We show that for every closed Riemannian manifold X there exists a positive number¶ ε0>0 such that for all 0< there exists some¶ δ>0 such that for every metric space Y with Gromov-Hausdorff distance to X less than¶ δ the geometric ε -complex |Yε| is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann.

Latschev, J (2001). *Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold.* Archiv der Mathematik, 77(6):522-528.

## Abstract

We show that for every closed Riemannian manifold X there exists a positive number¶ ε0>0 such that for all 0< there exists some¶ δ>0 such that for every metric space Y with Gromov-Hausdorff distance to X less than¶ δ the geometric ε -complex |Yε| is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann.

## Citations

## Altmetrics

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Gromov-Hausdorff distance; pseudo-metric space; simplicial complex |

Language: | English |

Date: | 2001 |

Deposited On: | 29 Nov 2010 16:27 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Springer |

ISSN: | 0003-889X |

Publisher DOI: | https://doi.org/10.1007/PL00000526 |

Related URLs: | http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1001.53026 http://www.ams.org/mathscinet-getitem?mr=1879057 |

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