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].