Diffeomorphism Stability and Codimension Three

TL;DR

This paper proves that under certain conditions, a converging sequence of Riemannian manifolds with bounded curvature and volume stabilizes in diffeomorphism type when singularities are of codimension at most three, extending Perelman's stability results.

Contribution

It establishes the diffeomorphism stability for sequences of manifolds with singularities along codimension ≤ 3, a significant strengthening of known stability theorems.

Findings

Sequences stabilize in diffeomorphism type under codimension ≤ 3 singularities.

Applications include stability of orbit spaces with low codimension singular strata.

Results extend the scope of Perelman's stability theorem to diffeomorphism classification.

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.