Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold
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We show that for every closed Riemannian manifold X there exists a positive number¶\( \varepsilon_0 > 0 \) such that for all 0< \( \varepsilon \leqq \varepsilon_0 \) there exists some¶\( \delta > 0 \) such that for every metric space Y with Gromov-Hausdorff distance to X less than¶\( \delta \) the geometric \( \varepsilon \)-complex \( |Y_\varepsilon| \) is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann .
KeywordsRiemannian Manifold Positive Answer Homotopy Equivalent Closed Riemannian Manifold
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