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\(K\)-theory and geometric topology. (English) Zbl 1116.55001

Friedlander, Eric M. (ed.) et al., Handbook of \(K\)-theory. Vol. 1 and 2. Berlin: Springer (ISBN 3-540-23019-X/hbk). 577-610 (2005).
This chapter of the handbook of \(K\)-theory is devoted to a survey of the contributions of \(K\)-theory to geometric topology, by which the author means the study of the topology of manifolds and their generalizations (simplicial and CW-complexes). Eight topics are selected and, for each, historical background and basic definitions are given, some major theorems (usually those whose proofs rest on \(K\)-theoretic ideas) are stated and a guide to the literature is provided. The topics are: Wall’s finiteness obstructions (including applications to the spherical space form problem), flat bundles, Whitehead and Reidemeister torsion (with material on simple homotopy theory and the \(s\)-cobordism theorem), controlled \(K\)-theory, equivariant and stratified topology (leading to the stratified \(s\)-cobordism theorem), Waldhausen’s \(A\)-theory, pseudo-isotopy theory (with a survey of the work of Hatcher and Wagoner, and of Kyoshi Igusa) and symbolic dynamics.
For the entire collection see [Zbl 1070.19002].

MSC:

55-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to algebraic topology
57-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to manifolds and cell complexes
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
19-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to \(K\)-theory
19L99 Topological \(K\)-theory
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