Magnitude, homology, and the Whitney twist

Abstract

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.

arXiv comment25 pages, 1 figure. Version 2: statement of Corollary 6.4 corrected, and minor edits throughout, following referee suggestions. To be published in Homology, Homotopy and Applications