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MiS Preprint
39/1998
Diffeomorphism finiteness, positive pinching, and second homotopy
Anton Petrunin and Wilderich Tuschmann
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 $|K|\leq C$ 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 $Ric \geq \delta$, $K\leq 1$, 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.
