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

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.

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.

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