Equivariant cohomology of cohomogeneity one actions

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 $H_G^{\ast }(M)$ is a Cohen-Macaulay module over $H^{\ast }(BG)$. 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 $H^{\mathrm{odd}}(M)=\{0\}$.

arXiv comment18 pages. v2: exposition improved; Remark 5.4, Lemma 5.5, and references added