Equivariant cohomology of cohomogeneity one actions: the topological case

TL;DR

This paper proves that for any cohomogeneity one action of a compact Lie group on a closed topological manifold, the equivariant cohomology is Cohen-Macaulay, extending previous smooth action results.

Contribution

It generalizes the Cohen-Macaulay property of equivariant cohomology from smooth to topological cohomogeneity one actions.

Findings

Equivariant cohomology is Cohen-Macaulay for topological actions.

The result extends previous smooth action theorems.

Relies on recent structure theorems for these actions.

Abstract

We show that for any cohomogeneity one continuous action of a compact connected Lie group $G$ on a closed topological manifold the equivariant cohomology equipped with its canonical $H^*(BG)$-module structure is Cohen-Macaulay. The proof relies on the structure theorem for these actions recently obtained by Galaz-Garcia and Zarei. We generalize in this way our previous result concerning smooth actions.