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

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 ≥ sec ≥ 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 mutu- ally non-homeomorphic. Finally, we construct examples of non-compact manifolds whose moduli spaces of complete metrics with sec ≤ 0 have infinitely many connected components.