Summary
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 examplesA l .
We also show that the formula doesnot hold for allK(π, 1) arrangements.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brieskorn, E.: Sur les groupes de tresses (d'après V.I. Arnold). Séminaire Bourbaki, 24e année 1971/72, Springer Lect. Notes. Berlin: Springer 1973
Hu, S.: Homotopy Theory. New York: Academic Press 1959
Jambu, M., Terao, H.: Free arrangements of hyperplanes and supersolvable lattices. Adv. Math.52, 248–258 (1984)
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)
Kohno, T.: Série de Poincaré-Koszul associée aux groupes de tresses pures. Invent. Math.
Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. New York: John Wiley 1966
Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math.56, 167–189 (1980)
Orlik, P., Solomon, L.: Coxeter Arrangements. Proc. Symp. Pure Math., Am. Math. Soc., Providence, R.I.: 1983
Terao, H.: Arrangements of hyperplanes and their freeness, I. J. Fac. Sci., Univ. Tokyo, Sect. IA Math.27, 293–312 (1980)
Terao, H.: The modular element of the lattice of an arrangement. (Preprint)
Author information
Authors and Affiliations
Additional information
Second author partially supported by NSF grant
Rights and permissions
About this article
Cite this article
Falk, M., Randell, R. The lower central series of a fiber-type arrangement. Invent Math 82, 77–88 (1985). https://doi.org/10.1007/BF01394780
Issue Date:
DOI: https://doi.org/10.1007/BF01394780