no. 4
Constructing generic smooth maps of a manifold into a surface with prescribed singular loci
Annales de l'Institut Fourier, Tome 45 (1995) no. 4, pp. 1135-1162.

On sait que l’ensemble des points singuliers S(f) d’une application différentiable générique f:M→N d’une variété à n dimensions dans une surface est une sous-variété à une dimension de M et que S(f) a une stratification naturelle introduite par l’indice absolu. Dans cet article nous donnons une caractérisation complète des sous-variétés à une dimension (stratifiées) qui se présentent comme un ensemble de points singuliers d’une application générique, en termes de classes d’homologie représentées par les sous-variétés.

It is known that the singular set S(f) of a generic smooth map f:M→N of an n-dimensional manifold into a surface is a closed 1-dimensional submanifold of M and that it has a natural stratification induced by the absolute index. In this paper, we give a complete characterization of those 1-dimensional (stratified) submanifolds which arise as the singular set of a generic map in terms of the homology class they represent.

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     title = {Constructing generic smooth maps of a manifold into a surface with prescribed singular loci},
     journal = {Annales de l'Institut Fourier},
     pages = {1135--1162},
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Saeki, Osamu. Constructing generic smooth maps of a manifold into a surface with prescribed singular loci. Annales de l'Institut Fourier, Tome 45 (1995) no. 4, pp. 1135-1162. doi : 10.5802/aif.1489. https://aif.centre-mersenne.org/articles/10.5802/aif.1489/

[Bi]M. Bina-Motlagh, Sur certaines applications génériques de S3 dans le plan, travail de diplôme, Université de Lausanne, 1973.

[BdR]O. Burlet and G. De Rham, Sur certaines applications génériques d'une variété close à 3 dimensions dans le plan, l'Enseign. Math., 20 (1974), 275-292. | MR | Zbl

[C]E. Chincaro, Bifurcaçõoes de applicaçõ de Whitney, Tese, IMPA, 1978.

[E1]J.M. Eliašberg, On singularities of folding type, Math. USSR Izv., 4 (1970), 1119-1134. | Zbl

[E2]J.M. Elišberg, Surgery of singularities of smooth mappings, Math. USSR Izv., 6 (1972), 1302-1326. | Zbl

[L1]H.I. Levine, The singularities, Sq1, Illinois J. Math., 8 (1964), 152-168. | MR | Zbl

[L2]H.I. Levine, Singularities of differentiable mappings, Proceedings of Liverpool Singularities, Symposium I, Edited by C.T.C. Wall, Lecture Notes in Math. vol. 192, Springer-Verlag, Berlin, Heidelberg, New York, 1971, 1-89. | Zbl

[L3]H.I. Levine, Elimination of cusps, Topology, 3 (suppl. 2) (1965), 263-296. | MR | Zbl

[MPS]W. Motta, P. Porto and O. Saeki, Stable maps of 3-manifolds into the plane and their quotient spaces, Proc. London Math. Soc., 71 (1995), 158-174. | MR | Zbl

[Po] P. Porto, Informações topológicas sobre o domínio de certas applicacões estáveis, Tese de Livre-Docência, Universidade de São Paulo, 1983.

[PF]P. Porto and Y. Furuya, On special generic maps from a closed manifold into the plane, Topology Appl., 35 (1990), 41-52. | MR | Zbl

[S1]O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl., 49 (1993), 265-293. | MR | Zbl

[S2]O. Saeki, Simple stable maps of 3-manifolds into surfaces, to appear in Topology. | Zbl

[T]R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 43-87. | EuDML | Numdam | MR | Zbl

[W]H. Whitney, On singularities of mappings of Euclidean spaces I ; mappings of the plane into the plane, Ann. of Math., 62 (1955), 374-410. | MR | Zbl

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