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On homotopy types of Vietoris-Rips complexes of metric gluings. (English) Zbl 07270220
The Vietoris-Rips complex is a fundamental tool in persistent homology theory or Topological Data Analysis (TDA). This complex can recover topological features of a sample underlying the data. Indeed, it was proved that if the underlying space is a closed Riemannian manifold \(M\), the scale parameter is sufficiently small, and a sample is sufficiently close to \(M\), then the Vietoris-Rips complex of the sample is homotopy equivalent to \(M\) [J.-C. Hausmann, Ann. Math. Stud. 138, 175–188 (1995; Zbl 0928.55003); J. Latschev, Arch. Math. 77, No. 6, 522–528 (2001; Zbl 1001.53026)]). In this paper, the authors study the Vietoris-Rips complexes of glued metric spaces at all scale parameter. In particular, it is proved that the Vietoris-Rips complex of the wedge sum of two pointed metric spaces \(\mathrm{VR}(X \vee Y;r)\) is homotopy equivalent to the wedge sum of the Vietoris-Rips complexes \(\mathrm{VR}(X;r) \vee \mathrm{VR}(Y;r)\) for all \(r>0\). More generally, the Vietoris-Rips complex of the glued space of two metric spaces along a common isometric subset is studied. These results enable us to compute the persistence homology of a glued space completely. Čech analogies are also studied. As an application, the Vietoris-Rips complexes of glued metric graphs are discussed.
MSC:
55N31 Persistent homology and applications, topological data analysis
55U10 Simplicial sets and complexes in algebraic topology
68T09 Computational aspects of data analysis and big data
55P15 Classification of homotopy type
05E45 Combinatorial aspects of simplicial complexes
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