an:07270220
Zbl 1455.55005
Adamaszek, Micha??; Adams, Henry; Gasparovic, Ellen; Gommel, Maria; Purvine, Emilie; Sazdanovic, Radmila; Wang, Bei; Wang, Yusu; Ziegelmeier, Lori
On homotopy types of Vietoris-Rips complexes of metric gluings
EN
J. Appl. Comput. Topol. 4, No. 3, 425-454 (2020).
00454861
2020
j
55N31 55U10 68T09 55P15 05E45
Vietoris-Rips complex; ??ech complex; metric space gluings; wedge sums; metric graphs; persistent homology
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 \(M\) [\textit{J.-C. Hausmann}, Ann. Math. Stud. 138, 175--188 (1995; Zbl 0928.55003); \textit{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 parameters. 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 persistent homology of a glued space completely. ??ech analogies are also studied. As an application, the Vietoris-Rips complexes of glued metric graphs are discussed.
Yuichi Ike (Kawasaki)
Zbl 0928.55003; Zbl 1001.53026