  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.
