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Equivariant homology and cohomology of groups. (English) Zbl 1083.18008

If \(\Gamma\) is a fix group of operators, \(\Gamma\)-equivariant extensions of a \(\Gamma\)-group \(G\) by a \(\Gamma\)-equivariant \(G\)-module \(A\) were classified by means of the cohomology of groups with operators studied by A. M. Cegarra, J. M. García-Calcines and J. A. Ortega [Homology Homotopy Appl. 4, 1–23 (2002; Zbl 1007.18013)]. Motivated by the classification of those \(\Gamma\)-equivariant extensions of \(G\) by \(A\) which are \(\Gamma\)-splitting, the author develops in this paper a different equivariant (co)homology theory of groups. This theory is introduced as relative \(\text{Tor}_n^{\mathcal F}\) and \(\text{Ext}^n_{\mathcal F}\) in the category of \(\Gamma\)-equivariant \(G\)-modules and it is described in term of suitable cocycles and, also, as a cotriple (co)homology theory. Then, suitable versions of Hopf formula for the second integral homology, universal coefficients theorems and universal central extensions are given in this equivariant setting. The classification of the splitting \(\Gamma\)-equivariant extensions of \(G\) by \(A\), in terms of the second equivariant cohomology group introduced in the paper, as well as some applications in algebraic \(K\)-theory and cohomology of topological spaces are finally established.

MSC:

18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18G50 Nonabelian homological algebra (category-theoretic aspects)
18G55 Nonabelian homotopical algebra (MSC2010)
19C09 Central extensions and Schur multipliers
20J06 Cohomology of groups
55N25 Homology with local coefficients, equivariant cohomology

Citations:

Zbl 1007.18013
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References:

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