Cohomogeneity one manifolds with singly generated rational cohomology II

TL;DR

This paper classifies smooth, simply connected manifolds with the rational cohomology of a sphere that admit cohomogeneity one actions, identifying specific geometric and topological types such manifolds can have.

Contribution

It provides a complete classification of such manifolds with cohomogeneity one actions, including explicit descriptions of their diffeomorphism types.

Findings

Manifolds are diffeomorphic to spheres, Brieskorn varieties, Wu manifold, or specific homogeneous spaces.

Identifies a four-parameter family of 7-manifolds with these properties.

Classifies all cohomogeneity one actions on these manifolds.

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 $SU(3)/SO(3)$, one of two homogeneous spaces of the form $\mathbf{G}_2/SU(2)$,or a member of a particular four-parameter family of $7$-manifolds.