The braid monodromy of plane algebraic curves and hyperplane arrangements.

*(English)*Zbl 0959.52018The 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)].

Reviewer: A.Lipovski (Beograd)

##### 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 |