Topological stability theorem for composite mappings

Isao Nakai

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 2, page 459-500
  • ISSN: 0373-0956

Abstract

top
We prove that generic convergent diagrams of proper smooth mappings are topologically stable. In proving global properties of diagrams we propose a generalization of the concept of singularity for diagrams, and we establish the geometry of composite mappings.

How to cite

top

Nakai, Isao. "Topological stability theorem for composite mappings." Annales de l'institut Fourier 39.2 (1989): 459-500. <http://eudml.org/doc/74838>.

@article{Nakai1989,
abstract = {We prove that generic convergent diagrams of proper smooth mappings are topologically stable. In proving global properties of diagrams we propose a generalization of the concept of singularity for diagrams, and we establish the geometry of composite mappings.},
author = {Nakai, Isao},
journal = {Annales de l'institut Fourier},
keywords = {critical sets; stratification; topological stability; maximal; trees; -stability},
language = {eng},
number = {2},
pages = {459-500},
publisher = {Association des Annales de l'Institut Fourier},
title = {Topological stability theorem for composite mappings},
url = {http://eudml.org/doc/74838},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Nakai, Isao
TI - Topological stability theorem for composite mappings
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 2
SP - 459
EP - 500
AB - We prove that generic convergent diagrams of proper smooth mappings are topologically stable. In proving global properties of diagrams we propose a generalization of the concept of singularity for diagrams, and we establish the geometry of composite mappings.
LA - eng
KW - critical sets; stratification; topological stability; maximal; trees; -stability
UR - http://eudml.org/doc/74838
ER -

References

top
  1. [A] V. I. ARNOLD, Evolution of wave fronts and equivariant Morse lemma, Comm. Pure Appl. Math., 29 (1976), 557-582. Zbl0343.58003MR55 #9148
  2. [Ba1] N. A. BAAS, Structural stability of composed mappings I-III, Preprint, Princeton, 1974. 
  3. [Ba2] N. A. BAAS, Hierarchical Systems, preprint, Univ. of Trondheim, 1976. 
  4. [Ba3] N. A. BAAS, On stability of composed mappings, preprint. 
  5. [Bu] M. A. BUCHNER, Stability of the cut locus in Dimension less than or Equal to 6, Invent. Math., Vol. 43-3 (1977), 199-233. Zbl0365.58010MR58 #2866
  6. [C] M. J. D. CARNEIRO, Singularities of envelopes of families of submanifolds in ℝN, Ann. Sc. Ec. Norm. Sup., 4e série, t. 1 (1983), 178-192. Zbl0525.58008MR85h:58023
  7. [Da] J. DAMON, Topological stability in the nice dimensions, Topology, 18 (1979), 129-142. Zbl0454.58003MR80h:58015
  8. [Du1] J.-P. DUFOUR, Sur la stabilité de diagrammes d'applications différentiables, Ann. Scient. Éc. Norm. Sup., 4e série, 10 (1977), 153-174. Zbl0354.58011
  9. [Du2] J.-P. DUFOUR, Triplets de fonctions et stabilité des enveloppes. C. R. Acad. Sci. Paris, Série I, t. 293 (16 nov. 1981), 509-512. Zbl0486.58005MR83j:58019
  10. [Du3] J.-P. DUFOUR, Familles de courbes planes différentiables, Topology, 22-4 (1983), 449-474. Zbl0521.58012MR84k:58034
  11. [Du4] J.-P. DUFOUR, Dynamique de multi-application du cercle, preprint. 
  12. [F] T. FUKUDA, Local topological properties of differentiable mappings, Inv. Math., 65 (1981), 227-250. Zbl0499.58008MR84e:58010
  13. [GG] M. GOLUBITSKY, V. GUILLEMIN, Stable mappings and their singularities, Graduate Text in Math. 14, Springer-Verlag. Zbl0294.58004MR49 #6269
  14. [Gi] C. GIBSON et al., Topological stability of smooth mappings, Lecture notes in Math. 552, Springer, Berlin, 1976. Zbl0377.58006MR55 #9151
  15. [L-T] D. T. LÊ, B. TEISSIER, Report on the problem session, Proceedings of Symposia in Pure Math., Vol. 40 (1983), Part. 2. Zbl0514.14001MR84k:32002
  16. [M1] J. N. MATHER, Stability of C∞-mappings : II. Infinitesimal stability implies stability, Ann. of Math., 89 (1969), 259-291. Zbl0177.26002MR41 #4582
  17. [M2] J. N. MATHER, Stability of C∞-mappings : V. Transversality. Advances in Mathematics, 192 (1971), 207-255. 
  18. [M3] J. N. MATHER, The nice dimensions, Springer Lecture notes in Math, 192 (1971), 207-253. Zbl0211.56105MR45 #2747
  19. [M4] J. N. MATHER, Stratification and mappings. Proc. Conference on Dynamical Systems (e.g. M. M. Peixoto, Academic Press, 1973), pp. 195-232. Zbl0286.58003MR51 #4306
  20. [N1] I. NAKAI, C∞-stability and the I-equivalence of diagrams of smooth mappings, Preprint Liverpool University, 1986. 
  21. [N2] I. NAKAI, Nice dimensions for the I0 equivalence of diagrams of map germs, Preprint Liverpool University, 1986. To appear in Pacific Journal of Math. Zbl0721.58008
  22. [P] A. DU PLESSIS, Genericity and smooth finite determinacy, pp. 295-312 in "Singularities", Proc. AMS Symp. in Pure Maths., Vol. 40 Part 1 (ed. P. Orlik), Amer Math. Soc. (1983). Zbl0523.58009MR85c:58016
  23. [Te] B. TEISSIER, The hunting of invariants in the geometry of discriminants, Real and complex singularities (ed. P. Holm, Sijthoff and Noordhoff, 1976), 556-677. 
  24. [To] J. TOUGERON, Idéaux des fonctions différentiables, Ergebnisse, Band 71, Springer-Verlag, 1972. Zbl0251.58001MR55 #13472
  25. [Th] R. THOM, Sur la théorie des enveloppes, J. Math. Pure et Appl., t. XL, fasc. 2 (1962). Zbl0105.16102MR25 #4454
  26. [W1] C. T. C. WALL, Stability, Pencils and Polytopes, Bull. London Math. Soc., 12 (1980), 401-421. Zbl0433.58006MR82h:58009
  27. [W2] C. T. C. WALL, Finite determinacy of smooth map germs, Bull. London Math. Soc., 13 (1981), 481-539. Zbl0451.58009MR83i:58020
  28. [W3] C. T. C. WALL, Determination of the semi-nice dimensions, Math. Proc. Cambridge Philoc. Soc., 97-1 (1983), 12, 79-88. Zbl0568.58008MR86c:58011

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.