Operations on Metric Thickenings

TL;DR

This paper introduces a category of simplicial metric thickenings that provides a framework to analyze the homotopy properties of complexes like Vietoris-Rips and Cech thickenings, especially regarding their behavior under products and wedge sums.

Contribution

It formalizes the concept of metric thickenings and proves their homotopy invariance under products and wedge sums for a broad class of complexes.

Findings

Product of metric thickenings is homotopy equivalent to the thickening of the product space.

Homotopy equivalence also holds for wedge sums of metric thickenings.

Applicable to complexes like Vietoris-Rips and Cech thickenings.

Abstract

Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris-Rips complex in applied topology. We formalize a remedy by introducing a category of simplicial metric thickenings whose objects have a natural realization as metric spaces. The properties of this category allow us to prove that, for a large class of thickenings including Vietoris-Rips and Cech thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.