On finite generation in magnitude (co)homology, and its torsion

TL;DR

This paper investigates the structural properties of magnitude (co)homology of graphs, showing polynomial growth and bounded torsion for graphs of bounded genus, and establishing a quasi-Groebner category for planar graphs.

Contribution

It applies the Sam-Snowden framework to graph homologies, proving polynomial rank growth and bounded torsion in magnitude cohomology for bounded genus graphs, and demonstrates the quasi-Groebner property for certain graph categories.

Findings

Magnitude cohomology has polynomial rank growth in graphs of bounded genus.

Torsion in magnitude cohomology is bounded for these graphs.

The category of planar graphs with contractions is quasi-Groebner.

Abstract

The aim of this paper is to apply the framework, which was developed by Sam and Snowden, to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton. More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs. We complement the work with a proof that the category of planar graphs of bounded genus and marked edges, with contractions, is quasi-Gr\"obner.