Constructing generic smooth maps of a manifold into a surface with prescribed singular loci

Osamu Saeki

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 4, page 1135-1162
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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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 45.4 (1995): 1135-1162. <http://eudml.org/doc/75149>.

@article{Saeki1995,
abstract = {It is known that the singular set $S(f)$ of a generic smooth map $f : M \rightarrow N$ 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.},
author = {Saeki, Osamu},
journal = {Annales de l'institut Fourier},
keywords = {generic map; singular set; absolute index},
language = {eng},
number = {4},
pages = {1135-1162},
publisher = {Association des Annales de l'Institut Fourier},
title = {Constructing generic smooth maps of a manifold into a surface with prescribed singular loci},
url = {http://eudml.org/doc/75149},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Saeki, Osamu
TI - Constructing generic smooth maps of a manifold into a surface with prescribed singular loci
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 4
SP - 1135
EP - 1162
AB - It is known that the singular set $S(f)$ of a generic smooth map $f : M \rightarrow N$ 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.
LA - eng
KW - generic map; singular set; absolute index
UR - http://eudml.org/doc/75149
ER -

References

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  1. [Bi]M. BINA-MOTLAGH, Sur certaines applications génériques de S3 dans le plan, travail de diplôme, Université de Lausanne, 1973. 
  2. [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. Zbl0299.58005MR51 #1846
  3. [C]E. CHINCARO, Bifurcaçõoes de applicaçõ de Whitney, Tese, IMPA, 1978. 
  4. [E1]J.M. ELIAŠBERG, On singularities of folding type, Math. USSR Izv., 4 (1970), 1119-1134. Zbl0226.57012
  5. [E2]J.M. ELIŠBERG, Surgery of singularities of smooth mappings, Math. USSR Izv., 6 (1972), 1302-1326. Zbl0273.57014
  6. [L1]H.I. LEVINE, The singularities, Sq1, Illinois J. Math., 8 (1964), 152-168. Zbl0124.38801MR28 #2560
  7. [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. Zbl0216.45803
  8. [L3]H.I. LEVINE, Elimination of cusps, Topology, 3 (suppl. 2) (1965), 263-296. Zbl0146.20001MR31 #756
  9. [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. Zbl0845.57025MR96a:57067
  10. [Po] P. PORTO, Informações topológicas sobre o domínio de certas applicacões estáveis, Tese de Livre-Docência, Universidade de São Paulo, 1983. 
  11. [PF]P. PORTO and Y. FURUYA, On special generic maps from a closed manifold into the plane, Topology Appl., 35 (1990), 41-52. Zbl0715.57015MR91c:57034
  12. [S1]O. SAEKI, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl., 49 (1993), 265-293. Zbl0768.57015MR93m:57031
  13. [S2]O. SAEKI, Simple stable maps of 3-manifolds into surfaces, to appear in Topology. Zbl0864.57028
  14. [T]R. THOM, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 43-87. Zbl0075.32104MR19,310a
  15. [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. Zbl0068.37101MR17,518d

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