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Edge stabilization in the homology of graph braid groups

Byung Hee An, Gabriel C Drummond-Cole and Ben Knudsen

Geometry & Topology 24 (2020) 421–469

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.

configuration spaces, braid groups, graphs, growth of Betti numbers, connectivity, homological stability
Mathematical Subject Classification 2010
Primary: 13D40, 20F36, 55R80
Secondary: 05C40
Received: 11 October 2018
Revised: 4 May 2019
Accepted: 7 June 2019
Published: 25 March 2020
Proposed: Benson Farb
Seconded: Haynes R Miller, Mark Behrens
Byung Hee An
Department of Mathematics Education
Teachers College
Kyungpook National University
South Korea
Gabriel C Drummond-Cole
Center for Geometry and Physics
Institute for Basic Science
South Korea
Ben Knudsen
Department of Mathematics
Harvard University
Cambridge, MA
United States