Vietoris-Rips Complexes of Planar Point Sets

Fix a finite set of points in Euclidean n-space E-n, thought of as a point-cloud sampling of a certain domain D subset of E-n. The Vietoris-Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural "shadow" projection map from the Vietoris-Rips complex to E-n that has as its image a more accurate n-dimensional approximation to the homotopy type of D. We demonstrate that this projection map is 1-connected for the planar case n = 2. That is, for planar domains, the Vietoris-Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Vietoris-Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to "quasi"-Vietoris-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-Vietoris-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three.