On $$\ell _p$$-Vietoris-Rips complexes

Abstract We study the concepts of the $$\ell _p$$ ℓ p -Vietoris-Rips simplicial set and the $$\ell _p$$ ℓ p -Vietoris-Rips complex of a metric space, where $$1\le p \le \infty .$$ 1 ≤ p ≤ ∞ . This theory unifies two established theories: for $$p=\infty ,$$ p = ∞ , this is the classical theory of Vietoris-Rips complexes, and for $$p=1,$$ 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 “ $$\ell _p$$ ℓ p -Vietoris-Rips spaces” are homotopy equivalent to the manifold; (3) we demonstrate that the $$\ell _p$$ ℓ p -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 $$\ell _p$$ ℓ p -Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on p ; and that the homology groups of the $$\ell _p$$ ℓ p -Vietoris-Rips spaces commute with filtered colimits of metric spaces.