Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture.
Eliyahu, Meital 1 ; Liberman, Eran 1 ; Schaps, Malka 1 ; Teicher, Mina 1
@article{10_2140_agt_2010_10_1285,
author = {Eliyahu, Meital and Liberman, Eran and Schaps, Malka and Teicher, Mina},
title = {The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups},
journal = {Algebraic and Geometric Topology},
pages = {1285--1304},
year = {2010},
volume = {10},
number = {3},
doi = {10.2140/agt.2010.10.1285},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1285/}
}
TY - JOUR AU - Eliyahu, Meital AU - Liberman, Eran AU - Schaps, Malka AU - Teicher, Mina TI - The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups JO - Algebraic and Geometric Topology PY - 2010 SP - 1285 EP - 1304 VL - 10 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1285/ DO - 10.2140/agt.2010.10.1285 ID - 10_2140_agt_2010_10_1285 ER -
%0 Journal Article %A Eliyahu, Meital %A Liberman, Eran %A Schaps, Malka %A Teicher, Mina %T The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups %J Algebraic and Geometric Topology %D 2010 %P 1285-1304 %V 10 %N 3 %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1285/ %R 10.2140/agt.2010.10.1285 %F 10_2140_agt_2010_10_1285
Eliyahu, Meital; Liberman, Eran; Schaps, Malka; Teicher, Mina. The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups. Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1285-1304. doi: 10.2140/agt.2010.10.1285
[1] , The fundamental group of the complement of an arrangement of complex hyperplanes, Topology 31 (1992) 757
[2] , , , Some analogs of Zariski’s theorem on nodal line arrangements, Algebr. Geom. Topol. 5 (2005) 691
[3] , , The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv. 72 (1997) 285
[4] , Homotopy types of line arrangements, Invent. Math. 111 (1993) 139
[5] , On parallel lines and free groups
[6] , Position of singularities and the fundamental group of the complement of a union of lines, Proc. Amer. Math. Soc. 124 (1996) 3299
[7] , Direct product of free groups as the fundamental group of the complement of a union of lines, Michigan Math. J. 44 (1997) 283
[8] , On the fundamental groups of complements of plane curves, PhD thesis, Bar-Ilan University (2001)
[9] , , , π1–classification of real arrangements with up to eight lines, Topology 42 (2003) 265
[10] , The theory of groups, Chelsea Publishing Co. (1976)
[11] , , Diffeomorphic types of the complements of arrangements of hyperplanes, Compositio Math. 92 (1994) 133
[12] , Presentations of groups, 22, Cambridge Univ. Press (1976)
[13] , On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933) 255
[14] , , Combinatorial group theory, 89, Springer (1977)
[15] , , , Combinatorial group theory.Presentations of groups in terms of generators and relations, Dover (1976)
[16] , , Braid group technique in complex geometry. I. Line arrangements in CP2, from: "Braids (Santa Cruz, CA, 1986)" (editors J S Birman, A Libgober), Contemp. Math. 78, Amer. Math. Soc. (1988) 425
[17] , , Product theorem of the fundamental group of a reducible curve, J. Math. Soc. Japan 30 (1978) 599
[18] , , Arrangements of hyperplanes, 300, Springer (1992)
[19] , The fundamental group of the complement of a union of complex hyperplanes, Invent. Math. 69 (1982) 103
[20] , , Rigidity of differentiable structure for new class of line arrangements, Comm. Anal. Geom. 13 (2005) 1057
Cité par Sources :