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arXiv · 1906.04028

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Žiga Virk
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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.

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Žiga Virk. 2019-06-10. Rips complexes as nerves and a Functorial Dowker-Nerve Diagram. https://doi.org/10.1007/s00009-021-01699-4

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