Given a finite sample of points from a metric space, what properties of the space
can one recover from the finite sample? In this regard, one of the most commonly
studied structures in Applied topology are Vietoris-Rips complexes, which associate
a topological structure on the data points to recover more information in terms of
homotopy type, homology groups, or persistent homology of the underlying space of
the finite sample. It can in general be difficult to determine the homotopy type of a
Vietoris-Rips complex, and only a little is understood about the homotopy types of
the Vietoris-Rips complexes. In particular, it has been conjectured that the planar
Rips complexes are homotopy equivalent to the wedge of spheres.
In this talk, we first give a brief overview of topological data analysis, motivating
the use of persistent homology as a suitable tool to solve the problem of extracting
global topological information from a discrete sample of points. With describing what
is known about the homotopy types of planar Rips complexes, we show that the above
conjecture holds for the classes of Rips complexes with pseudo-manifold structure.
Finally, we classify all possible finite closed pure Vietoris-Rips complexes of dimension
2 with certain homology condition.
