On the cohomology rings of tree braid groups

Abstract

Let Γ be a finite connected graph. The (unlabelled) configuration space U C n Γ of n points on Γ is the space of n -element subsets of Γ . The n -strand braid group of Γ , denoted B n Γ , is the fundamental group of U C n Γ . We use the methods and results of [Daniel Farley, Lucas Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005) 1075–1109. Electronic] to get a partial description of the cohomology rings H ∗ ( B n T ) , where T is a tree. Our results are then used to prove that B n T is a right-angled Artin group if and only if T is linear or n < 4 . This gives a large number of counterexamples to Ghrist’s conjecture that braid groups of planar graphs are right-angled Artin groups.