Equivariant cohomology of cohomogeneity one actions

We show that if G × M → 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 ⁎ ( B G ) . Moreover, this module is free if and only if the rank of at least one isotropy group is equal to rank G. We deduce as corollaries several results concerning the usual (de Rham) cohomology of M, such as a new proof of the following obstruction to the existence of a cohomogeneity one action: if M admits a cohomogeneity one action, then χ ( M ) > 0 if and only if H odd ( M ) = { 0 } .