Dualities in equivariant Kasparov theory

TL;DR

This paper explores duality isomorphisms in equivariant Kasparov theory, extending classical dualities and connecting them to Lefschetz invariants and the Baum-Connes conjecture, with implications for operator algebras and noncommutative geometry.

Contribution

It generalizes Kasparov's dualities to a broader setting, introduces an equivariant Lefschetz invariant, and relates dualities to the Baum-Connes assembly map for groupoids.

Findings

Established duality isomorphisms for equivariant bivariant K-theory.

Connected dualities to Lefschetz invariants of self-maps.

Applied dualities to describe the Baum-Connes assembly map.

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.