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

摘要

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].