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Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold. (English) Zbl 1001.53026
For a pseudo-metric space \((X,d)\) and a positive number \(\varepsilon\) we can define an abstract simplicial complex \(X_\varepsilon\), whose vertices are the points of \(X\) and whose \(k\)-simplices consist of all \((k+1)\)-tuples of points in \(X\) with pairwise distances less than \(\varepsilon\). J.-C. Hausmann proved that for a closed Riemannian manifold \(X\) and a sufficiently small \(\varepsilon\) the geometric realization \(|X_\varepsilon |\) of the above complex \(X_\varepsilon\) is homotopy equivalent to \(X\). The author considers a metric space \(Y\) with small Gromov-Hausdorff distance to \(X\) and proves the following theorem: Let \(X\) be a closed Riemannian manifold. Then there exists \(\varepsilon_0 >0\) such that for every \(0<\varepsilon \leq \varepsilon_0\) there exists a \(\delta >0\) such that the geometric realization \(|Y_\varepsilon |\) of the complex \(Y_\varepsilon\), of any metric space \(Y\) which has Gromov-Hausdorff distance less than \(\delta\) to \(X\) is homotopy equivalent to \(X\). For the proof the author uses the notion of \(\varepsilon\)-crushing and \(\varepsilon\)-crushability for a metric space.

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
55U10 Simplicial sets and complexes in algebraic topology
53B20 Local Riemannian geometry
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