Discrete Morse theory and graph braid groups

By Daniel S. Farley, Lucas Sabalka. 2004

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If is any finite graph, then the unlabelled configuration space of n points on , denoted U C n , is the space of n-element subsets of . The braid group of on n strands is the fundamental group of U C n . We apply a discrete version of Morse theory to these U C n , for any n and any , and provide a clear description of the critical cells i n 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 strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in of degree at least 3 (and k is thus independent of n). AMS Classification 20F65, 20F36; 57M15, 57Q05, 55R80

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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