Tokyo Journal of Mathematics

Whitney's Umbrellas in Stable Perturbations of a Map Germ $({\bf R}^n, 0)\to ({\bf R}^{2n- 1}, 0)$

Mariko OHSUMI

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Abstract

Let $f:({\bf R}^n, 0)\to ({\bf R}^p, 0)$ be a $C^{\infty}$ map-germ. We are interested in whether the number modulo 2 of stable singular points of codimension $n$ that appear near the origin in a generic perturbation of $f$ is a topological invariant. In this paper we concentrate on investigating the problem when $p$ is $2n- 1$, where stable singular points of codimension $n$ are only Whitney's umbrellas, and give a positive answer to the problem.

Article information

Source
Tokyo J. Math., Volume 29, Number 2 (2006), 475-493.

Dates
First available in Project Euclid: 1 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1170348180

Digital Object Identifier
doi:10.3836/tjm/1170348180

Mathematical Reviews number (MathSciNet)
MR2284985

Zentralblatt MATH identifier
1135.58016

Subjects
Primary: 32S30: Deformations of singularities; vanishing cycles [See also 14B07] 32S50: Topological aspects: Lefschetz theorems, topological classification, invariants 58C25: Differentiable maps
Secondary: 58K15: Topological properties of mappings 58K60: Deformation of singularities 58K65: Topological invariants

Citation

OHSUMI, Mariko. Whitney's Umbrellas in Stable Perturbations of a Map Germ $({\bf R}^n, 0)\to ({\bf R}^{2n- 1}, 0)$. Tokyo J. Math. 29 (2006), no. 2, 475--493. doi:10.3836/tjm/1170348180. https://projecteuclid.org/euclid.tjm/1170348180


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References

  • J. M. Boardman, Singularities of differentiable maps, Inst. Hautes \~Etudes Sci. Publ. Math. 33 (1967), 21–57.
  • T. Fukuda, Local Topological properties of differentiable mappings. I, Invent. Math. 65 (1981), 227–250.
  • T. Fukuda and G. Ishikawa, On the number of cusps of stable perturbations of a plane-to-plane singlarity, Tokyo J. Math. 10 (1987), 375–384.
  • T. Fukui, J. Nuño Ballesteros and M. Saia, Counting singularities in stable perturbations of map germs, Sūrikaisekikenkyūsho kōkyūroku 926 (1995), 1–20.
  • T. Fukui, J. Nuño Ballesteros and M. Saia, On the number of singlarities in generic deformations of map germs, J. London Math. Soc. (2) 58 (1998), 141–152.
  • T. Fukui and J. Weyman, Cohen-Macaulay properties of Thom-Boardman strata I, Morin's ideal, Proc. London Math. Soc. (3) 80 (2000), 257–303.
  • T. Fukui and J. Weyman, Cohen-Macaulay properties of Thom-Boardman strata II, The defining ideals of $\Sigma^{i, j}$, Proc. London Math. Soc. (3) 87 (2003), 137–163.
  • T. Gaffney and D. Mond, Cusps and double folds of germs of analytic maps ${\bf C}^2\to {\bf C}^2$, J. London Math. Soc. (2) 43 (1991), 185–192.
  • J. Mather, Stability of $C^{\infty}$-mappings I. The division theorem, Ann. of Math. 87 (1968), 89–104.
  • J. Mather, Stability of $C^{\infty}$-mappings II. Infinitesimal stability implies stability, Ann. of Math. 89 (1969), 254–291.
  • J. Mather, Stability of $C^{\infty}$-mappings III. Finitely determined map-germs, Publ. Math. Inst. Hautes Etudes Sci. 35 (1968), 127–156.
  • J. Mather, Stability of $C^{\infty}$-mappings IV. Classification of stable germs by ${\bf R}$-algebras, Publ. Math. Inst. Hautes Etudes Sci. 37 (1969), 223–248.
  • J. Mather, Stability of $C^{\infty}$-mappings V. Transversality, Advances in Math. 4 (1970), 301–336.
  • J. Mather, Stability of $C^{\infty}$-mappings VI. The nice dimensions, Springer Lecture Notes in Math. 192 (1971), 207–253.
  • J. Mather, Stratifications and mappings, Proceedings of the Dynamical Sytems Conference, Salvador, Academic Press (1971).
  • J. Mather, How to stratify mappings and jet spaces, Springer Lecture Notes in Math. 535 (1976), 128–176.
  • J. W. Milnor, Singular Points of Complex Hypersurfaces, Priceton University Press (1968).
  • D. M. Q. Mond, On the classification of germs of maps from ${\bf R}^2$ to ${\bf R}^3$, Proc. London Math. Soc. (3) 50 (1985), 333–369.
  • D. M. Q. Mond, Vanishing cycles for analytic maps, Singularity theory and its applications, Lecture Notes in Math. 1462, Springer (1991), 221–234.
  • B. Morin, Calcul jacobien, Ann. Sci. \~Ecole Norm. Sup. 8 (1975), 1–98.
  • J. Nuño Ballesteros and M. Saia, An invariant for map germs (preprint, 1995).
  • J. Nuño Ballesteros and M. Saia, `Multiplicity of Boardman strata and deformations of map germs', Glasgow Math. J. 40 (1998), 21–32.
  • T. Nishimura, Singular points and Mather's theory, Mathematical of singular points vol. 2 Singularities and bifurcation xPart I, Kyōritu Publisher (in Japanese) (2002).
  • R. Thom, Les singularités des applications différéntables, Ann. Inst. Fourier 6 (1955), 43–87.
  • R. Thom, Un lemme sur les applications différéntiables, Bol. Soc. Math. Mexic. 2nd series 1 (1956), 59–71.
  • C. T. C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), 481–539.
  • C. T. C. Wall, Topological invariance of the Milnor number mod $2$, Topology 22 (1983), 345–350.
  • H. Whitney, Differentiable manifolds, Ann. of Math. 37 (1936), 645–680.
  • H. Whitney, The general type of singularity of a set of $2n- 1$ smooth functions of $n$ variables, Duke Math. J. 10 (1943), 161–172.
  • H. Whitney, The self-intersections of a smooth $n$-manifolds in $2n$-space, Ann. of Math. 45 (1944), 220–246.
  • H. Whitney, The singularities of a smooth $n$-manifold in $(2n- 1)$-space, Ann. of Math. 45 (1944), 247–293.
  • H. Whitney, On singularities of mappings of Euclidean spaces I. Mappings of the plane into the plane, Ann. of Math. 62 (1955), 374–410.