Let G be a locally compact group, let X be a universal proper G-space, and letX be a G-equivariant compactification of X that is H -equivariantly contractible for each compact subgroup H ⊆ G. Let ∂X =X \ X. Assuming the Baum-Connes conjecture for G with coefficients C and C(∂X), we construct an exact sequence that computes the map on K-theory induced by the embedding C * r G → C(∂X) r G. This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincaré duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic χ(G\X) is non-zero, then the unit element of C(∂X) r G is a torsion element of order |χ(G\X)|. Furthermore, we get a new proof of a theorem of Lück and Rosenberg concerning the class of the de Rham operator in equivariant K-homology. (1991): 19K35, 46L80 Mathematics Subject Classification

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.The stable Higson corona c(X) has better functoriality properties than the C * -algebra C * (X) that figures in the usual coarse Baum-Connes assembly map (see [13,19,20,[24][25][26][27]). The assignment X → c(X) is functorial from the coarse category of coarse spaces to the category of C * -algebras and C * -algebra homomorphisms. The analogous statement for the coarse C * -algebra C * (X) is true only after passing to K-theory. Moreover, the C * -algebra c(X) is designed to be closely related to certain bivariant Kasparov groups. This is the source of a homotopy invariance result, which implies our assertion for scalable spaces. Another advantage of the stable Higson corona and the coarse co-assembly map is their relationship with alternative approaches to the Novikov conjecture, namely, almost flat K-theory (see [4]) and the Lipschitz approach of [5].The map µ * is an isomorphism for any discrete group G that has a dual Dirac morphism, without any hypothesis on BG. This is how we are going to prove isomorphism of µ * for groups that uniformly embed in a Hilbert space: we show that such groups have a dual Dirac morphism. Actually, already the existence of an approximate dual Dirac morphism implies that µ * is an isomorphism. Using results of Kasparov and Skandalis [16], it follows that the coarse co-assembly map is an isomorphism for groups acting properly by isometries on bolic spaces. The usual coarse Baum-Connes conjecture for a group G is equivalent to the Baum-Connes conjecture with coefficients ∞ (G) [25]. Despite this, it is not known whether the existence of an action of G on a bolic space implies the coarse Baum-Connes conjecture for G. The existence of a dual Dirac morphism only implies split injectivity of the coarse Baum-Connes assembly map.Given the above observations, we expect the coarse co-assembly map to become a useful tool in connection with the Novikov conjecture. However, at the moment we have no examples of groups for which our method proves the Novikov conjecture while others fail. We also remark that we do not know whether the map µ * is an isomorphism for the standard counterexamples to the coarse Baum-Connes conjecture. Coarse spacesWe begin by recalling the notion of a coarse space and some related terminology (see [13,20]). Then we introduce σ-coarse spaces, which are useful to deal with the Rips complex construction.Let X be a set. We define the diagonal ∆ X , the transpose of E ⊆ X × X, and the composition of E 1 , E 2 ⊆ X × X byDefinition 2.1. A coarse structure on X is a collection E of subsets E ⊆ X × X-called ent...

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented maps by a class of K-oriented normal maps, which are maps together with a certain factorisation. Our construction does not use any special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant K-theories agree if there is such a duality isomorphism.Comment: The article was split into two parts to make it more accessible. Some results were added and som notation is changed, notably normal maps are now called normally non-singular maps. Thus this is essentially a new article, superseding the first versio

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map

Abstract. Let A be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then A −1 ⋊G is KK-theoretically Poincaré dual to`A⊗ C 0 (X) Cτ (X)´⋊G, where A −1 is the inverse of A in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov's duality theorem. As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifold M and Poincaré dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum-Connes conjecture.