Bivariant K-theory via correspondences

Abstract

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.

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 version

References in corpus (5)

• On the Equivalence of Geometric and Analytic K-Homology
• Equivariant representable K-theory
• Dualities in equivariant Kasparov theory
• Equivariant embedding theorems and topological index maps
• Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Cited by in corpus (14)

• Dualities in equivariant Kasparov theory
• Equivariant embedding theorems and topological index maps
• A Geometric Description of Equivariant K-Homology for Proper Actions
• Positive Scalar Curvature and Poincare Duality for Proper Actions
• Geometric K-homology with coefficients I
• Geometric construction of classes in van Daele's K-theory
• Dynamical correspondences for Smale spaces
• Localization techniques in circle-equivariant KK-theory
• Cobordism bicycles of vector bundles
• Equivariant correspondences and the Borel-Bott-Weil theorem
• Enriched categories of correspondences and characteristic classes of singular varieties
• The ring structure of twisted equivariant $KK$-theory for noncompact Lie groups
• Motivic Bivariant Characteristic Classes
• Baum-Connes and the Fourier-Mukai transform