We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ 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 "$\ell_p$-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the $\ell_p$-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 $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces.

翻译：本文研究度量空间的$\ell_p$-Vietoris-Rips单纯集与$\ell_p$-Vietoris-Rips复形概念，其中$1\leq p \leq \infty$。该理论统一了两个既有理论：当$p=\infty$时对应经典的Vietoris-Rips复形理论，当$p=1$时则对应于模糊幅度同调理论。我们证明了若干在Vietoris-Rips复形中已知结论在一般情况下的推广：（1）证明了对应版本持续同调的稳定性定理；（2）对于紧致黎曼流形及充分小的尺度参数，证明了所有"$\ell_p$-Vietoris-Rips空间"均与流形同伦等价；（3）论证了$\ell_p$-Vietoris-Rips空间在取度量完备化操作下具有同伦不变性。此外，我们证明了当尺度参数趋于零时，$\ell_p$-Vietoris-Rips空间同调群的极限与$p$无关；并且$\ell_p$-Vietoris-Rips空间的同调群与度量空间的滤过余极限可交换。