Inventiones mathematicae

, Volume 82, Issue 1, pp 77–88

The lower central series of a fiber-type arrangement

  • Michael Falk
  • Richard Randell


For a certain class (“fiber-type”) of arrangements, including the supersolvable ones of Jambu and Terao [3], we prove a formula relating the Poincaré polynomial of the complement with the ranks of successive quotients in the lower central series of the fundamental group. Such a formula was proved by Kohno [5] for the single family of examplesAl.

We also show that the formula doesnot hold for allK(π, 1) arrangements.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brieskorn, E.: Sur les groupes de tresses (d'après V.I. Arnold). Séminaire Bourbaki, 24e année 1971/72, Springer Lect. Notes. Berlin: Springer 1973Google Scholar
  2. 2.
    Hu, S.: Homotopy Theory. New York: Academic Press 1959Google Scholar
  3. 3.
    Jambu, M., Terao, H.: Free arrangements of hyperplanes and supersolvable lattices. Adv. Math.52, 248–258 (1984)Google Scholar
  4. 4.
    Kohno, T.: On the holonomy Lie algebra and the nilpotent tower of the fundamental group of the complement of hypersurfaces. Nagoya Math. J.92, 21–37 (1983)Google Scholar
  5. 5.
    Kohno, T.: Série de Poincaré-Koszul associée aux groupes de tresses pures. Invent. Math.Google Scholar
  6. 6.
    Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. New York: John Wiley 1966Google Scholar
  7. 7.
    Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math.56, 167–189 (1980)Google Scholar
  8. 8.
    Orlik, P., Solomon, L.: Coxeter Arrangements. Proc. Symp. Pure Math., Am. Math. Soc., Providence, R.I.: 1983Google Scholar
  9. 9.
    Terao, H.: Arrangements of hyperplanes and their freeness, I. J. Fac. Sci., Univ. Tokyo, Sect. IA Math.27, 293–312 (1980)Google Scholar
  10. 10.
    Terao, H.: The modular element of the lattice of an arrangement. (Preprint)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Michael Falk
    • 1
  • Richard Randell
    • 1
  1. 1.Department of MathematicsThe University of IowaIowa CityUSA

Personalised recommendations