Diffeomorphism finiteness, positive pinching, and second homotopy
Anton Petrunin and Wilderich Tuschmann
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Submission date: 07. Nov. 1999 (revised version: November 1999)
published in: Geometric and functional analysis, 9 (1999) 4, p. 736-774
DOI number (of the published article): 10.1007/s000390050101
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 manifoldswith finite second homotopy groups which admit a Riemannian metricwith sectional curvature bounded in absolute value by and diameter uniformly bounded from above by D containsonly finitely many diffeomorphism types.
2. Given any m and any ,there exists a positive constant such that the injectivity radius of any simply connectedcompact m-dimensional Riemannian manifold with finite second homotopygroup and , , is bounded from below by .
In an appendix we discuss Riemannian megafolds,a generalized notion of Riemannian manifolds,and their use and usefulness in collapsing with bounded curvature.