Some applications of collapsing with bounded curvature

In my talk I will discuss the following results which were obtained in joint work with Wilderich Tuschmann. 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 $\vert K \vert\le C$ and diameter $\le D$ 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 Ricci curvature $Ric\ge\delta$, $K\le 1$, is bounded from below by $i_0(m,\delta)$. I also intend to discuss Riemannian megafolds, a generalized notion of Riemannian manifolds, and their use and usefulness in the proof of these results.