Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold

Abstract. 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 [4].