Journal of the Mathematical Society of Japan

On the enhancement to the Milnor number of a class of mixed polynomials

Kazumasa INABA

Full-text: Open access

PDF File (252 KB)

Abstract

The enhancement to the Milnor number is an invariant of the homotopy classes of fibered links in the sphere $S^{2n-1}$ and belongs to $\mathbb{Z}/r\mathbb{Z}$, where $r=0$ if $n=2$ and $r=2$ if $n=2$. Mixed polynomials are polynomials in complex variables $z_1,\dots,z_n$ and their conjugates $\bar{z}_1,\dots,\bar{z}_n$. M. Oka showed that mixed polynomials have Milnor fibrations under the strongly non-degeneracy condition. In this present paper, we study fibered links which are defined by a certain class of mixed polynomials which admit Milnor fibrations and show that any element of $\mathbb{Z}/r\mathbb{Z}$ is realized by the enhancement to the Milnor number of such a fibered link.

Article information

Source
J. Math. Soc. Japan, Volume 66, Number 1 (2014), 25-36.

Dates
First available in Project Euclid: 24 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1390600835

Digital Object Identifier
doi:10.2969/jmsj/06610025

Mathematical Reviews number (MathSciNet)
MR3161391

Zentralblatt MATH identifier
1296.14028

Subjects
Primary: 14J17: Singularities [See also 14B05, 14E15]
Secondary: 37C27: Periodic orbits of vector fields and flows 58K45: Singularities of vector fields, topological aspects

Keywords
mixed polynomial enhanced Milnor number fibered link

Citation

INABA, Kazumasa. On the enhancement to the Milnor number of a class of mixed polynomials. J. Math. Soc. Japan 66 (2014), no. 1, 25--36. doi:10.2969/jmsj/06610025. https://projecteuclid.org/euclid.jmsj/1390600835


Export citation

References

  • J. L. Cisneros-Molina, Join theorem for polar weighted homogeneous singularities, In: Singularities II, (eds. J.-P. Brasselet, J. L. Cisneros-Molina, D. Massey, J. Seade and B. Teissier), Contemp, Math., 475, Amer. Math. Soc., Providence, RI, 2008, pp.,43–59.
    Mathematical Reviews (MathSciNet): MR2454370
    Digital Object Identifier: doi:10.1090/conm/475/09274
  • A. H. Durfee, Fibered knots and algebraic singularities, Topology, 13 (1974), 47–59.
    Mathematical Reviews (MathSciNet): MR336750
    Digital Object Identifier: doi:10.1016/0040-9383(74)90037-8
  • D. Eisenbud and W. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Stud., 110, Princeton University Press, Princeton, NJ, 1985.
    Mathematical Reviews (MathSciNet): MR817982
  • Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math., 98 (1989), 623–637.
    Mathematical Reviews (MathSciNet): MR1022310
    Digital Object Identifier: doi:10.1007/BF01393840
  • E. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, In: Proceedings of the International Congress of Mathematicians. Vol.,II, Beijing, 2002, (eds. L. Tatsien), Higher Ed. Press, Beijing, 2002, pp.,405–414.
    Mathematical Reviews (MathSciNet): MR1957051
  • S.-T. Hu, Homotopy Theory, Pure Appl. Math. (Amst.), 8, Academic Press, New York, London, 1959.
  • M. Ishikawa, On the contact structure of a class of real analytic germs of the form $f\bar{g}$, In: Singularities–Niigata-Toyama 2007, (eds. J.-P. Brasselet, S. Ishii, T. Suwa and M. Vaquie), Adv. Stud. Pure Math., 56, 2009, pp.,201–223.
    Mathematical Reviews (MathSciNet): MR2604084
  • R. Jacoby, One-parameter transformation groups of the three-sphere, Proc. Amer. Math. Soc., 7 (1956), 131–142.
    Mathematical Reviews (MathSciNet): MR75531
    Digital Object Identifier: doi:10.1090/S0002-9939-1956-0075531-0
  • L. H. Kauffman and W. D. Neumann, Products of knots, branched fibrations and sums of singularities, Topology, 16 (1977), 369–393.
    Mathematical Reviews (MathSciNet): MR488073
    Digital Object Identifier: doi:10.1016/0040-9383(77)90042-8
  • J. Milnor, Singular Points of Complex Hypersurfaces, Ann. Math. Stud., 61, Princeton University Press, Princeton, NJ, 1968.
    Mathematical Reviews (MathSciNet): MR239612
  • W. Neumann and L. Rudolph, Unfoldings in knot theory, Math. Ann., 278 (1987), 409–439.
    Mathematical Reviews (MathSciNet): MR909235
    Digital Object Identifier: doi:10.1007/BF01458078
  • W. Neumann and L. Rudolph, The enhanced Milnor number in higher dimensions, In: Differential Topology, Siegen, Germany, 1987, (ed. U. Koschorke), Lecture Notes in Math., 1350, Springer-Verlag, 1988, pp.,109–121.
    Mathematical Reviews (MathSciNet): MR979336
    Digital Object Identifier: doi:10.1007/BFb0081471
  • W. Neumann and L. Rudolph, Difference index of vectorfields and the enhanced Milnor number, Topology, 29 (1990), 83–100.
    Mathematical Reviews (MathSciNet): MR1046626
    Digital Object Identifier: doi:10.1016/0040-9383(90)90026-G
  • M. Oka, Topology of polar weighted homogeneous hypersurfaces, Kodai Math. J., 31 (2008), 163–182.
    Mathematical Reviews (MathSciNet): MR2435890
    Digital Object Identifier: doi:10.2996/kmj/1214442793
    Project Euclid: euclid.kmj/1214442793
  • M. Oka, Non-degenerate mixed functions, Kodai Math. J., 33 (2010), 1–62.
    Mathematical Reviews (MathSciNet): MR2732230
    Digital Object Identifier: doi:10.2996/kmj/1270559157
    Project Euclid: euclid.kmj/1270559157
  • A. Pichon, Real analytic germs $f\bar{g}$ and open-book decompositions of the 3-sphere, Internat. J. Math., 16 (2005), 1–12.
    Mathematical Reviews (MathSciNet): MR2115674
    Digital Object Identifier: doi:10.1142/S0129167X05002710
  • A. Pichon and J. Seade, Fibred multilinks and singularities $f\bar{g}$, Math. Ann., 342 (2008), 487–514.
    Mathematical Reviews (MathSciNet): MR2430986
    Digital Object Identifier: doi:10.1007/s00208-008-0234-3
  • M. A. S. Ruas, J. Seade and A. Verjovsky, On real singularities with a Milnor fibration, In: Trends in Singularities, (eds. A. Libgober and M. Tibăr), Trends Math., Birkhäuser, Basel, 2002, pp.,191–213.
    Mathematical Reviews (MathSciNet): MR1900787
    Digital Object Identifier: doi:10.1007/978-3-0348-8161-6_9
  • L. Rudolph, Isolated critical points of mappings from ${\bf R}^{4}$ to ${\bf R}^{2}$ and a natural splitting of the Milnor number of a classical fibered link. Part 1: Basic theory; examples, Comment. Math. Helv., 62 (1987), 630–645.
    Mathematical Reviews (MathSciNet): MR920062
    Digital Object Identifier: doi:10.1007/BF02564467
  • J. Seade, Fibred links and a construction of real singularities via complex geometry, Bol. Soc. Brasil. Mat. (N.S.), 27 (1996), 199–215.
    Mathematical Reviews (MathSciNet): MR1418933
    Digital Object Identifier: doi:10.1007/BF01259360
  • J. Seade, On the Topology of Isolated Singularities in Analytic Spaces, Progr. Math., 241, Birkhäuser Verlag, Basel, 2006.
    Mathematical Reviews (MathSciNet): MR2186327
  • W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc., 52 (1975), 345–347.
    Mathematical Reviews (MathSciNet): MR375366
    Digital Object Identifier: doi:10.1090/S0002-9939-1975-0375366-7

Mathematical Society of Japan

Journal of the Mathematical Society of Japan

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more