Dualities in equivariant Kasparov theory

Abstract

We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov's first and second Poincare duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the description of bivariant Kasparov theory for commutative C*-algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities apply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum-Connes assembly map via localisation of categories.

Significant changes, mainly variants of the duality isomorphisms with different support conditions and applications

References in corpus (4)

• On the Equivalence of Geometric and Analytic K-Homology
• Bivariant K-theory via correspondences
• Categorical aspects of bivariant K-theory
• Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Cited by in corpus (12)

• Homological algebra in bivariant K-theory and other triangulated categories
• C*-Algebras over Topological Spaces: The Bootstrap Class
• Equivariant representable K-theory
• Bivariant K-theory via correspondences
• Equivariant embedding theorems and topological index maps
• Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
• Groups with Spanier-Whitehead duality
• Equivariant Lefschetz maps for simplicial complexes and smooth manifolds
• Categorical approach to the Baum-Connes conjecture for étale groupoids
• Localization techniques in circle-equivariant KK-theory
• Twisted Geometric K-homology for Proper actions of discrete groups
• An equivariant Poincaré duality for proper cocompact actions by matrix groups