Non-negative pinching, moduli spaces and bundles with infinitely many souls

Abstract

We show that in each dimension n ≥ 10, there exist infinite sequences of homotopy equivalent, but mutually non-homeomorphic closed simply connected Riemannian n-manifolds with $0 \leq \rm{sec} \leq 1$, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete non-negatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of non- compact manifolds whose moduli spaces of complete metrics with sec $\geq 0$ have infinitely many connected components.