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

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.

💡 Research Summary

The paper establishes a precise link between the KH‑assembly map, as introduced by Bartels and Lück, and the algebraic KK‑theory developed by Cortiñas and Thom. After a concise review of Weibel’s homotopy K‑theory (KH) and the definition of the KH‑assembly map for a group G acting on a G‑CW‑complex X, the authors turn to the algebraic KK‑theory framework, which mirrors Kasparov’s analytic KK‑theory but operates purely in an algebraic setting, replacing C∗‑algebras with suitable rings and bimodules.

The central result (Theorem 3.1) asserts that a natural transformation β_A: KK_alg(ℤ