MiS Preprint Repository

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.