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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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##### References:
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