Fix a finite set of points in Euclidean n-space \euc^n, thought of as a
point-cloud sampling of a certain domain D\subset\euc^n. 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
\euc^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 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.Comment: 16 pages, 8 figure