Equivariant cohomology of cohomogeneity one actions

TL;DR

This paper investigates the structure of equivariant cohomology for cohomogeneity one actions of compact Lie groups on manifolds, revealing conditions for freeness and implications for the manifold's topology.

Contribution

It proves that the equivariant cohomology is Cohen-Macaulay and characterizes when it is free, linking group ranks to topological properties of the manifold.

Findings

Equivariant cohomology is Cohen-Macaulay for cohomogeneity one actions.

Freeness of the module occurs iff a rank condition on isotropy groups is met.

Obstruction to cohomogeneity one actions related to the Euler characteristic and odd cohomology.

Abstract

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce as corollaries several results concerning the usual (de Rham) cohomology of $M$, such as the following obstruction to the existence of a cohomogeneity one action: if $M$ admits a cohomogeneity one action, then $\chi(M)>0$ if and only if $H^{\rm odd}(M)=\{0\}$.