The $KH$-Isomorphism Conjecture and Algebraic $KK$-theory

TL;DR

This paper connects the $KH$-assembly map with algebraic $KK$-theory, showing similarities to the Baum-Connes map and applying known methods to the $KH$-isomorphism conjecture.

Contribution

It provides a new $KK$-theoretic description of the $KH$-assembly map, bridging it with established algebraic $KK$-theory frameworks.

Findings

$KH$-assembly map described via algebraic $KK$-theory

Methods from Baum-Connes conjecture apply to $KH$-isomorphism conjecture

Elementary cases show similar proof techniques can be used

Abstract

In this article we prove that the $KH$-asembly map, as defined by Bartels and L{\"u}ck, can be described in terms of the algebraic $KK$-theory of Cortinas and Thom. The $KK$-theory description of the $KH$-assembly map is similar to that of the Baum-Connes assembly map. In very elementary cases, methods used to prove the Baum-Connes conjecture also apply to the $KH$-isomorphism conjecture.