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Generalized Tate cohomology. (English) Zbl 0876.55003

Mem. Am. Math. Soc. 543, 178 p. (1995).
Let \(G\) be a compact Lie group, \(EG\) a contractible free \(G\)-space and let \(\widetilde{E}G\) be the unreduced suspension of \(EG\) with one of the cone points as basepoint. Let \(k_G\) be a \(G\)-spectrum. Let \(X_+\) denote the disjoint union of \(X\) and a \(G\)-fixed basepoint. Define the \(G\)-spectra \(f(k_G)= k_G\wedge EG_+\), \(c(k_G)= F(EG_+,k_G)\), and \(t(k_G)= F(EG_+,k_G)\wedge \widetilde{E}G\).
The last of these is the \(G\)-spectrum representing the generalized Tate homology and cohomology theories associated to \(k_G\). Here \(F(EG_+,k_G)\) is the function space spectrum. The authors develop the properties of these theories, illustrating the manner in which they generalise the classical Tate-Swan theories. The results obtained extend those of the first author [Proc. Edinb. Math. Soc., II. Ser. 30, No. 3, 435-443 (1987; Zbl 0608.57029)].
Numerous calculations and applications to stable homotopy are given, using the norm cofibration \(f(k_G)\to c(k_G)\to t(k_G)\) and the Tate versions of the Atiyah-Hirzebruch spectra sequence. For example, when \(G\) is the circle the Tate theories are related to cyclic homology.

MSC:

55N15 Topological \(K\)-theory
19L47 Equivariant \(K\)-theory
55P42 Stable homotopy theory, spectra
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55P91 Equivariant homotopy theory in algebraic topology
55Q10 Stable homotopy groups
55Q45 Stable homotopy of spheres
55Q91 Equivariant homotopy groups
55T25 Generalized cohomology and spectral sequences in algebraic topology
55N91 Equivariant homology and cohomology in algebraic topology
20J06 Cohomology of groups
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
18G40 Spectral sequences, hypercohomology
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