Edge stabilization in the homology of graph braid groups

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.

arXiv comment49 pages, 19 figures. Mild revision for clarity, figures added. Accepted for publication in Geometry & Topology, may vary slightly from published version

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Cited by in corpus (8)

• Non-abelian Quantum Statistics on Graphs
• Geometric presentations of braid groups for particles on a graph
• The categorical graph minor theorem
• Asymptotic homology of graph braid groups
• Deletion and contraction in configuration spaces of graphs
• Heisenberg homology of ribbon graphs
• Configuration space in a product
• An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs