Torsion in the Magnitude homology of graphs

TL;DR

This paper investigates the torsion phenomena in magnitude homology of finite graphs, showing that any finitely generated abelian group can appear as a subgroup and providing explicit computations for outerplanar graphs.

Contribution

It demonstrates the universality of torsion in magnitude homology and offers detailed calculations for specific classes of graphs.

Findings

Any finitely generated abelian group can be realized as a subgroup of a graph's magnitude homology.

Torsion of any prime order can occur in magnitude homology.

Complete computations of magnitude homology for outerplanar graphs are provided.

Abstract

Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular, that torsion of a given prime order can appear in the magnitude homology of a graph and that there are infinitely many such graphs. Finally, we provide complete computations of magnitude homology of outerplanar graphs and focus on the ranks of the groups along the main diagonal of magnitude homology.