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Persistence stability for geometric complexes. (English) Zbl 1320.55003
The authors of the article define a persistence module \({\mathbb V}\) to be a direct system, indexed by the real numbers, in the category of vector spaces; that is, a collection of vector spaces \(\{V_a\mid a\in{\mathbb R}\}\) and linear maps \(v_a^b:V_a\rightarrow V_b\) for \(a\leq b\), such that \(v_b^c\circ v_a^b=v_a^c\) and \(v_a^a\) is the identity. A homomorphism \(\Phi\in\text{Hom}^\epsilon({\mathbb U},{\mathbb V})\) of degree \(\epsilon\) between persistence modules is defined to be a collection of linear maps \(\phi_a:U_a\rightarrow V_{a+\epsilon}\) such that \(v_{a+\epsilon}^{b+\epsilon}\circ\phi_a=\phi_b\circ u_a^b\), whenever \(a\leq b\). In particular the shift map \(1_{\mathbb V}^\epsilon\in\text{Hom}^\epsilon({\mathbb V},{\mathbb V})\), which denotes the collection of maps \(v_a^{a+\epsilon}\), is a homomorphism of degree \(\epsilon\). Persistence modules \({\mathbb U}\) and \({\mathbb V}\) are said to be \(\epsilon\)-interleaved if there are \(\Phi\in\text{Hom}^\epsilon({\mathbb U},{\mathbb V})\) and \(\Psi\in\text{Hom}^\epsilon({\mathbb V},{\mathbb U})\) such that \(\Psi\circ\Phi=1_{\mathbb U}^{2\epsilon}\) and \(\Phi\circ\Psi=1_{\mathbb V}^{2\epsilon}\).
Of primary concern in the article is the Vietoris-Rips complex of a metric space \(X\). Recall that this is the simplicial complex \(\text{Rips}(X,a)\) with vertex set \(X\), and whose simplices are finite sets of vertices such that the distance between any two pairs of vertices is at most \(a\). For a fixed homological dimension, the homology groups \(\{H(\text{Rips}(X,a))\mid a\in{\mathbb R}\}\) form a persistence module \(H(\text{Rips}(X))\). It is shown that for any two metric spaces \(X\) and \(Y\), the persistence modules \(H(\text{Rips}(X))\) and \(H(\text{Rips}(Y))\) are \(\epsilon\)-interleaved whenever the Gromov-Hausdorff distance between \(X\) and \(Y\) is less than \(\epsilon/2\). A similar result is shown to hold for the Čech complex of \(X\), as well as for other related simplicial complexes.
Using the techniques developed in the article, the authors deduce other properties of the Vietoris-Rips and Čech persistence modules, such as tameness, stability of persistence diagrams, and properties of their first and second homology groups.

MSC:
55N05 Čech types
57Q05 General topology of complexes
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