The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions

TL;DR

This paper investigates the behavior of Baum-Connes and Mishchenko-Kasparov assembly maps in group extensions, establishing conditions under which their properties are preserved and applying these results to various conjectures in topology and operator algebras.

Contribution

It provides new criteria for the injectivity, surjectivity, and isomorphism of assembly maps in group extensions, and demonstrates the stability of several conjectures under group extensions.

Findings

Assembly maps are preserved under certain group extension conditions.

Strong Novikov and Baum-Connes conjectures are stable under direct products and extensions.

New examples for the rational analytic and strong Novikov conjectures are obtained.

Abstract

The Baum-Connes assembly map with coefficients $e_{\ast}$ and the Mishchenko-Kasparov assembly map with coefficients $\mu_{\ast}$ are two homomorphisms from the equivariant $K$-homology of classifying spaces of groups to the $K$-theory of reduced crossed products. In this paper, we investigate these two assembly maps for group extensions $1\rightarrow N \rightarrow \Gamma \xrightarrow{q} \Gamma/ N \rightarrow 1$. Firstly, under the assumption that $e_{\ast}$ is isomorphic for $q^{-1}(F)$ for any finite subgroup $F$ of $\Gamma/N$, we prove that $e_{\ast}$ is injective, surjective and isomorphic for $\Gamma$ if they are also true for $\Gamma/N$, respectively. Secondly, under the assumption that $e_{\ast}$ is rationally isomorphic for $N$, we verify that $\mu_{\ast}$ is rationally injective for $\Gamma$ if it is also rationally injective for $\Gamma/N$. Finally, when $\Gamma$ is an isometric semi-direct product $N\rtimes G$, we confirm that $e_{\ast}$ is injective, surjective and isomorphic for $\Gamma$ if they also hold for $G$ and $\Gamma$ satisfies three partial conjectures along $N$, respectively. As applications, we show that the strong Novikov conjecture, the surjective assembly conjecture and the Baum-Connes conjecture with coefficients are closed under direct products, central extensions of groups and extensions by finite groups. Meanwhile, we also show that the rational analytic Novikov conjecture with coefficients is preserved under extensions of finite groups. Besides, we employ these results to obtain some new examples for the rational analytic and the strong Novikov conjecture beyond the class of coarsely embeddable groups.