Volume 24, issue 1 (2020)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
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
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.

Keywords
configuration spaces, braid groups, graphs, growth of Betti numbers, connectivity, homological stability
Mathematical Subject Classification 2010
Primary: 13D40, 20F36, 55R80
Secondary: 05C40
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
Authors
Byung Hee An
Department of Mathematics Education
Teachers College
Kyungpook National University
Daegu
South Korea
Gabriel C Drummond-Cole
Center for Geometry and Physics
Institute for Basic Science
Pohang
South Korea
https://drummondcole.com/gabriel/academic/
Ben Knudsen
Department of Mathematics
Harvard University
Cambridge, MA
United States