On the cohomology rings of tree braid groups

2006/02/28 by Daniel Farley, Lucas Sabalka · 38 citations
Computer Science · Mathematics · #Artin group #Braid #Braid group #Cohomology #Combinatorics #Coxeter group #Discrete mathematics #Fundamental group #Geometric and Algebraic Topology #Graph #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #Topological and Geometric Data Analysis #math.AT #math.GR #msc:20F36 #msc:20F65 #msc:55R80 #msc:57M15

paper · pdf · doi:10.1016/j.jpaa.2007.04.011

published in Journal of Pure and Applied Algebra 212(1), 53-71 (Elsevier BV) · 25 pages, 7 figures. Revised version, accepted by the Journal of Pure and Applied Algebra

arxiv created 2007/03/27 · openalex publication_date 2007/05/06 · arxiv updated 2010/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let Γ be a finite connected graph. The (unlabelled) configuration space UCⁿ Γ of n points on Γ is the space of n-element subsets of Γ. The n-strand braid group of Γ, denoted B_nΓ, is the fundamental group of UCⁿ Γ. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" 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.

Citations

• Configuration spaces and braid groups on graphs in robotics
• Metric Spaces of Non-Positive Curvature
• Braid groups and right angled Artin groups