Diffeomorphism finiteness, positive pinching, and second homotopy

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| ≤ C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. 2. Given any m and any δ > 0, there exists a positive constant i₀= i₀(m,δ) > 0 such that the injectivity radius of any simply connected compact m-dimensional Riemannian manifold with finite second homotopy group and Ric ≥ δ, K ≤ 1, is bounded from below by i₀(m, δ). In an appendix we discuss Riemannian megafolds, a generalized notion of Riemannian manifolds, and their use (and usefulness) in collapsing with bounded curvature.