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Rips complexes as nerves and a functorial Dowker-nerve diagram. (English) Zbl 07321615
Summary: 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.
MSC:
05E45 Combinatorial aspects of simplicial complexes
55U10 Simplicial sets and complexes in algebraic topology
55U05 Abstract complexes in algebraic topology
57N16 Geometric structures on manifolds of high or arbitrary dimension
57N65 Algebraic topology of manifolds
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