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 Čech thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.

Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

“…Remark 6.2 Corollaries 5.2 and 5.8 of[5] establish results analogous to Theorem 6.1 for the products and metric gluings of Vietoris-Rips metric thickenings.…”

“…This, taken together with the comments above leads to the following conjecture: 5 The cost of the empty matching upper bounds the bottleneck distance.…”

“…However, FillRad.M / D 1 2 arccos. 1=.n C 1// 4 , whereas FillRad.N ;ı / Ä C n ı by inequality (5). This means that (7) cannot hold in general, even when the manifolds M and N have the same dimension.…”

“…The reverse inequality follows from Proposition 9.7 and Remarks 9.3 and 9.27 relating the spread to the filling radius of spheres. Indeed, by basic properties of the bottleneck distance, 5 for every integer k 0,…”