Discrete Morse theory and graph braid groups

Abstract

If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(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 UC^n(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).

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-44.abs.html

References in corpus (2)

• Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups
• Euler characteristic of the configuration space of a complex

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• Discrete Morse Theory for Weighted Simplicial Complexes
• Abrams's stable equivalence for graph braid groups
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• Stability phenomena in the homology of tree braid groups
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• Topology of configuration space of two particles on a graph, II
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• Topological complexity of unordered configuration spaces of certain graphs
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• An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs
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