Algebraic & Geometric Topology

Line arrangements and direct products of free groups

Kristopher Williams

Full-text: Open access


We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct product of free groups, then the complements of the arrangements are homotopy equivalent. For any such arrangement A, we also construct an arrangement A such that A is a complexified-real arrangement, the intersection lattices of the arrangements are isomorphic, and the complements of the arrangements are diffeomorphic.

Article information

Algebr. Geom. Topol., Volume 11, Number 1 (2011), 587-604.

Received: 8 October 2010
Revised: 9 December 2010
Accepted: 22 December 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C30: Planar arrangements of lines and pseudolines
Secondary: 32S22: Relations with arrangements of hyperplanes [See also 52C35] 14F35: Homotopy theory; fundamental groups [See also 14H30]

line arrangement fundamental group hyperplane arrangement direct product of free groups homotopy type


Williams, Kristopher. Line arrangements and direct products of free groups. Algebr. Geom. Topol. 11 (2011), no. 1, 587--604. doi:10.2140/agt.2011.11.587.

Export citation


  • A D R Choudary, A Dimca, Ş Papadima, Some analogs of Zariski's theorem on nodal line arrangements, Algebr. Geom. Topol. 5 (2005) 691–711
  • D C Cohen, A I Suciu, The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv. 72 (1997) 285–315
  • M Eliyahu, E Liberman, M Schaps, M Teicher, The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups, Algebr. Geom. Topol. 10 (2010) 1285–1304
  • C J Eschenbrenner, M J Falk, Orlik–Solomon algebras and Tutte polynomials, J. Algebraic Combin. 10 (1999) 189–199
  • M Falk, Homotopy types of line arrangements, Invent. Math. 111 (1993) 139–150
  • M Falk, Combinatorial and algebraic structure in Orlik–Solomon algebras, from: “Combinatorial geometries (Luminy, 1999)”, European J. Combin. 22 (2001) 687–698
  • M J Falk, N J Proudfoot, Parallel connections and bundles of arrangements, from: “Arrangements in Boston: a Conference on Hyperplane Arrangements (1999)”, Topology Appl. 118 (2002) 65–83
  • K-M Fan, Direct product of free groups as the fundamental group of the complement of a union of lines, Michigan Math. J. 44 (1997) 283–291
  • T Jiang, S S-T Yau, Diffeomorphic types of the complements of arrangements of hyperplanes, Compositio Math. 92 (1994) 133–155
  • A Libgober, On the homotopy type of the complement to plane algebraic curves, J. Reine Angew. Math. 367 (1986) 103–114
  • P Orlik, H Terao, Arrangements of hyperplanes, Grund. der Math. Wissenschaften 300, Springer, Berlin (1992)
  • G Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement
  • N White (editor), Theory of matroids, Encyclopedia of Math. and its Appl. 26, Cambridge Univ. Press (1986)
  • N White (editor), Combinatorial geometries, Encyclopedia of Math. and its Applications 29, Cambridge Univ. Press (1987)