1996/08/02 by Daniel C. Cohen, Alexander I. Suciu · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic Geometry and Number Theory #Braid #Braid group #Braid theory #Combinatorics #Complement (music) #Fundamental group #Geometric and Algebraic Topology #Homomorphism #Hyperplane #Mathematics #Monodromy #Pure mathematics #alg-geom #math.AG #msc:05B35 #msc:14H30 #msc:20F36 #msc:32S25 #msc:52B30 #msc:57M05
paper · pdf · doi:10.1007/s000140050017
published as Commentarii Mathematici Helvetici 72 (1997), no. 2, 285-315. · 27 pages with 7 figures, author-supplied DVI file available at ftp://ftp.math.neu.edu/Pub/faculty/Suciu_Alex/papers/bmono.dvi AMSTeX v 2.1, pictex, edge-vertex-graphs
arxiv created 1996/08/02 · openalex publication_date 1997/06/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group Bn. Using Hansen's polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated "braided wiring diagram." The ensuing presentation of the fundamental group of the complement is shown to be Tietze-I equivalent to the Randell-Arvola presentation. Work of Libgober then implies that the complement of a line arrangement is homotopy equivalent to the 2-complex modeled on either of these presentations. Finally, we prove that the braid monodromy of a line arrangement determines the intersection lattice. Examples of Falk then show that the braid monodromy carries more information than the group of the complement, thereby answering a question of Libgober.