Chern character for totally disconnected groups

TL;DR

This paper develops a bivariant Chern character for equivariant KK-theory of totally disconnected groups, linking it to cosheaf homology and extending previous work for profinite groups.

Contribution

It introduces a new bivariant Chern character for totally disconnected groups' equivariant KK-theory, connecting it to cosheaf homology and generalizing Baum-Schneider's Chern character.

Findings

Isomorphism between complexified KK-theory and cosheaf homology for these groups

Extension of Baum-Schneider's Chern character to a broader class of groups

Provides new tools for understanding the Baum-Connes conjecture in this context

Abstract

In this paper we construct a bivariant Chern character for the equivariant KK-theory of a totally disconnected group with values in bivariant equivariant cohomology in the sense of Baum and Schneider. We prove in particular that the complexified left hand side of the Baum-Connes conjecture for a totally disconnected group is isomorphic to cosheaf homology. Moreover, it is shown that our transformation extends the Chern character defined by Baum and Schneider for profinite groups.