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1996 | 33 | 1 | 287-296
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Thom polynomials for open Whitney umbrellas of isotropic mappings

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A smooth mapping $f:L^n → (M^{2n},ω)$ of a smooth n-dimensional manifold L into a smooth 2n-dimensional symplectic manifold (M,ω) is called isotropic if f*ω vanishes. In the last ten years, the local theory of singularities of isotropic mappings has been rapidly developed by Arnol'd, Givental' and several authors, while it seems that the global theory of their singularities has not been well studied except for the work of Givental' [G1] in the case of dimension 2 (cf. [A], [Au], [I2], [I-O]). In the present paper, we are concerned with typical singularities with corank 1 of isotropic maps $f:L^n → (M^{2n},ω)$ (arbitrary dimension n), so-called open Whitney umbrellas of higher order, investigated by Givental' [G2], Ishikawa [I1] and Zakalyukin [Z], and our purpose is to give their topological invariants from the viewpoint of "Thom polynomial theory" (cf. [T], [P], [K], [AVGL]). These are obtained as a variant of Porteous' formulae on Thom polynomials for $A_k$-singularities [P]. Throughout this paper, manifolds are assumed to be paracompact Hausdorff spaces and of class $C^{∞}$, and maps are also of class $C^{∞}$.
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  • Department of Mathematics, Kagoshima University, Kagoshima, 890 Japan
  • [A] V. I. Arnol'd, Singularities of Caustics and Wave Fronts, Kluwer Acad. Publ., 1990.
  • [AVGL] V. I. Arnol'd, V. A. Vasil'ev, V. V. Goryunov and O. P. Lyashko, Dynamical Systems VI, Singularity Theory I, Encyclopedia Math. Sci. 6, Springer, 1993.
  • [Au] M. Audin, Quelques remarques sur les surfaces lagrangiennes de Givental, J. Geom. Phys. 7 (1990), 583-598.
  • [G1] A. B. Givental', Lagrange imbeddings of surfaces and unfolded Whitney umbrella, Funktsional. Anal. Prilozhen. 20 (3) (1986), 35-41 (in Russian); English transl.: Funct. Anal. Appl. 20 (1986), 197-203.
  • [G2] A. B. Givental', Singular Lagrange varieties and their Lagrangian mappings, Itogi Nauki Tekh., Sovrem. Probl. Mat. 33, VINITI, Moscow, 1988 (in Russian); English transl.: J. Soviet Math. 52 (1990).
  • [GG] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Grad. Texts in Math. 14, Springer, Berlin, 1973.
  • [I1] G. Ishikawa, The local model of an isotropic map-germ arising from one-dimensional symplectic reduction, Math. Proc. Cambridge Philos. Soc. 111 (1992), 103-112.
  • [I2] G. Ishikawa, Maslov class of an isotropic map-germ arising from one dimensional symplectic reduction, in: Recent Developments in Differential Geometry, Adv. Stud. Pure Math. 22, Kinokuniya, 1993, 53-68.
  • [IO] G. Ishikawa and T. Ohmoto, Local invariants of singular surfaces in an almost complex four-manifold, Ann. Global Anal. Geom. 11 (1993), 125-133.
  • [K] L. Kleiman, The enumerative geometry of singularities, in: Real and Complex Singularities, Sijthoff and Noordhoff, 1977, 297-396.
  • [MS] J. Milnor and J. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, N.J., and Univ. of Tokyo Press, Tokyo, 1974.
  • [P] I. R. Porteous, Simple singularities, in: Lecture Notes in Math. 192, Springer 1972, 286-307.
  • [R] F. Ronga, Le calcul des classes duals et singularités de Boardman d'ordre deux, Comment. Math. Helv. 47 (1972), 15-35.
  • [T] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 43-87.
  • [W] A. Weinstein, Lectures on Symplectic Manifolds, Regional Conf. Ser. in Math. 29, Amer. Math. Soc., 1977.
  • [Z] V. M. Zakalyukin, Generating ideals of singular Lagrange varieties, in: Theory of Singularities and its Applications, V. I. Arnol'd (ed.), Adv. Soviet Math. 1 (1990), 201-210.
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