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The braid monodromy of plane algebraic curves and hyperplane arrangements. (English) Zbl 0959.52018
The authors present a detailed and unified view of the braid monodromy and the fundamental group of the complement of an algebraic curve \(C\) in \(\mathbb{C}^2\) and an arrangement \(A\) of complex or real lines in \(\mathbb{C}^2\), previously obtained by various methods. The braid monodromy of a complex plane algebraic curve was studied in [B. G. Moishezon, Algebraic Geometry, Lect. Notes Math. 862, Springer-Verlag, 107-192 (1981; Zbl 0476.14005)] and used in [A. Libgober, Proc. Symp. Pure Math. 46, No. 2, 29-45 (1987; Zbl 0703.14007)] to determine the braid monodromy presentation of the homotopy group \(\pi_1(\mathbb{C}^2\setminus C)\). On the other hand, one algorithm for computing the presentation of the homotopy group of the complex arrangement’s complement was found in [W. A. Arvola, Topology 31, No. 4, 757-765 (1992; Zbl 0772.57001)]. The authors develop the technique of “braided wiring diagrams”. This technique is then used to show that the braid monodromy presentation and the Arvola presentation are Tietze-I equivalent, therefore implying that the homotopy class of the complement is determined by the 2-complex associated to either of these presentations. The authors also compare the amount of information contained in the arrangement’s fundamental group, intersection lattice and braid monodromy: they prove that braid-equivalence of braid monodromies implies lattice isomorphism. Using the example from [M. Falk, Invent. Math. 111, No. 1, 139-150 (1993; Zbl 0772.52011)], of two arrangements having isomorphic fundamental groups and nonisomorphic lattices, thus non-braid-equivalent braid monodromies, the authors give an answer to a question posed in [A. Libgober, Invent. Math. 95, No. 1, 25-30 (1989; Zbl 0674.14015)].

MSC:
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
32S22 Relations with arrangements of hyperplanes
20F36 Braid groups; Artin groups
05B35 Combinatorial aspects of matroids and geometric lattices
32S25 Complex surface and hypersurface singularities
57M05 Fundamental group, presentations, free differential calculus
14H30 Coverings of curves, fundamental group
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55Q52 Homotopy groups of special spaces
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