Vietoris-Rips Homology Theory for Semi-Uniform Spaces

TL;DR

This paper extends Vietoris-Rips homology to semi-uniform spaces, broadening its applicability beyond graphs and metric spaces, and establishes foundational properties like the Eilenberg-Steenrod axioms in this new setting.

Contribution

It introduces a generalized Vietoris-Rips homology for semi-uniform spaces and proves fundamental axioms, enhancing the theoretical framework of topological data analysis.

Findings

Defines Vietoris-Rips homology for semi-uniform spaces

Proves Eilenberg-Steenrod axioms in this context

Provides a natural homotopy concept for semi-uniform spaces

Abstract

While the Vietoris-Rips complex is now widely used in both topological data analysis and the theory of hyperbolic groups, many of the fundamental properties of its homology have remained elusive. In this article, we define the Vietoris-Rips homology for semi-uniform spaces, which generalizes the classical theory for graphs and metric spaces, and provides a natural, general setting for the construction. We then prove a version of the Eilenberg-Steenrod axioms in this setting, giving a natural definition of homotopy for semi-uniform spaces in the process.