Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Abstract

Using ideas of the Dowker duality we prove that the Rips complex at scale r is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. These results are incorporated into a systematic theory of filtrations arising from covers. As a result we provide a general framework for reconstruction of spaces by Rips complexes, a short proof of the reconstruction result of Hausmann, and completely classify reconstruction scales for metric graphs. Furthermore we introduce a new extraction method for homology of a space based on nested Rips complexes at a single scale, which requires no conditions on neighboring scales nor the Euclidean structure of the ambient space.

Operations on Metric Thickenings

“…We will frequently return to this example. In particular, the homotopy type of the Vietoris-Rips complex of various spaces is widely studied [2,3,9,10,16,17,35,36,37]. By formulating a category of simplicial metric thickenings which includes Vietoris-Rips thickenings, we are able to compute the homotopy type of Vietoris-Rips thickenings of spaces constructed from limit and colimit operations.…”

Metric thickenings and group actions

“…More recently, in applied and computational topology, Vietoris-Rips and Čech complexes have been used to recover the "shape" of a dataset. Indeed, there are theoretical guarantees that if X is a sufficiently nice sample from an unknown underlying space M , then one can recover the homotopy types, homology groups, or approximate persistent homology of M from X [14,15]. In data analysis contexts, instead of letting r be arbitrarily small (as for (co)homology theories), and instead of letting r be sufficiently large (as in geometric group theory), we instead are interested in an intermediate range of scale parameters r. Indeed, if r is smaller than the distance between any two data points in X, then VR(X; r) = X is a disjoint union of points.…”

Metric thickenings and group actions

“…More recently, in applied and computational topology, Vietoris-Rips and Čech complexes have been used to recover the "shape" of a dataset. Indeed, there are theoretical guarantees that if X is a sufficiently nice sample from an unknown underlying space M , then one can recover the homotopy types, homology groups, or approximate persistent homology of M from X [25,37]. In data analysis contexts, instead of letting r be arbitrarily small (as for (co)homology theories), and instead of letting r be sufficiently large (as in geometric group theory), we instead are interested in an intermediate range of scale parameters r. Indeed, if r is smaller than the distance between any two data points in X, then VR(X; r) = X is a disjoint union of points.…”