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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.

##### MSC:
 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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