  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.
