Edge stabilization in the homology of graph braid groups

TL;DR

This paper studies the homology of graph braid groups, introducing a stabilization map that increases particles on edges, proving finite generation of the homology module, and analyzing polynomial growth of Betti numbers.

Contribution

It introduces a new stabilization map for graph configuration spaces, proves the homology module is finitely generated, and analyzes the polynomial growth of Betti numbers.

Findings

Homology module is finitely generated over the polynomial ring of edges.

Betti numbers grow polynomially with degree explicitly calculated.

The chain-level action contains more information than the homology action.

Abstract

We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges.