# zbMATH — the first resource for mathematics

The Vietoris-Rips complexes of a circle. (English) Zbl 1366.05124
Summary: Given a metric space $$X$$ and a distance threshold $$r>0$$, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of $$X$$ of diameter less than $$r$$. A theorem of Jean-Claude Hausmann states that if $$X$$ is a Riemannian manifold and $$r$$ is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of $$r$$, even though these complexes arise naturally in applications using persistent homology. We show that as $$r$$ increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, etc., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Čech complex of the circle (i.e., the nerve complex of the covering of a circle by all arcs of a fixed length) also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, etc., until finally it is contractible.

##### MSC:
 05E45 Combinatorial aspects of simplicial complexes 55U10 Simplicial sets and complexes in algebraic topology
Full Text:
##### References:
 [1] doi:10.1007/s11856-012-0166-1 · Zbl 1275.05041 [2] doi:10.1007/s00454-016-9803-5 · Zbl 1354.05149 [3] doi:10.1016/j.aam.2016.08.007 · Zbl 1379.37011 [4] doi:10.1016/j.jcta.2011.06.008 · Zbl 1234.05237 [5] ; Björner, Handbook of combinatorics, II, 1819, (1995) [6] doi:10.1090/S0273-0979-09-01249-X · Zbl 1172.62002 [7] doi:10.1007/s00454-009-9209-8 · Zbl 1231.05306 [8] doi:10.1007/s10711-013-9937-z · Zbl 1320.55003 [9] doi:10.1007/978-1-4613-9586-7_3 [10] ; Hausmann, Prospects in topology. Ann. of Math. Stud., 138, 175, (1995) [11] doi:10.1073/pnas.2.11.630 [12] doi:10.7151/dmgt.1424 · Zbl 1173.05041 [13] doi:10.1016/j.disc.2008.02.037 · Zbl 1215.05163 [14] doi:10.1016/S0021-9800(67)80075-9 [15] doi:10.1007/978-3-540-71962-5 [16] doi:10.1007/PL00000526 · Zbl 1001.53026 [17] doi:10.1016/j.disc.2008.04.003 · Zbl 1228.05218 [18] ; Matoušek, Contrib. Discrete Math., 3, 37, (2008) [19] ; Solomon, Geometric probability. Regional Conf. Series in Appl. Math., 28, (1978) [20] doi:10.1007/BF01447877 · JFM 53.0552.01 [21] doi:10.1515/crll.1999.035 · Zbl 0995.55004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.