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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].

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

Citations:

Zbl 1468.55001
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