Fix a finite set of points in Euclidean
n-space
\mathbbEn\mathbb{E}^{n}
, thought of as a point-cloud sampling of a certain domain
D Ì \mathbbEnD\subset\mathbb{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
\mathbbEn\mathbb{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.
EWC supported by NSF MSPA-MCS # 0528086.
VdS supported by DARPA SPA # 30759.
JE supported by NSF MSPA-MCS # 0528086.
RG supported by DARPA SToMP # HR0011-07-1-0002 and NSF MSPA-MCS # 0528086.