## Abstract

Fix a finite set of points in Euclidean n-space E^{n}, thought of as a point-cloud sampling of a certain domain D ⊂ 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.

Original language | English (US) |
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Pages (from-to) | 75-90 |

Number of pages | 16 |

Journal | Discrete and Computational Geometry |

Volume | 44 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2010 |

## Keywords

- Quasi-Rips complex
- Rips complex
- Topology

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics