ABSTRACT
The chapter explains quickest and probably for a homotopy theorist most convenient approach to assembly maps is via homotopy colimits. The Farrell-Jones Conjecture and the Baum-Connes Conjecture are very powerful conjectures and are the main motivation for the study of assembly maps. One of the basic features of a homology theory is excision. It often comes from the fact that a representing cycle can be arranged to have arbitrarily good control. There is a more general version of the Farrell-Jones Conjecture, the so called Full Farrell-Jones Conjecture, where one allows coefficients in additive categories and the passage to finite wreath products, It implies the Farrell-Jones Conjectures. There is an important transformation from algebraic K-theory to topological cyclic homology, the so called cyclotomic trace. Topological K-theory and the Baum-Connes Conjecture make sense and are studied also for topological groups, e.g., reductive p-adic groups and Lie groups.
