Cohomology of discrete groups and their finite subgroups. (English) Zbl 0982.55010

Akita, Toshiyuki

J. Math. Soc. Japan 52, No. 4, 869-875 (2000).

The author starts from the result that if \(G\) is a finite group, and the Abelian subgroups are suitably ordered then \(H^*(G,\mathbb{Z})\cong \varprojlim_A H^*(A, \mathbb{Z})\) modulo a nilpotent kernel and cokernel. By means of a Leray spectral sequence argument he extends this result to discrete groups of finite virtual cohomological dimension, and replaces the coefficients \(\mathbb{Z}\) by an arbitrary commutative ring \(k\) with unit. He also shows that it suffices to take the limit over a category \({\mathcal J}\) of finite subgroups which satisfy the conditions (i) for every finite subgroup \(H\) there exists \(K\) in \({\mathcal J}\) with \(H\subseteq K\) and (ii) \({\mathcal J}\) is closed under conjugation and taking intersections. There is an analogous result for Tate-Farrell cohomology and (presumably) for the more general cohomology theory applying to a wider class of discrete groups.

Reviewer: C.B.Thomas (Cambridge)

MSC:

55R40	Homology of classifying spaces and characteristic classes in algebraic topology
20J06	Cohomology of groups
20J05	Homological methods in group theory
55R35	Classifying spaces of groups and \(H\)-spaces in algebraic topology

Keywords:

group cohomology