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Edge stabilization in the
homology of graph braid groups
Byung Hee An, Gabriel C Drummond-Cole and Ben Knudsen |
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Geometry & Topology 24 (2020) 421–469
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Abstract |
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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. |
Keywords
configuration spaces, braid groups, graphs, growth of Betti
numbers, connectivity, homological stability
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Mathematical Subject Classification 2010
Primary: 13D40, 20F36, 55R80
Secondary: 05C40
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References |
Publication
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
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Authors |
| Byung Hee An | |
| Department of Mathematics
Education Teachers College Kyungpook National University Daegu South Korea |
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| Gabriel C Drummond-Cole | |
| Center for Geometry and
Physics Institute for Basic Science Pohang South Korea |
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| https://drummondcole.com/gabriel/academic/ | |
| Ben Knudsen | |
| Department of Mathematics Harvard University Cambridge, MA United States |
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