Homology, Homotopy and Applications

Volume 26 (2024)

Number 1

Magnitude, homology, and the Whitney twist

Pages: 105 – 130

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n1.a7


Emily Roff (School of Mathematics, University of Edinburgh, Scotland, United Kingdom)


Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the magnitude of graphs is its behaviour with respect to an operation known as the Whitney twist.We give a homological account of magnitude’s invariance under Whitney twists, extending the previously known result to encompass a substantially wider class of gluings. As well as providing a new tool for the computation of magnitudes, this is the first new theorem about magnitude to be proved using magnitude homology.


magnitude, graph, Whitney twist

2010 Mathematics Subject Classification

05C31, 05C76, 55N35

Copyright © 2024, Emily Roff. Permission to copy for private use granted.

Received 4 November 2022

Received revised 26 March 2023

Accepted 27 March 2023

Published 21 February 2024