Diffeomorphism Stability and Codimension Three

Abstract

Given $k\in \mathbb{R},$ $v,$ $D>0,$ and $n\in \mathbb{N},$ let ${\left\{M_{\alpha }\right\}}_{\alpha =1}^{\infty }$ be a Gromov-Hausdorff convergent sequence of Riemannian $n$--manifolds with sectional curvature $\ge k,$ volume $>v,$ and diameter $\le D.$ Perelman's Stability Theorem implies that all but finitely many of the $M_{\alpha }$s are homeomorphic. The Diffeomorphism Stability Question asks whether all but finitely many of the $M_{\alpha }$s are diffeomorphic. We answer this question affirmatively in the special case when all of the singularities of the limit space occur along smoothly and isometrically embedded Riemannian manifolds of codimension $\le 3$. We then describe several applications. For instance, if the limit space is an orbit space whose singular strata are of codimension at $\le 3,$ then all but finitely many of the $M_{\alpha }$s are diffeomorphic.

The paper is in final form and is to appear in the Journal of Geometric Analysis