Finitary Group Cohomology and Eilenberg-Mac Lane Spaces

TL;DR

This paper investigates the conditions under which groups have Eilenberg-Mac Lane spaces with finitely many cells in high dimensions, focusing on groups with cohomology almost everywhere finitary within a specific class.

Contribution

It establishes that groups in Kropholler's class LHF with certain cohomological properties admit Eilenberg-Mac Lane spaces with finitely many high-dimensional cells.

Findings

Groups in LHF with cohomology almost everywhere finitary have controlled Eilenberg-Mac Lane spaces.

The converse holds universally for any group.

Open question remains for groups outside the studied class.

Abstract

We say that a group G has cohomology almost everywhere finitary if and only if the nth cohomology functors of G commute with filtered colimits for all sufficiently large n. In this paper, we show that if G is a group in Kropholler's class LHF with cohomology almost everywhere finitary, then G has an Eilenberg--Mac Lane space K(G,1) which is dominated by a CW-complex with finitely many n-cells for all sufficiently large n. It is an open question as to whether this holds for arbitrary G. We also remark that the converse holds for any group G.