Magnitude Homology, Diagonality, Medianness, K\"unneth and Mayer-Vietoris

TL;DR

This paper extends key formulas and concepts of magnitude homology from graphs to metric spaces, demonstrating stability properties and applications to median spaces, and proposes a new framework for betweenness spaces.

Contribution

It generalizes the K"unneth and Mayer-Vietoris formulas and diagonality notions to metric spaces, and introduces a new perspective on magnitude homology in betweenness spaces.

Findings

K"unneth and Mayer-Vietoris formulas extend to metric spaces.

Median spaces are shown to be diagonal with vanishing magnitude homology.

Proposes a new definition of magnitude homology for betweenness spaces.

Abstract

Magnitude homology of graphs is introduced by Hepworth and Willerton in arXiv:1505.04125 . Magnitude homology of arbitrary metric spaces by Leinster and Shulman in arXiv:1711.00802v2 . We verify that the K\"unneth and Mayer-Vietoris formulas proved in arXiv:1505.04125 for graphs extend naturally to the metric setting. The same is done for the notion of diagonality, also originating from arXiv:1505.04125 . Stability of this notion under products, retracts, filtrations is verified, and as an application, it is shown that median spaces are diagonal; in particular, any Menger convex median space has vanishing magnitude homology. Finally, we argue for a definition of magnitude homology in the context of "betweenness spaces" and develop some of its properties.