The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions
Jianguo Zhang
TL;DR
This work analyzes how the two central assembly maps in noncommutative geometry—the Baum–Connes assembly map e_* and the Mishchenko–Kasparov assembly map μ_*—behave under group extensions and isometric semidirect products. By developing refined equivariant localization algebras, Mayer–Vietoris techniques, and Green imprimitivity frameworks, the authors prove new closure results for the strong Novikov conjecture, the surjective assembly conjecture, the Baum–Connes conjecture with coefficients, and their rational analytic variants under extensions by finite or central subgroups and direct products. They establish three main theorems: (i) BC-type closure under extensions with finite-subgroup BC data and quotient SNC/SAC/BCC; (ii) rational-analytic Novikov stability under rational BC and quotient RANC with Künneth hypotheses; (iii) a comprehensive suite of results for isometric semi-direct products, including a two-direction localization formalism and a third main theorem ensuring SNC/SAC/BCC/RANC along both factors. The results yield new examples of groups satisfying the rational analytic and strong Novikov conjectures beyond coarsely embeddable groups and provide robust tools for verifying assembly-map conjectures in complex group-structural settings, with substantial implications for higher-index theory and noncommutative topology.
Abstract
The Baum-Connes assembly map with coefficients $e_{\ast}$ and the Mishchenko-Kasparov assembly map with coefficients $μ_{\ast}$ 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 $1\rightarrow N \rightarrow Γ\xrightarrow{q} Γ/ N \rightarrow 1$. Firstly, under the assumption that $e_{\ast}$ is isomorphic for $q^{-1}(F)$ for any finite subgroup $F$ of $Γ/N$, we prove that $e_{\ast}$ is injective, surjective and isomorphic for $Γ$ if they are also true for $Γ/N$, respectively. Secondly, under the assumption that $e_{\ast}$ is rationally isomorphic for $N$, we verify that $μ_{\ast}$ is rationally injective for $Γ$ if it is also rationally injective for $Γ/N$. Finally, when $Γ$ is an isometric semi-direct product $N\rtimes G$, we confirm that $e_{\ast}$ is injective, surjective and isomorphic for $Γ$ if they also hold for $G$ and $Γ$ 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.