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.

19 pages, some issues resolved

References in corpus (2)

• 1-Dimensional Intrinsic Persistence of Geodesic Spaces
• Approximations of 1-Dimensional Intrinsic Persistence of Geodesic Spaces and Their Stability

Cited by in corpus (11)

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• Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
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• The Persistent Topology of Optimal Transport Based Metric Thickenings
• Critical edges in Rips complexes and persistence
• Relative persistent homology
• Persistent Homology with Selective Rips complexes detects geodesic circles
• Contractibility of the Rips complexes of Integer lattices via local domination
• Geometric Bounds for Persistence
• Rigidity of terminal simplices in persistent homology