Groups Which Cannot be Realized as Fundamental Groups of the Complements to Hypersurface in CN

  • Anatoly S. Libgober

Abstract

Finding restrictions imposed on a group by the fact that it can appear as a fundamental group of a smooth algebraic variety is an important problem particularly attributed to J.P. Serre. It has rather different aspects in characteristic p and zero and here we will address exclusively the latter case. Most restrictions described in the literature seem to rely on Hodge theory or some clever use of it (cf. [2]). A prototype of such restrictions is evenness of rk1/π′1Q)where π′1 is the commutator subgroup of the fundamental group 1rl. In the case of open non-singular varieties one can apply mixed Hodge theory. This was done by J. Morgan who obtained restrictions on the nilpotent quotients of the fundamental groups [13]. Here I shall describe a different (but also by no means complete) type of restriction on the fundamental groups of open varieties which are complements to hypersurfaces in C n It is implicitly contained in previous work on Alexander polynomial of plane curves [4]. For example many knot groups cannot occur as fundamental groups of the complement to an algebraic curve. This gives automatically the same restrictions on the fundamental groups of complements to arbitrary hypersurfaces in C n as follows from the well-known argument using Zariski Lefschetz type theorem: For a generic plane H relative to given hypersurface V in C n the natural map \(\pi _1 (H - H \cap V,p_0 ) \to \pi _1 (C^n - V,p_0 )(p_0 \in H)\)is an isomorphism, i.e., possible fundamental groups of the complement to hypersurfaces in C n are precisely the fundamental groups of the complements to plane curves. Therefore from now on I shall work with plane curves only.

Keywords

Manifold Tame 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Abhyankar, Tame coverings and the fundamental groups of algebraic varieties (Par VI) Amer. J. of Math., 81, 1959, pp. 46–94.MathSciNetCrossRefGoogle Scholar
  2. F.E.A. Johnson, and E.G. Rees, On the fundamental group for a complex algebraic manifold, Bull. London Math. Soc.,19, 1987, pp. 463–466.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    T. Kohno, An algebraic computation of the Alexander polynomial of plane algebraic curve, Proc. Japan Acad. Ser. A. Math., 59, 1983, pp. 94–97.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    A. Libgober, Alexander polynomials of plane algebraic curves and cyclic multiple planes, Duke Math. J., 49, 1982, pp. 833–851.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    A. Libgober, Alexander invariants of plane algebraic curves,Proc. Symp. Pure Math., 40, Amer. Math. Soc., Providence, RI, Part 2, pp. 135–144.Google Scholar
  6. [6]
    A. Libgober, Homotyopy groups of the complements to algebraic hypersurfaces, Bull. A. M. S., 1985.Google Scholar
  7. [7]
    J. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Etudes Sci. Publ. Math. 48, 1978, pp. 137–204.MATHCrossRefGoogle Scholar
  8. [8]
    R. Randell, Minor fibres and the Alexander polynomials of plane algebraic curves, Proc. Symp. Pure Math., 40, Providence, RI, Part 2, pp. 415–420.Google Scholar
  9. [9]
    D. Sullivan, On the intersection form of compact 3-manifold, Topology, 14, pp. 275–277.Google Scholar
  10. [10]
    van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math., 55, 1933.Google Scholar
  11. [11]
    O. Zariski, On the irregularity of cyclic multiple Planes, Ann. of Math., 32, 1981, pp. 309–318MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Anatoly S. Libgober

There are no affiliations available

Personalised recommendations