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Transformation groups and algebraic \(K\)-theory. (English) Zbl 0679.57022

Lecture Notes in Mathematics, 1408, Subseries: Mathematica Gottingensis. Berlin etc.: Springer-Verlag. xii, 443 p. DM 69.00 (1989).
This book is devoted to the application of algebraic \(K\)-theory to problems in transformation groups. The first segment of the book gives the geometric construction of the invariants. The invariants considered include the finiteness obstruction, Whitehead torsion, and the Euler characteristic. The second segment is devoted to the same invariants from an algebraic viewpoint. Here one studies modules over a category and appropriate chain complexes to form algebraic \(K\)-groups. The final segment is devoted to further extension of the algebra and application of these ideas to geometry. Here one finds an examination of the homological algebra, introduction of equivariant Reidemeister torsion, Swan homomorphisms, and transfers.
Reviewer: R.E.Stong

MSC:

57S17 Finite transformation groups
57R67 Surgery obstructions, Wall groups
57Q12 Wall finiteness obstruction for CW-complexes
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
19-02 Research exposition (monographs, survey articles) pertaining to \(K\)-theory
19Jxx Obstructions from topology
57S15 Compact Lie groups of differentiable transformations
55R12 Transfer for fiber spaces and bundles in algebraic topology
55R91 Equivariant fiber spaces and bundles in algebraic topology
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57R91 Equivariant algebraic topology of manifolds
57Q91 Equivariant PL-topology
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