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Finite jumps in Milnor number imply vanishing folds

Author: Donal B. O’Shea
Journal: Proc. Amer. Math. Soc. 87 (1983), 15-18
MSC: Primary 14B07; Secondary 32B30, 32G11
MathSciNet review: 677221
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Abstract: Let $ \left\{ {{X_t}} \right\}$ be a family of isolated hypersurface singularities in which the Milnor number is not constant. It is proved that there must be a vanishing fold centered at any $ t = {t_0}$ at which the Milnor number of the $ {X_t}$ changes discontinuously. This is much stronger than the condition that the Whitney conditions fail.

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  • [1] Joël Briançon and Jean-Paul Speder, La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 6, Aiii, A365–A367 (French, with English summary). MR 0425165
  • [2] J. Briançon and J.-P. Speder, Les conditions de Whitney impliquent “𝜇(*) constant”, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 2, xi, 153–163 (French, with English summary). MR 0419825
  • [3] François Bruhat and Henri Cartan, Sur la structure des sous-ensembles analytiques réels, C. R. Acad. Sci. Paris 244 (1957), 988–990 (French). MR 0086108
  • [4] Charles Ehresmann, Sur les espaces fibrés différentiables, C. R. Acad. Sci. Paris 224 (1947), 1611–1612 (French). MR 0020774
  • [5] Lê Dũng Tráng and C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), no. 1, 67–78. MR 0399088
  • [6] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
  • [7] -, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1969.
  • [8] D. O'Shea, Vanishing folds in $ \mu $-constant families, Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R. I. (to appear).
  • [9] Bernard Teissier, Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972) Soc. Math. France, Paris, 1973, pp. 285–362. Astérisque, Nos. 7 et 8 (French). MR 0374482

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Article copyright: © Copyright 1983 American Mathematical Society