Discrete Morse theory and graph braid groups
Daniel Scott Farley, Lucas Sabalka · 2005
Abstract If Γ is any finite graph, then the unlabelled configuration space of n points on Γ, denoted UC n Γ, is the space of n-element subsets of Γ. The braid group of Γ on n strands is the fundamental group of UC n Γ. We apply a discrete version of Morse theory to these UC 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 UC 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). AMS Classification 20F65, 20F36; 57M15, 57Q05, 55R80