Homology, Homotopy and Applications

Volume 23 (2021)

Number 2

Magnitude homology, diagonality, and median spaces

Pages: 121 – 140

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a7


Rémi Bottinelli (Institut de Mathématiques, Université de Neuchâtel, Switzerland)

Tom Kaiser (Institut de Mathématiques, Université de Neuchâtel, Switzerland)


We verify that the Künneth and Mayer–Vietoris formulae for magnitude homology of graphs, proven by Hepworth and Willerton, generalise naturally to the metric setting. Similarly, we extend the notion of diagonality of graphs to metric spaces, and verify its stability under products, retracts, and filtrations. As an application, we show that median spaces are diagonal; in particular any Menger convex median space has vanishing magnitude homology.


magnitude, metric space

2010 Mathematics Subject Classification


Copyright © 2021, Rémi Bottinelli and Tom Kaiser. Permission to copy for private use granted.

The first author was supported by the Swiss National Science Foundation project no. PP00P2-144681/1.

Received 27 April 2020

Received revised 26 May 2020

Accepted 2 October 2020

Published 21 April 2021