Discrete Morse theory and graph braid groups

2004/10/31 by Daniel Farley, Lucas Sabalka · 2 citations
Mathematics · #math.GR #math.AT #math.GT #msc:20F65 #msc:20F36 #msc:57M15 #msc:57Q05 #msc:55R80

paper · pdf · doi:10.2140/agt.2005.5.1075

published as Algebr. Geom. Topol. 5 (2005) 1075-1109 · Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-44.abs.html

arxiv created 2005/09/04 · arxiv updated 2014/09/30

Abstract

If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UCⁿ(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UCⁿ(Gamma). We apply a discrete version of Morse theory to these UCⁿ(Gamma), for any n and any Gamma, 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 UCⁿ(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).

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