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Geometriae Dedicata

, Volume 158, Issue 1, pp 87–108 | Cite as

New open-book decompositions in singularity theory

  • Haydée Aguilar-Cabrera
Original Paper

Abstract

In this article, we study the topology of real analytic germs \({F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)}\) given by \({F(x,y,z)=\overline{xy}(x^p+y^q)+z^r}\) with \({p,q,r \in \mathbb{N}, p,q,r \geq 2}\) and (p, q) = 1. Such a germ gives rise to a Milnor fibration \({\frac{F}{\mid F \mid}\colon \mathbb{S}^5\setminus L_F \to \mathbb{S}^1}\). We describe the link L F as a Seifert manifold and we show that in many cases the open-book decomposition of \({\mathbb{S}^5}\) given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in \({\mathbb{C}^3}\).

Keywords

Milnor fibration Real singularities Open-book decompositions Seifert manifolds 

Mathematics Subject Classification (2000)

32S55 32S25 57M27 

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References

  1. 1.
    Artin M.: On isolated rational singularities of surfaces. Am. J. Math. 88, 129–136 (1966)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Barth W., Peters C., van de Ven A.: Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4. Springer, Berlin (1984)Google Scholar
  3. 3.
    Church P.T., Lamotke K.: Non-trivial polynomial isolated singularities. Nederl. Akad. Wetensch. Proc. Ser. A 78, Indag. Math. 37, 149–154 (1975)MathSciNetGoogle Scholar
  4. 4.
    Cisneros-Molina, J.L.: Join theorem for polar weighted homogeneous singularities. In: Singularities II, Contemp. Math., vol. 475, pp. 43–59. Amer. Math. Soc., Providence, RI (2008)Google Scholar
  5. 5.
    Durfee A.H.: The signature of smoothings of complex surface singularities. Math. Ann. 232, 85–98 (1978)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Durfee A.H.: Neighborhoods of algebraic sets. Trans. Am. Math. Soc. 276(2), 517–530 (1983)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Jankins M., Neumann W.D.: Lectures on Seifert Manifolds, Brandeis Lecture Notes, vol. 2. Brandeis University, Waltham, MA (1983)Google Scholar
  8. 8.
    Larrión F., Seade J.: Complex surface singularities from the combinatorial point of view. Topol. Appl. 66(3), 251–265 (1995)MATHCrossRefGoogle Scholar
  9. 9.
    Laufer, H.B.: On μ for surface singularities. In: Several Complex Variables (Proc. Sympos. Pure Math., vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), pp. 45–49. Am. Math. Soc., Providence, R. I. (1977)Google Scholar
  10. 10.
    Looijenga E.: A note on polynomial isolated singularities. Nederl. Akad. Wetensch. Proc. Ser. A 74, Indag. Math. 33, 418–421 (1971)MathSciNetGoogle Scholar
  11. 11.
    Looijenga, E., Wahl, J.: Quadratic functions and smoothing surface singularities. Topology 25(3), 261–291 (1986) doi: 10.1016/0040-9383(86)90044-3. URL: http://dx.doi.org/10.1016/0040-9383(86)90044-3 Google Scholar
  12. 12.
    Milnor J.: Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, N.J (1968)Google Scholar
  13. 13.
    Neumann W.D.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Am. Math. Soc. 268(2), 299–344 (1981)MATHCrossRefGoogle Scholar
  14. 14.
    Neumann, W.D., Raymond, F.: Seifert manifolds, plumbing, μ-invariant and orientation reversing maps. In: Algebraic and Geometric Topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977), Lecture Notes in Math., vol. 664, pp. 163–196. Springer, Berlin (1978)Google Scholar
  15. 15.
    Oka M.: On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials. Topology 12, 19–32 (1973)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Oka, M.: Topology of polar weighted homogeneous hypersurfaces. Kodai Math. J. 31(2), 163–182 (2008) doi: 10.2996/kmj/1214442793. URL http://dx.doi.org/10.2996/kmj/1214442793
  17. 17.
    Oka M.: Non-degenerate mixed functions. Kodai Math. J. 33(1), 1–62 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Orlik P.: Seifert Manifolds, Lecture Notes in Mathematics, vol. 291. Springer, Berlin (1972)Google Scholar
  19. 19.
    Pichon A.: Real analytic germs \({f\overline g}\) and open-book decompositions of the 3-sphere. Internat. J. Math. 16(1), 1–12 (2005)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pichon A., Seade J.: Real singularities and open-book decompositions of the 3-sphere. Ann. Fac. Sci. Toulouse Math.(6) 12(2), 245–265 (2003)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Pichon A., Seade J.: Fibred multilinks and singularities \({f\bar{g}}\). Mathematische Annalen 342(3), 487–514 (2008)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Ruas, M.A.S., Seade, J., Verjovsky, A.: On real singularities with a Milnor fibration. In: Trends in Singularities, Trends Math., pp. 191–213. Birkhäuser, Basel (2002)Google Scholar
  23. 23.
    Seade, J.: A cobordism invariant for surface singularities. In: Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, pp. 479–484. Am. Math. Soc., Providence, R.I. (1983)Google Scholar
  24. 24.
    Seade J.: Open book decompositions associated to holomorphic vector fields. Bol. Soc. Mat. Mexicana(3) 3(2), 323–335 (1997)MathSciNetMATHGoogle Scholar
  25. 25.
    Seade J.: On the Topology of Isolated Singularities in Analytic Spaces, Progress in Mathematics, vol. 241. Birkhäuser Verlag, Basel (2006)Google Scholar
  26. 26.
    Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities. In: Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, pp. 513–536. Am. Math. Soc., Providence, RI (1983)Google Scholar

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Authors and Affiliations

  1. 1.Instituto de MatemáticasUnidad Cuernavaca, Universidad Nacional Autónoma de MéxicoCuernavacaMéxico
  2. 2.Institut de Mathématiques de LuminyUniversité de la MéditerranéeMarseilleFrance

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