Vietoris thickenings and complexes are weakly homotopy equivalent

Patrick Gillespie

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arXiv · 2303.01019

Vietoris thickenings and complexes are weakly homotopy equivalent

Patrick Gillespie

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Abstract

Characterizing the homotopy types of the Vietoris--Rips complexes of a metric space $X$ is in general a difficult problem. The Vietoris--Rips metric thickening, a metric space analogue of the Vietoris--Rips complex, was introduced as a potentially more amenable object of study with several advantageous properties, yet the relationship between its homotopy type and that of the Vietoris--Rips complex was not fully understood. We show that for any metric space $X$ and threshold $r>0$, the natural bijection between the (open) Vietoris--Rips complex and Vietoris--Rips metric thickening is a weak homotopy equivalence.

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BibTeX RIS

Patrick Gillespie. 2023-03-02. Vietoris thickenings and complexes are weakly homotopy equivalent. https://arxiv.org/abs/2303.01019

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