The equivariant cohomology for semidirect product actions

TL;DR

This paper extends the understanding of equivariant cohomology for compact Lie group actions, especially for semi-direct products, by identifying conditions under which cohomology reduces to subgroups, aiding in the analysis of syzygy orders.

Contribution

It generalizes the reduction of equivariant cohomology to maximal tori to broader classes of groups, including semi-direct products with specific subgroup conditions.

Findings

Reduction of equivariant cohomology to subgroups under certain conditions

Analysis of semi-direct product groups with elementary abelian 2-subgroups

Application to syzygy order of equivariant cohomology with involutions

Abstract

The rational Borel equivariant cohomology for actions of a compact connected Lie group is determined by restriction of the action to a maximal torus. We show that a similar reduction holds for any compact Lie group $G$ when there is a closed subgroup $K$ such that the cohomology of the classifying space $BK$ is free over the cohomology of $BG$ for field coefficients. We study the particular case when $G$ is a semi-direct product and $K$ is its maximal elementary abelian 2-subgroup for cohomology with coefficients in a field of characteristic two. This provides a different approach to investigate the syzygy order of the equivariant cohomology of a space with a torus action and a compatible involution, and we relate this description with results for 2-torus actions.