Rips Complexes as Nerves and a Functorial Dowker-Nerve Diagram

• Autores: Ziga Virk
• Localización: Mediterranean journal of mathematics, ISSN 1660-5446, Vol. 18, Nº. 2, 2021
• Idioma: inglés
• DOI: 10.1007/s00009-021-01699-4
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• Resumen

  • 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.