Persistent homology with selective Rips complexes detects geodesic circles
The study of manifolds by combining geometry and topology has a long history; in particular, homology or homotopy classes of closed geodesics have been explored in [\textit{H. Huber}, Math. Ann. 138, 1--26 (1959; Zbl 0089.06101), \textit{J. Souček}, Commentat. Math. Univ. Carol. 25, 265--272 (1984; Zbl 0552.58013); \textit{I. A. Taĭmanov}, Russ. Math. Surv. 40, No. 6, 143--144 (1986; Zbl 0633.53067); \textit{P. Frosini}, Atti Semin. Mat. Fis. Univ. Modena 47, No. 2, 271--292 (1999; Zbl 0935.55007)] to mention a few. The research exposed in the present paper is along the same lines, facing the question: Which geometric properties does Persistent Homology (PH) encode and how does it encode them? The answer is looked for through geodesic spaces, in continuation of the preceding papers [\textit{Ž. Virk}, Rev. Mat. Complut. 32, No. 1, 195--213 (2019; Zbl 1412.55018); J. Topol. Anal. 12, No. 1, 169--207 (2020; Zbl 1443.55002)].\N\NThe main object of interest is \textit{bottleneck loops} in a metric space \(X\). A loop \(\alpha\) of finite length is a bottleneck loop if it is the shortest representative of its (free) homotopy class in some neighborhood of \(\alpha\) itself in \(X\). The key idea is to use PH, but not on normal (Vietoris-)Rips complexes as usual [\textit{V. de Silva} and \textit{R. Ghrist}, Algebr. Geom. Topol. 7, 339--358 (2007; Zbl 1134.55003)]. The new notion is that of a \textit{selective} Rips complex sRips\((X, r_1, n, r_2)\). Given positive real numbers \(r_1 \ge r_2\) and a natural number \(n\), a finite set \(\sigma\) of points is a simplex in sRips\((X, r_1, n, r_2)\) if it is a simplex in Rips\((X, r_1)\) and can be partitioned into \(n+1\) clusters of diameter \(< r_2\) (Def. 4.1). Through several lemmas, the author reaches Thm. 5.1, that connects bottleneck loops with persistent 2-cycles of the homology of sRips\((X, r_1, n, r_2)\) on an Abelian group \(G\). Corollaries 5.3 and 5.4 make the connection clearer. Conditions, for these propositions to hold also in the case of Rips complexes, are examined. Some examples show how selective Rips complexes may actually be more sensitive.
scientific article; zbMATH DE number 7900014
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