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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
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