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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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.

Received:
07.11.99
Published:
07.11.99

Related publications

inJournal
1999 Repository Open Access
Anton Petrunin and Wilderich Tuschmann

Diffeomorphism finiteness, positive pinching, and second homotopy

In: Geometric and functional analysis, 9 (1999) 4, pp. 736-774