2023/03/02 by Patrick Gillespie, Gillespie, Patrick · 2 citations
Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2303.01019
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.