Diffeomorphism Finiteness, Positive Pinching, and Second Homotopy

Abstract.

Our main results can be stated as follows:¶¶1. For any given numbers m, C and D, the class of m-dimensional simply connected closed smooth manifolds with finite second homotopy groups which admit a Riemannian metric with sectional curvature bounded in absolute value by $|K| \le C$ and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types.¶2. Given any m and any $\delta > 0$, there exists a positive constant $i_0 = i_0(m,\delta) > 0$ such that the injectivity radius of any simply connected compact m-dimensional Riemannian manifold with finite second homotopy group and Ric $\ge \delta, K \le 1$ is bounded from below by $i_0(m,\delta)$.¶¶In an appendix we discuss Riemannian megafolds, a generalized notion of Riemannian manifolds, and their use (and usefulness) in collapsing with bounded curvature.