Vietoris-Rips and Cech Complexes of Metric Gluings

We study Vietoris–Rips and Čech complexes of metric wedge sums and metric gluings. We show that the Vietoris–Rips (resp. Čech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris–Rips (resp. Čech) complexes. We also provide generalizations for certain metric gluings, i.e. 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. 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.