On the equivariant cohomology of cohomogeneity one Alexandrov spaces

TL;DR

This paper characterizes cohomogeneity one actions of compact Lie groups on Alexandrov spaces where the action is Cohen--Macaulay, extending previous manifold results to singular Alexandrov spaces using rational homotopy theory.

Contribution

It generalizes Cohen--Macaulay action characterization from manifolds to Alexandrov spaces, revealing new non-Cohen--Macaulay actions and introducing a rational homotopy approach.

Findings

Identifies conditions for Cohen--Macaulay actions on Alexandrov spaces.

Shows existence of non-Cohen--Macaulay actions in the singular setting.

Develops a rational homotopy model for equivariant cohomology.

Abstract

We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group $G$ for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen--Macaulay. In fact, we present results in a slightly more general context. We extend the methods in this field by a conceptual approach on equivariant cohomology via rational homotopy theory using an explicit rational model for a double mapping cylinder.