Equivariant cohomology of cohomogeneity one actions

Details Equivariant cohomology of cohomogeneity one actions Equivariant cohomology of cohomogeneity one actions

2011-10-28

arxiv.org/abs/1110.6318v2

Mathematics

Differential Geometry

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

Scientific paper

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 $H^{\rm odd}(M)=\{0\}$.

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