The quotient criterion for syzygies in equivariant cohomology for elementary abelian $2$-group actions

TL;DR

This paper introduces a criterion to determine the syzygy order of equivariant cohomology for manifolds with elementary abelian 2-group actions, extending previous results from torus actions to the real case.

Contribution

It develops a quotient criterion for syzygy orders in equivariant cohomology specifically for elementary abelian 2-group actions, adapting the torus action framework to the real setting.

Findings

Provides a criterion based on face filtration cohomology

Extends the quotient criterion to real group actions

Connects equivariant cohomology with face complex structures

Abstract

Let $G$ be a elementary abelian $2$-group and $X$ be a manifold with a locally standard action of $G$. We provide a criterion to determine the syzygy order of the $G$-equivariant cohomology of $X$ with coefficients over a field of characteristic two using a complex associated to the cohomology of the face filtration of the manifold with corners $X/G$. This result is the real version of the quotient criterion for locally standard torus actions developed by M. Franz in arXiv:1205.4462.