 0 . Since Br(G/G 0 ) is trivial by assumption, c must be a coboundary. We conclude that E is a split extension.
Proof. The last equality follows from the fact that C 0 (E) and C 0 (M (0) , Cliff(E ⊗ R C)) are M-equivariantly KK-equivalent [7]. The second equality is just the definition.
To prove the first equality, let us suppose for instance that n = dim E is even, the proof for n odd being analogous. We have to compare the graded D-D bundle A ′ E with the graded central extension S 1 → Γ → Γ which is pulled back from S 1 → P in c (n) → O(n). By naturality, we can just assume that Γ = M = O(n), and that E = R n is endowed with the canonical action of O(n).
Denote by α : O(n) → PÛ (Ĥ) the canonical action of O(n) on Cℓ n = Cliff(E C ). To show that the central extension associated to the graded D-D bundle Cliff(E C ) → · is (S 1 → P in c (n) → O(n)), it suffices to prove that there exists a lifting α: It follows that β(−1) = −Id, hence β induces a lift β : Spin c (n) →Û (Ĥ) which is S 1 -equivariant. This means that the restriction of E to SO(n) is isomorphic to S 1 → Spin c (n) → SO(n). To conclude that the restriction of E to O(n) is isomorphic to S 1 → P in c (n) → O(n), we apply Lemma 2.4 to G = O(n) and G 0 = SO(n).

Poincaré duality 3.1 Kasparov's constructions
Let M be a compact manifold (actually, Poincaré duality can be generalized to arbitrary manifolds [8], but in this paper we confine ourselves to compact ones for simplicity). We suppose that M is endowed with a Riemannian metric which is invariant by the action of a locally compact group G. Given any vector bundle A over any manifold M , we denote by C A (M ) the space of continuous sections vanishing at infinity. We will also write C A whenever there is no ambiguity. We denote by τ the complexified cotangent bundle of M .
In [8], Kasparov constructed two elements and D ∈ KK G (C τ (M ), C) (in this paper, we will use Le Gall's [9] notation KK M ⋊G (·, ·) for equivariant KK-theory with respect to the groupoid M ⋊ G, rather than Kasparov's RKK G (M ; ·, ·), but of course both are equivalent).
Let us recall the construction of θ and D.
Let Let us explain the construction of θ. Denoting by ρ the distance function on M , let r > 0 be so small that for all (x, y) in U = {(x, y) ∈ M × M | ρ(x, y) < r}, there exists a unique geodesic from x to y.
For every C(M ×M )-algebra A, we denote by A U the C * -algebra C 0 (U )A. Then the element θ is defined as

Constructions in twisted K-theory
In this subsection, we construct an element θ A ∈ KK M ⋊G (C(M ), C A⊗ C A⊗A op ) for any graded D-D bundle A over (M × G ⇒ M ), i.e. for any G-equivariant graded D-D bundle over M . We may assume that A is stabilized, i.e. that A ∼ = A⊗K(Ĥ ⊗ L 2 (G)). First, let us denote by p t (x, y) the geodesic segment joining x to y at constant speed (0 ≤ t ≤ 1).
Using p t , we see that p t : U → M is a G-equivariant homotopy equivalence. Unfortunately, this does not imply that Br(U ⋊ G) and Br(M ⋊ G) are isomorphic for arbitrary G, hence we will make the following Assumption. In the sequel of this paper, and unless stated otherwise, G will be a compact Lie group acting smoothly on a compact manifold M .
In that case, . As a consequence, there is a continuous, G-equivariant family of isomorphisms u t,x,y : A x ∼ → A pt(x,y) .
Of course, the u t 's are not unique, but this will not be important as far as K-theory is concerned as we will see.
We then define θ A as

Twisted K-homology
Given a C * -algebra A endowed with an action of a locally compact group G, the Gequivariant K-homology of A, K * G (A), is defined by KK * G (A, C). If A is a G-equivariant graded D-D bundle over M , we define K G,A * (M ) by K * G (C A (M )).

Remark 3.3
The map µ does not depend of the choice of the isomorphisms u t,x,y , hence ν doesn't either.
The rest of the paper is devoted to the proof of Theorem 3.1.

Proof of ν • µ = Id
For all α ∈ KK M ⋊G (C(M ) ⊗ A, C A (M )⊗B), we have Suppose shown that .
We postpone the proof of (b) until subsection 3.9 3.9 Proof of (4) Let us first recall the proof when A is trivial [8,Lemma 4.5]. We want to show that for Write α = [(E, T )] where C(M, A)E = E and T is G-continuous. Then both products can be written as where F i is of the form M We want to show that α Let us just explain the homotopy between the two modules, the homotopy between the F i 's being obtained using Kasparov's technical theorem in the same way as in [8,Lemma 4.5].
The left-hand side is and the right-hand side is where we recall that p 1 : U → M is the second projection (x, y) → y. 1 is the Morita equivalence between C 0 (U ) and p * 0 C A⊗C 0 (U ) p * 1 C A op obtained by composing the Morita equivalence p * 0 H between C 0 (U ) and p * 0 (C A⊗A op ) with the isomorphism p * 0 A op ∼ = p * 1 A op . Using the map p t : U → M instead of p 1 , consider (with obvious notations) the homotopy E⊗ C A σ M,C A (F t )⊗ C 0 (U ) H ′ t⊗C p * t (A⊗A op ) (U ) p * t H op . For t = 1, we get (6). For t = 0, we get E⊗ C A (C A⊗ C τ ) U⊗C 0 (U ) p * 0 H⊗ C p * 0 (A⊗A op ) (U ) p * 0 H op where the right C p * 0 (A⊗A op ) (U )-structure on E⊗ C A (C A⊗ C τ ) U⊗C 0 (U ) p * 0 H is defined as follows: C p * 0 A acts on (C A⊗ C τ ) U by the obvious action, and C p * 0 A op acts on p * 0 H. In other words, it is the tensor product of (5) with β A over C A , where β A is the C A -C A -bimodule In the expression above, the right C A⊗A op -module structure on C A⊗C(M ) H is defined as follows: ∀a ∈ C A , ∀b ∈ C A op ∀ξ ⊗ η ∈ C A⊗C(M ) H, (ξ ⊗ η) · (a ⊗ b) = (−1) |η| |a| ξa ⊗ ηb.