Diffeomorphism Stability and Codimension Three

Given k is an element of R, v, D>0, and n is an element of N, let {M-alpha}(alpha=1)(infinity) be a Gromov-Hausdorff convergent sequence of Riemannian n-manifolds with sectional curvature >= k, volume >v, and diameter <= 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 Riemannian manifolds of codimension <= 3. We then describe several applications. For instance, if the limit space is an orbit space whose singular strata are of codimension <= 3, then all but finitely many of the M(alpha)s are diffeomorphic.

Automated summary

This paper discusses the convergent sequence of Riemannian n-manifolds under certain conditions and answers the Diffeomorphism Stability Question. The results show that in special cases, all but finitely many of the M(alpha)s are diffeomorphic.