## Abstract

Let M_{i} →^{dGH} X denote a sequence of n-manifolds converging to a compact metric space, X, in the Gromov-Hausdorff topology such that the sectional curvature is bounded in absolute value and dim(X) < n. We prove the following stability result: If the fundamental groups of M_{i} are torsion groups of uniformly bounded exponents and the second twisted Betti numbers of M_{i} vanish, then there is a manifold, M, and a sequence of diffeomorphisms from M to a subsequence of {M_{i}} such that the distance functions of the pullback metrics converge to a pseudo-metric in C^{0}-norm. Furthermore, M admits a foliation with leaves diffeomorphic to flat manifolds (not necessarily compact) such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics. These results lead to a few interesting applications.

Original language | English (US) |
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Pages (from-to) | 61-109 |

Number of pages | 49 |

Journal | Inventiones Mathematicae |

Volume | 150 |

Issue number | 1 |

DOIs | |

State | Published - 2002 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)