Volume 5, issue 3 (2005)

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

Volume 20
Issue 7, 3219–3760
Issue 6, 2687–3218
Issue 5, 2145–2685
Issue 4, 1601–2143
Issue 3, 1073–1600
Issue 2, 531–1072
Issue 1, 1–529

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
Discrete Morse theory and graph braid groups

Daniel Farley and Lucas Sabalka

Algebraic & Geometric Topology 5 (2005) 1075–1109

arXiv: math.GR/0410539


If Γ is any finite graph, then the unlabelled configuration space of n points on Γ, denoted UCnΓ, is the space of n–element subsets of Γ. The braid group of Γ on n strands is the fundamental group of UCnΓ.

We apply a discrete version of Morse theory to these UCnΓ, for any n and any Γ, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UCnΓ strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Γ of degree at least 3 (and k is thus independent of n).

graph braid groups, configuration spaces, discrete Morse theory
Mathematical Subject Classification 2000
Primary: 20F65, 20F36
Secondary: 57M15, 57Q05, 55R80
Forward citations
Received: 26 October 2004
Accepted: 28 June 2005
Published: 31 August 2005
Daniel Farley
Department of Mathematics
University of Illinois at Urbana-Champaign
Champaign IL 61820
Lucas Sabalka
Department of Mathematics
University of Illinois at Urbana-Champaign
Champaign IL 61820