Lück, Wolfgang Assembly maps. (English) Zbl 1473.18013 Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 851-890 (2020). The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].Summary: 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.For the entire collection see [Zbl 1468.55001]. Cited in 1 ReviewCited in 3 Documents MSC: 18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) 46L80 \(K\)-theory and operator algebras (including cyclic theory) 55P91 Equivariant homotopy theory in algebraic topology 57N99 Topological manifolds 18-02 Research exposition (monographs, survey articles) pertaining to category theory 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology Keywords:assembly maps; equivariant homology and homotopy theory; Farrell- Jones Conjecture; Baum-Connes Conjecture Citations:Zbl 1468.55001 PDFBibTeX XMLCite \textit{W. Lück}, in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 851--890 (2020; Zbl 1473.18013) Full Text: DOI arXiv