The equivariant cohomology for cyclic group actions on some polyhedral products

TL;DR

This paper investigates the structure of equivariant cohomology for finite cyclic group actions on polyhedral products, focusing on module properties and conditions for freeness in positive characteristic fields.

Contribution

It extends the understanding of equivariant cohomology for cyclic groups, providing new criteria for freeness and analyzing module structures in positive characteristic settings.

Findings

Criteria for freeness over the cohomology ring established

Module structure of equivariant cohomology characterized

Applications to cyclic group actions on polyhedral products

Abstract

The equivariant cohomology for actions of compact connected abelian groups and elementary abelian p-groups have been widely studied in the last decades. We study some of these results on actions of finite cyclic groups over a field of positive characteristic. In particular, we study the module structure of the equivariant cohomology over the cohomology of the classifying space and we provide a criterion for freeness over this ring and the polynomial subring of it. We apply these results to canonical actions of finite cyclic groups on polyhedral products arising from the boundary of a polygo