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 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 \(\le 3\), then all but finitely many of the \(M_{\alpha }\)s are diffeomorphic.

给定\（k \ in \ mathbb {R} \），v，\（D> 0 \）和\（n \ in \ mathbb {N} \），让\（\ left \ {M _ {\ alpha} \ right \} _ {\ alpha = 1} ^ {\ infty} \）是具有截面曲率\（\ ge k \），体积\（> v \）和Riemannian n流形的Gromov–Hausdorff收敛序列。直径\（\ le D \）。佩雷尔曼的稳定性定理暗示所有\（M _ {\ alpha} \）都是同胚的。微分同构稳定性问题询问除了有限数量之外的所有\（M _ {\ alpha} \）s是微晶的。在极限空间的所有奇点都沿着余维\（\ le 3 \）的黎曼流形出现的特殊情况下，我们肯定地回答这个问题。然后，我们描述几个应用程序。例如，如果极限空间是其奇异层为余维\（\ le 3 \）的轨道空间，则除有限外，所有\（M _ {\ alpha} \）都是微晶的。