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Rips Complexes as Nerves and a Functorial Dowker-Nerve Diagram

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Abstract

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.

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Authors and Affiliations

  1. University of Ljubljana, Ljubljana, Slovenia

    Žiga Virk

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  1. Žiga Virk
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Correspondence to Žiga Virk.

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Research was supported by Slovenian Research Agency grants No. N1-0114, P1-0292, J1-8131, and N1-0064. The author would like to thank the referee for careful reading and helpful suggestions.

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Virk, Ž. Rips Complexes as Nerves and a Functorial Dowker-Nerve Diagram. Mediterr. J. Math. 18, 58 (2021). https://doi.org/10.1007/s00009-021-01699-4

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  • DOI: https://doi.org/10.1007/s00009-021-01699-4

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