Comparison of equivariant cohomological dimensions

TL;DR

This paper compares three definitions of equivariant cohomological dimension for groups with operators, providing examples of strict inequalities and characterizing groups of dimension one, with implications for topological complexity.

Contribution

It introduces a comparative analysis of Takasu, Adamson, and Bredon equivariant cohomological dimensions, including examples and characterizations of low-dimensional groups.

Findings

Examples of strict inequalities in cohomological dimensions

Characterization of groups with equivariant cohomological dimension one

Topological complexity not always given by relative cohomological dimension

Abstract

We compare three definitions of the equivariant cohomological dimension of a group with operators, coming from Takasu, Adamson and Bredon relative group cohomologies, giving examples of strict inequality in all cases where it can occur. We prove and make use of Stallings--Swan type results which characterise the groups of equivariant cohomological dimension one. Some of our examples are relevant to Farber's problem which asks for an algebraic characterisation of the topological complexity of discrete groups. In particular, the topological complexity of a group is not in general given by a relative cohomological dimension of the product relative to the diagonal subgroup.