Discrete Morse theory and graph braid groups

Farley, Daniel; Sabalka, Lucas

doi:10.2140/agt.2005.5.1075

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Discrete Morse theory and graph braid groups

Daniel Farley and Lucas Sabalka

Algebraic & Geometric Topology 5 (2005) 1075–1109

DOI: 10.2140/agt.2005.5.1075

arXiv: math.GR/0410539

Abstract

If $Γ$ is any finite graph, then the unlabelled configuration space of $n$ points on $Γ$ , denoted $U C^{n} Γ$ , is the space of $n$ –element subsets of $Γ$ . The braid group of $Γ$ on $n$ strands is the fundamental group of $U C^{n} Γ$ .

We apply a discrete version of Morse theory to these $U C^{n} Γ$ , 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 $U C^{n} Γ$ 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 $n$ ).

Keywords

graph braid groups, configuration spaces, discrete Morse theory

Mathematical Subject Classification 2000

Primary: 20F65, 20F36

Secondary: 57M15, 57Q05, 55R80

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Publication

Received: 26 October 2004

Accepted: 28 June 2005

Published: 31 August 2005

Authors

Daniel Farley
Department of Mathematics University of Illinois at Urbana-Champaign Champaign IL 61820 USA
http://www.math.uiuc.edu/~farley/
Lucas Sabalka
Department of Mathematics University of Illinois at Urbana-Champaign Champaign IL 61820 USA
http://www.math.uiuc.edu/~sabalka/

Volume 5, issue 3 (2005)