Using ideas related to 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 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.
We prove that the space of persistence diagrams on n points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension 2n. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when the number of points is not bounded, the corresponding spaces of persistence diagrams do not have finite asymptotic dimension. Furthermore, in the case of the bottleneck distance, the corresponding space does not coarsely embed into Hilbert space.
In 1995 Jean-Claude Hausmann proved that a compact Riemannian manifold X is homotopy equivalent to its Rips complex Rips(X, r) for small values of parameter r. He then conjectured that the connectivity of Rips complexes is a monotone function in r, a statement which has been supported by all known examples up to present. In this paper, we prove that S^3 equipped with a certain Riemannian metric is a counterexample to Hausmann's conjecture. Our proof combines the Stability Theorem of persistent homology, a persistent version of Hausmann's Theorem, and an approximation theorem of Ferry and Okun.