Journal of the Mathematical Society of Japan

Generic smooth maps with sphere fibers

Osamu SAEKI and Keiichi SUZUOKA

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Abstract

In this paper, we study various topological properties of generic smooth maps between manifolds whose regular fibers are disjoint unions of homotopy spheres. In particular, we show that if a closed 4 -manifold admits such a generic map into a surface,then it bounds a 5 -manifold with nice properties. As a corollary, we show that each regular fiber of such a generic map of the 4 -sphere into the plane is a homotopy ribbon 2 -link and that any spun 2 -knot of a classical knot can be realized as a component of a regular fiber of such a map.

Article information

Source
J. Math. Soc. Japan, Volume 57, Number 3 (2005), 881-902.

Dates
First available in Project Euclid: 14 September 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1158241939

Digital Object Identifier
doi:10.2969/jmsj/1158241939

Mathematical Reviews number (MathSciNet)
MR2139738

Zentralblatt MATH identifier
1091.57021

Subjects
Primary: 57R45: Singularities of differentiable mappings
Secondary: 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx] 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}

Keywords
generic map stable map Stein factorization regular fiber special generic map homotopy ribbon 2-link 4-manifold nonsingular stable map

Citation

SAEKI, Osamu; SUZUOKA, Keiichi. Generic smooth maps with sphere fibers. J. Math. Soc. Japan 57 (2005), no. 3, 881--902. doi:10.2969/jmsj/1158241939. https://projecteuclid.org/euclid.jmsj/1158241939


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References

  • O. Burlet and G. de Rham, Sur certaines applications génériques d'une variété close à $3$ dimensions dans le plan, Enseign. Math., 20 (1974), 275–292.
  • T. Cochran, Ribbon knots in $S^4$, J. London Math. Soc.,(2), 28 (1983), 563–576.
  • J. M. Èlia\usberg, On singularities of folding type, Math. USSR-Izv., 4 (1970), 1119–1134.
  • J. M. Èlia\usberg, Surgery of singularities of smooth mappings, Math. USSR-Izv., 6 (1972), 1302–1326.
  • T. Fukuda, Topology of folds, cusps and Morin singularities, In “A Fete of Topology”, (eds. Y. Matsumoto, T. Mizutani and S. Morita), Academic Press, 1987, pp.,331–353.
  • Y. K. S. Furuya, Sobre aplicações genéricas $M^4 \to {\bf R}^2$, PhD Thesis, ICMSC, University of São Paulo, 1986.
  • M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Grad. Texts in Math., Vol.,14, Springer, New York-Heidelberg-Berlin, 1973.
  • J. Hempel, $3$-Manifolds, Ann. of Math. Stud., No.,86, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1976.
  • J. T. Hiratuka, A fatorização de Stein e o número de singularidades de aplicações estáveis (in Portuguese), PhD Thesis, University of São Paulo, 2001.
  • T. Kanenobu, Non-ribbon $n$-knots with Seifert manifolds homeomorphic to punctured $S^n \times S^1$, Math. Sem. Notes, 10 (1982), 69–74.
  • R. C. Kirby, The topology of $4$-manifolds, Lecture Notes in Math., Vol.,1374, Springer, Berlin, 1989.
  • M. Kobayashi and O. Saeki, Simplifying stable mappings into the plane from a global viewpoint, Trans. Amer. Math. Soc., 348 (1996), 2607–2636.
  • M. Kreck, $h$-cobordisms between $1$-connected $4$-manifolds, Geom. Topol., 5 (2001), 1–6.
  • L. Kushner, H. Levine and P. Porto, Mapping three-manifolds into the plane I, Bol. Soc. Mat. Mexicana, 29 (1984), 11–33.
  • H. Levine, Classifying immersions into $\R^4$ over stable maps of $3$-manifolds into $\R^2$, Lecture Notes in Math., Vol.,1157, Springer, Berlin, 1985.
  • J. N. Mather, Generic projections, Ann. of Math.,(2), 98 (1973), 226–245.
  • J. Milnor, Lectures on the $h$-cobordism theorem, Math. Notes, Princeton Univ. Press, Princeton, N.J., 1965.
  • B. Moishezon and M. Teicher, Existence of simply connected algebraic surfaces of general type with positive and zero indices, Proc. Nat. Acad. Sci. USA, 83 (1986), 6665–6666.
  • B. Moishezon and M. Teicher, Simply-connected algebraic surfaces of positive index, Invent. Math., 89 (1987), 601–643.
  • B. Morin, Formes canoniques des singularités d'une application différentiable, C. R. Acad. Sci. Paris, 260 (1965), 5662–5665, 6503–6506.
  • P. Porto and Y. K. S. Furuya, On special generic maps from a closed manifold into the plane, Topology Appl., 35 (1990), 41–52.
  • D. Rolfsen, Knots and links, Math. Lecture Series, No.,7, Publish or Perish, Inc., Berkeley, Calif., 1976.
  • O. Saeki, Notes on the topology of folds, J. Math. Soc. Japan, 44 (1992), 551–566.
  • O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl., 49 (1993), 265–293.
  • N. Shibata, On non-singular stable maps of $3$-manifolds with boundary into the plane, Hiroshima Math. J., 30 (2000), 415–435.
  • S. Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc., 10 (1959), 621–626.
  • K. Suzuoka, On the topology of sphere type fold maps of $4$-manifolds into the plane, PhD Thesis, Univ. of Tokyo, March 2002.
  • C. T. C. Wall, Diffeomorphisms of $4$-manifolds, J. London Math. Soc., 39 (1964), 131–140.
  • J. H. C. Whitehead, The immersion of an open $3$-manifold in Euclidean $3$-space, Proc. London Math. Soc.,(3), 11 (1961), 81–90.
  • H. Whitney, On singularities of mappings of Euclidean spaces: I, mappings of the plane into the plane, Ann. of Math., 62 (1955), 374–410.
  • T. Yanagawa, On ribbon $2$-knots: the $3$-manifold bounded by the $2$-knots, Osaka J. Math., 6 (1969), 447–464.