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On complex line arrangements and their boundary manifolds

  • V. FLORENS (a1), B. GUERVILLE-BALLÉ (a2) and M.A. MARCO-BUZUNARIZ (a3)
Abstract
Abstract

Let ${\mathcal A}$ be a line arrangement in the complex projective plane $\mathds{C}\mathds{P}^2$. We define and describe the inclusion map of the boundary manifold, the boundary of a closed regular neighbourhood of ${\mathcal A}$, in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computations provide a new minimal presentation of the fundamental group of the complement.

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[1]Artal E. Topology of arrangements and position of singularities. Annales de la fac. des sciences de Toulouse, to appear. (2014).
[2]Artal E., Carmona J., Cogolludo–Agustín J.I. and Marco M.A.Topology and combinatorics of real line arrangements. Compos. Math. 141 6 (2005), 15781588.
[3]Artal E., Carmona J., Cogolludo–Agustín J.I. and Marco M.A.Invariants of combinatorial line arrangements and Rybnikov's example. In Singularity theory and its applications, Izumiya S., Ishikawa G., Tokunaga H., Shimada I. and Sano T., Eds. Adv. Stud. Pure Math. vol. 43 (Mathematical Society of Japan, Tokyo, 2007).
[4]Artal E., Florens V. and Guerville–Ballé B. A topological invariant of line arrangements arXiv:1407.3387 (2014).
[5]Arvola W.The fundamental group of the complement of an arrangement of complex hyperplanes. Topology 31 4 (1992), 757765.
[6]Cohen D. and Suciu A.The boundary manifold of a complex line arrangement. In Groups, Homotopy and Configuration Spaces. Geom. Topol. Monogr. vol. 13 (Geom. Topol. Publ., Coventry, 2008), pp. 105146.
[7]Guerville–Ballé B. Topological invariants of line arrangements. PhD. thesis. Université de Pau et des Pays de l'Adour and Universidäd de Zaragoza (2013).
[8]Guerville–Ballé B. An arithmetic Zariski 4-tuple of twelve lines, arXiv:1411.2300 (2014).
[9]Hironaka E.Boundary manifolds of line arrangements. Math. Ann. 319 1 (2001), 1732.
[10]Libgober A.On the homotopy type of the complement to plane algebraic curves. J. Reine Angew. Math. 367 (1986), 103114.
[11]MacLane S.Some interpretations of abstract linear dependence in terms of projective geometry. Amer. J. Math. 58 1 (1936), 236240.
[12]Neumann W.A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Amer. Math. Soc. 268 2 (1981), 299344.
[13]Orlik P. and Terao H.Arrangements of Hyperplanes. Grundlehren Math. Wiss. vol. 300 (Springer-Verlag, Berlin, 1992).
[14]Rybnikov G. On the fundamental group of the complement of a complex hyperplane arrangement. Preprint available at arXiv:math.AG/9805056 (1998).
[15]Waldhausen F.Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II. Invent. Math. 3 (1967), 308333; ibid. 4 (1967), 87–117.
[16]Westlund E. The boundary manifold of an arrangement. PhD. thesis. University of Wisconsin Madison (1997).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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