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 $\geq k,$ volume $>v,$ and diameter $\leq D.$ 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 $\leq 3$. We then describe several applications. For instance, if the limit space is an orbit space whose singular strata are of codimension at $\leq 3,$ then all but finitely many of the $M_{\alpha }$s are diffeomorphic.