The graph minor theorem in topological combinatorics

TL;DR

This paper explores topological combinatorics through graph complexes, demonstrating that their homology torsion is universally bounded for graphs of bounded genus, extending the graph minor theorem algebraically.

Contribution

It introduces an algebraic perspective on the graph minor theorem, showing bounded torsion in homology of certain complexes across graphs of bounded genus.

Findings

Homology torsion is universally bounded for these complexes.

Results extend the graph minor theorem algebraically.

Applicable to various graph complexes in topological combinatorics.

Abstract

We study a variety of natural constructions from topological combinatorics, including matching complexes as well as other graph complexes, from the perspective of the graph minor category of \parencite{MiProRa}. We prove that these complexes must have universally bounded torsion in their homology across all graphs of bounded genus. One may think of these results as arising from an algebraic version of the graph minor theorem of Robertson and Seymour \parencite{RSXX,RSXXIII}.