We study the C*-algebra crossed product C 0 (X)⋊ G of a locally compact group G acting
properly on a locally compact Hausdorff space X. Under some mild extra conditions, which
are automatic if G is discrete or a Lie group, we describe in detail, and in terms of the action …

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\rtimes G is KK-theoretically Poincaré dual to\big (\mathcal A ̂ ⊗ …

For a large class of word hyperbolic groups G the cross product C^*-algebra arising from the
action of G on its Gromov boundary is shown to satisfy Poincare duality in K-theory. This
class strictly contains fundamental groups of compact, negatively curved manifolds. The …

We interpret certain equivariant Kasparov groups as equivariant representable K-theory
groups and compute these via a classifying space and as K-theory groups of suitable σ-C*-
algebras. We also relate equivariant vector bundles to these σ-C*-algebras and provide …

Let G be a locally compact group, let X be a universal proper G-space, and let be a G-
equivariant compactification of X that is H-equivariantly contractible for each compact
subgroup. Let. Assuming the Baum-Connes conjecture for G with coefficients and C (∂ X) …

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. Our construction uses no special features of equivariant K-theory …

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 …

We study several duality isomorphisms between equivariant bivariant K-theory groups,
generalising Kasparov's first and second Poincaré duality isomorphisms. We use the first
duality to define an equivariant generalisation of Lefschetz invariants of generalised self …

Let G be a torsion-free discrete group with a finite-dimensional classifying space B G. We
show that G has a dual-Dirac morphism if and only if a certain coarse (co-) assembly map is
an isomorphism. Hence the existence of a dual-Dirac morphism for such groups is a metric …

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 …