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2005 Discrete Morse theory and graph braid groups
Daniel Farley, Lucas Sabalka
Algebr. Geom. Topol. 5(3): 1075-1109 (2005). DOI: 10.2140/agt.2005.5.1075


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).


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Daniel Farley. Lucas Sabalka. "Discrete Morse theory and graph braid groups." Algebr. Geom. Topol. 5 (3) 1075 - 1109, 2005.


Received: 26 October 2004; Accepted: 28 June 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1134.20050
MathSciNet: MR2171804
Digital Object Identifier: 10.2140/agt.2005.5.1075

Primary: 20F36, 20F65
Secondary: 55R80, 57M15, 57Q05

Rights: Copyright © 2005 Mathematical Sciences Publishers


Vol.5 • No. 3 • 2005
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