Topological K-homology for non-proper actions and assembly maps

TL;DR

This paper introduces a new model for equivariant topological K-homology applicable to all group actions, enabling broader analysis of assembly maps and advancing the study of the Baum-Connes conjecture.

Contribution

It develops a concrete model for G-equivariant K-homology for all actions, facilitating new approaches to assembly maps and conjectures in topology and algebra.

Findings

Constructed a model applicable to all G-actions.

Established a transitivity principle for assembly maps.

Enhanced methods for verifying the Baum-Connes conjecture.

Abstract

For a countable discrete group $G$, we construct a new and concrete model for the equivariant topological $K$-homology theory of $G$, which is defined for all $G$-actions, not just for proper $G$-actions. The construction of our model combines some of the ideas from the localization algebra approach to the coarse Baum-Connes conjecture and the controlled algebra approach to the Farrell-Jones conjecture. It brings new ideas into the study of the Baum-Connes conjecture. First, as the model is defined for all $G$-actions, we are able to define relative assembly maps. We prove, by using an induction structure of our theory, a transitivity principle that can be used to verify when a relative assembly map is an isomorphism. This promotes the study of the original assembly map relative to the family of finite subgroups to assembly maps relative to any family of subgroups of a group. Second, our new model is suitable for the use of the method of controlled topology and controlled algebra, which plays an important role in proving the coarse Baum-Connes conjecture and the Farrell-Jones conjecture for a large class of groups.