Bivariant K-theory via correspondences

TL;DR

This paper develops a topological equivariant bivariant K-theory using correspondences, extending the concept beyond smooth maps, and explores duality isomorphisms linking it to K-theory with support conditions.

Contribution

It introduces a new correspondence-based framework for equivariant bivariant K-theory that generalizes existing theories without relying on smooth structures.

Findings

Constructs bivariant extensions for arbitrary equivariant cohomology theories.

Formulates conditions for duality isomorphisms in geometric bivariant K-theory.

Shows cases where bivariant K-theory reduces to K-theory with support conditions.

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.