On Homotopy Types of Vietoris-Rips Complexes of Metric Gluings

TL;DR

This paper investigates the homotopy types of Vietoris-Rips complexes formed from metric gluings, providing new insights into their topological structure and persistent homology, especially for metric graphs glued along short paths.

Contribution

It establishes that Vietoris-Rips complexes of metric wedge sums are homotopy equivalent to wedge sums of complexes and generalizes this to gluings along isometric subsets, with applications to metric graphs.

Findings

Vietoris-Rips complex of a wedge sum is homotopy equivalent to the wedge sum of complexes.

Homotopy types of complexes can be determined for graphs glued along short paths.

The persistent homology of complexes for a broad class of metric graphs is described.

Abstract

We study Vietoris-Rips complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips complexes. We also provide generalizations for when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path (compared to lengths of certain loops in the input graphs). As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.