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For a countable discrete group $G$, we construct a new and concrete model forthe equivariant topological $K$-homology theory of $G$, which is defined forall $G$-actions, not just for proper $G$-actions. The construction of our modelcombines some of the ideas from the localization algebra approach to the coarseBaum-Connes conjecture and the controlled algebra approach to the Farrell-Jonesconjecture. 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 definerelative assembly maps. We prove, by using an induction structure of ourtheory, a transitivity principle that can be used to verify when a relativeassembly map is an isomorphism. This promotes the study of the originalassembly map relative to the family of finite subgroups to assembly mapsrelative to any family of subgroups of a group. Second, our new model issuitable for the use of the method of controlled topology and controlledalgebra, which plays an important role in proving the coarse Baum-Connesconjecture and the Farrell-Jones conjecture for a large class of groups.

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