Thom polynomials for open Whitney umbrellas of isotropic mappings

Toru Ohmoto

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 287-296
  • ISSN: 0137-6934

Abstract

top
A smooth mapping f : L n → ( M 2 n , 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 2 n , 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 ∞ .

How to cite

top

Ohmoto, Toru. "Thom polynomials for open Whitney umbrellas of isotropic mappings." Banach Center Publications 33.1 (1996): 287-296. <http://eudml.org/doc/262716>.

@article{Ohmoto1996,
abstract = {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 $C^\{∞\}$.},
author = {Ohmoto, Toru},
journal = {Banach Center Publications},
keywords = {isotropic mappings; Thom polynomials},
language = {eng},
number = {1},
pages = {287-296},
title = {Thom polynomials for open Whitney umbrellas of isotropic mappings},
url = {http://eudml.org/doc/262716},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Ohmoto, Toru
TI - Thom polynomials for open Whitney umbrellas of isotropic mappings
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 287
EP - 296
AB - 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 $C^{∞}$.
LA - eng
KW - isotropic mappings; Thom polynomials
UR - http://eudml.org/doc/262716
ER -

References

top
  1. [A] V. I. Arnol'd, Singularities of Caustics and Wave Fronts, Kluwer Acad. Publ., 1990. 
  2. [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. 
  3. [Au] M. Audin, Quelques remarques sur les surfaces lagrangiennes de Givental, J. Geom. Phys. 7 (1990), 583-598. Zbl0728.57020
  4. [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. 
  5. [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). 
  6. [GG] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Grad. Texts in Math. 14, Springer, Berlin, 1973. 
  7. [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. Zbl0761.58015
  8. [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. Zbl0798.58029
  9. [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. Zbl0831.57019
  10. [K] L. Kleiman, The enumerative geometry of singularities, in: Real and Complex Singularities, Sijthoff and Noordhoff, 1977, 297-396. 
  11. [MS] J. Milnor and J. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, N.J., and Univ. of Tokyo Press, Tokyo, 1974. 
  12. [P] I. R. Porteous, Simple singularities, in: Lecture Notes in Math. 192, Springer 1972, 286-307. 
  13. [R] F. Ronga, Le calcul des classes duals et singularités de Boardman d'ordre deux, Comment. Math. Helv. 47 (1972), 15-35. Zbl0236.58003
  14. [T] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 43-87. 
  15. [W] A. Weinstein, Lectures on Symplectic Manifolds, Regional Conf. Ser. in Math. 29, Amer. Math. Soc., 1977. 
  16. [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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.