Equivariant homology and cohomology of groups

Abstract

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ -equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.It is well known that the study of groups with operators has many important applications in algebra and topology. The category of groups enriched with an action by automorphisms of a given group provides a suitable setting for the investigation of an extensive list of subjects with recognized mathematical interest. See, for instance, recent results in equivariant stable homotopy theory [6] and articles devoted to equivariant algebraic K-theory [13,29,24]. The origin of the equivariant investigation in homological algebra, particularly in extension theory of groups, goes back to the article of Whitehead [36]. It should be noted that recently a theory of cohomology of groups with operators was developed [8], motivated by E-mail address: hvedri@rmi.acnet.ge (H. Inassaridze).0166-8641/$ -see front matter 

Hopf formulas for equivariant integral homology of groups

“…Analogues of much of the theory of the classical homological algebra of groups have been obtained in that equivariant setting such as (co)chain and cotriple presentations, the Hopf formula, exact (co)homology sequences, and the Tate cohomology of groups. Moreover, the relationship with the equivariant cohomology of spaces was established, and it was shown that the well known isomorphism of Milnor's K-group K 2 (A) with the second integral homology H 2 (E(A)) of the elementary group E(A) of a unital ring A remains true for arbitrary rings by taking into account the action on E(A) of the Steinberg group St(Z) of the ring Z of integers (see [11, Corollary 24]).As promised we continue the investigation of equivariant homology theory of H. Inassaridze [11]. Namely, using the purely algebraic method of n-foldČech derived functors developed in [7,12], we provide higher Hopf type formulas for the equivariant integral group homology (Theorem 12), extending the Hopf formula for the second equivariant homology [11] and recovering the Brown-Ellis higher Hopf formulas [2], when the group actions are trivial.…”

“…Moreover, the relationship with the equivariant cohomology of spaces was established, and it was shown that the well known isomorphism of Milnor's K-group K 2 (A) with the second integral homology H 2 (E(A)) of the elementary group E(A) of a unital ring A remains true for arbitrary rings by taking into account the action on E(A) of the Steinberg group St(Z) of the ring Z of integers (see [11, Corollary 24]).…”

“…Analogues of much of the theory of the classical homological algebra of groups have been obtained in that equivariant setting such as (co)chain and cotriple presentations, the Hopf formula, exact (co)homology sequences, and the Tate cohomology of groups. Moreover, the relationship with the equivariant cohomology of spaces was established, and it was shown that the well known isomorphism of Milnor's K-group K 2 (A) with the second integral homology H 2 (E(A)) of the elementary group E(A) of a unital ring A remains true for arbitrary rings by taking into account the action on E(A) of the Steinberg group St(Z) of the ring Z of integers (see [11, Corollary 24]).…”

“…Namely, using the purely algebraic method of n-foldČech derived functors developed in [7,12], we provide higher Hopf type formulas for the equivariant integral group homology (Theorem 12), extending the Hopf formula for the second equivariant homology [11] and recovering the Brown-Ellis higher Hopf formulas [2], when the group actions are trivial.…”

“…We begin by formulating in the Γ-equivariant framework some well known notions and recall some facts from [11] needed for our future purpose.…”