Almost Nonnegative Curvature and Cohomogeneity One

We show that any close cohomogeneity one manifold supports metrics of almost non­-negative sectional curvature which are moreover invariant under the cohomogeneity one action, thereby establishing a conjecture of Grove and Ziller in the almost nonnegatively curved setting. Applications of our result include that there are infinitely many dimensions in which there exist almost nonnegatively curved topological spheres and homotopy lens as well as
homotopy real projective spaces, all differentiably distinct from the standard ones.