An application of the theory of FI-algebras to graph configuration spaces

TL;DR

This paper explores the algebraic structure of graph configuration spaces' homology groups using FI-algebras, revealing stable behaviors of syzygies across certain graph families.

Contribution

It applies FI-algebra theory to analyze the module structure of graph configuration spaces, demonstrating stability properties of syzygies in specific graph families.

Findings

Homology groups form finitely generated graded modules over polynomial rings.

Syzygies exhibit stable behaviors in certain graph families.

Provides new insights into algebraic structures of graph configuration spaces.

Abstract

Recent work of An, Drummond-Cole, and Knudsen, as well as the author, has shown that the homology groups of configuration spaces of graphs can be equipped with the structure of a finitely generated graded module over a polynomial ring. In this work we study this module structure in certain families of graphs using the language of FI-algebras recently explored by Nagel and R\"omer. As an application we prove that the syzygies of the modules in these families exhibit a range of stable behaviors.