Paschke duality and assembly maps

TL;DR

This paper constructs a natural transformation called the Paschke transformation between geometric and analytic G-equivariant K-homology, showing it is an equivalence under certain conditions and relating it to the Baum-Connes conjecture.

Contribution

It introduces the Paschke transformation linking coarse geometric and analytic G-equivariant K-homology, and compares the Davis-Lück and Kasparov assembly maps.

Findings

Paschke transformation is an equivalence under finiteness conditions.

Provides a direct comparison between Davis-Lück and Kasparov assembly maps.

Establishes a connection relevant to the Baum-Connes conjecture.

Abstract

We construct a natural transformation between two versions of $G$-equivariant $K$-homology with coefficients in a $G$-$C^{*}$-category for a countable discrete group $G$. Its domain is a coarse geometric $K$-homology and its target is the usual analytic $K$-homology. Following classical terminology, we call this transformation the Paschke transformation. We show that under certain finiteness assumptions on a $G$-space $X$, the Paschke transformation is an equivalence on $X$. As an application, we provide a direct comparison of the homotopy theoretic Davis-L\"uck assembly map with Kasparov's analytic assembly map appearing in the Baum-Connes conjecture.