Cohomogeneity one manifolds with singly generated rational cohomology II

Abstract

We classify cohomogeneity one actions on smooth, simply connected, closed manifolds with the rational cohomology of a sphere. In particular, we show that such a manifold is diffeomorphic to a sphere, a Brieskorn variety, the Wu manifold $\mathrm{SU}(3)/\mathrm{SO}(3)$, one of two homogeneous spaces of the form ${\mathbf{G}}_2/\mathrm{SU}(2)$, or a member of a particular four-parameter family of $7$-manifolds.