Greenlees, J. P. C.; May, J. P. 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. Reviewer: V.P.Snaith (MR 96e:55006) Cited in 8 ReviewsCited in 85 Documents 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 Keywords:compact Lie group; suspension; \(G\)-spectrum; Tate homology; cohomology Citations:Zbl 0618.57015; Zbl 0608.57029 PDFBibTeX XMLCite \textit{J. P. C. Greenlees} and \textit{J. P. May}, Generalized Tate cohomology. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0876.55003) Full Text: DOI