Operations on Metric 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.

In Proceedings ACT 2020, arXiv:2101.07888

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Cited by in corpus (1)

• Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius