2024/11/04 by Sergei O. Ivanov, Xiaomeng Xu, Ivanov, Sergei O. +1
Computer Science · Mathematics · #cs.CG #math.AT #math.MG
paper · pdf · doi:10.48550/arxiv.2411.01857
published as Journal of Applied and Computational Topology 10 (2026), Article 15
arxiv created 2025/02/27 · arxiv updated 2026/07/30
We study the concepts of the ℓp-Vietoris-Rips simplicial set and the ℓp-Vietoris-Rips complex of a metric space, where 1≤ p ≤ ∞. This theory unifies two established theories: for p=∞, this is the classical theory of Vietoris-Rips complexes, and for p=1, this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the "ℓp-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the ℓp-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the ℓp-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on p; and that the homology groups of the ℓp-Vietoris-Rips spaces commute with filtered colimits of metric spaces.