Cohomogeneity one manifolds with singly generated rational cohomology II

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 \operatorname{SU}(3)/{\operatorname{SO}}(3) , one of two homogeneous spaces of the form \mathbf{G}_{2}/{\operatorname{SU}}(2) , or a member of a particular four-parameter family of 7 -manifolds.