Rips Complexes of Planar Point Sets

Fix a finite set of points in Euclidean n-space En, thought of as a point-cloud sampling of a certain domainD ⊂ En. The 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 Rips complex to En that has as its image a more accurate ndimensional 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 Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a 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’-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-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.