If is any finite graph, then the unlabelled configuration space of points on , denoted , is the space of –element subsets of . The braid group of on strands is the fundamental group of .
We apply a discrete version of Morse theory to these , for any and any , 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 strong deformation retracts onto a CW complex of dimension at most , where is the number of vertices in of degree at least 3 (and is thus independent of ).
"Discrete Morse theory and graph braid groups." Algebr. Geom. Topol. 5 (3) 1075 - 1109, 2005. https://doi.org/10.2140/agt.2005.5.1075