no. 2
Topological stability theorem for composite mappings
Annales de l'Institut Fourier, Tome 39 (1989) no. 2, pp. 459-500.

Nous démontrons que les diagrammes convergents génériques des applications différentiables sont topologique stables. En démontrant quelques propriétés globales des diagrammes, nous proposons une généralisation du concept de la singularité pour diagrammes, et nous montrons la géométrie des applications composées.

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.

@article{AIF_1989__39_2_459_0,
     author = {Nakai, Isao},
     title = {Topological stability theorem for composite mappings},
     journal = {Annales de l'Institut Fourier},
     pages = {459--500},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {2},
     year = {1989},
     doi = {10.5802/aif.1174},
     mrnumber = {91e:58020},
     zbl = {0673.58025},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1174/}
}
TY  - JOUR
AU  - Nakai, Isao
TI  - Topological stability theorem for composite mappings
JO  - Annales de l'Institut Fourier
PY  - 1989
SP  - 459
EP  - 500
VL  - 39
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1174/
DO  - 10.5802/aif.1174
LA  - en
ID  - AIF_1989__39_2_459_0
ER  - 
%0 Journal Article
%A Nakai, Isao
%T Topological stability theorem for composite mappings
%J Annales de l'Institut Fourier
%D 1989
%P 459-500
%V 39
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1174/
%R 10.5802/aif.1174
%G en
%F AIF_1989__39_2_459_0
Nakai, Isao. Topological stability theorem for composite mappings. Annales de l'Institut Fourier, Tome 39 (1989) no. 2, pp. 459-500. doi : 10.5802/aif.1174. http://www.numdam.org/articles/10.5802/aif.1174/

[A] V. I. Arnold, Evolution of wave fronts and equivariant Morse lemma, Comm. Pure Appl. Math., 29 (1976), 557-582. | MR | Zbl

[Ba1] N. A. Baas, Structural stability of composed mappings I-III, Preprint, Princeton, 1974.

[Ba2] N. A. Baas, Hierarchical Systems, preprint, Univ. of Trondheim, 1976.

[Ba3] N. A. Baas, On stability of composed mappings, preprint.

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[Da] J. Damon, Topological stability in the nice dimensions, Topology, 18 (1979), 129-142. | MR | Zbl

[Du1] J.-P. Dufour, Sur la stabilité de diagrammes d'applications différentiables, Ann. Scient. Éc. Norm. Sup., 4e série, 10 (1977), 153-174. | Numdam | Zbl

[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. | MR | Zbl

[Du3] J.-P. Dufour, Familles de courbes planes différentiables, Topology, 22-4 (1983), 449-474. | MR | Zbl

[Du4] J.-P. Dufour, Dynamique de multi-application du cercle, preprint.

[F] T. Fukuda, Local topological properties of differentiable mappings, Inv. Math., 65 (1981), 227-250. | MR | Zbl

[GG] M. Golubitsky, V. Guillemin, Stable mappings and their singularities, Graduate Text in Math. 14, Springer-Verlag. | MR | Zbl

[Gi] C. Gibson et al., Topological stability of smooth mappings, Lecture notes in Math. 552, Springer, Berlin, 1976. | MR | Zbl

[L-T] D. T. Lê, B. Teissier, Report on the problem session, Proceedings of Symposia in Pure Math., Vol. 40 (1983), Part. 2. | MR | Zbl

[M1] J. N. Mather, Stability of C∞-mappings : II. Infinitesimal stability implies stability, Ann. of Math., 89 (1969), 259-291. | MR | Zbl

[M2] J. N. Mather, Stability of C∞-mappings : V. Transversality. Advances in Mathematics, 192 (1971), 207-255.

[M3] J. N. Mather, The nice dimensions, Springer Lecture notes in Math, 192 (1971), 207-253. | MR | Zbl

[M4] J. N. Mather, Stratification and mappings. Proc. Conference on Dynamical Systems (e.g. M. M. Peixoto, Academic Press, 1973), pp. 195-232. | MR | Zbl

[N1] I. Nakai, C∞-stability and the I-equivalence of diagrams of smooth mappings, Preprint Liverpool University, 1986.

[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. | Zbl

[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). | MR | Zbl

[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.

[To] J. Tougeron, Idéaux des fonctions différentiables, Ergebnisse, Band 71, Springer-Verlag, 1972. | MR | Zbl

[Th] R. Thom, Sur la théorie des enveloppes, J. Math. Pure et Appl., t. XL, fasc. 2 (1962). | MR | Zbl

[W1] C. T. C. Wall, Stability, Pencils and Polytopes, Bull. London Math. Soc., 12 (1980), 401-421. | MR | Zbl

[W2] C. T. C. Wall, Finite determinacy of smooth map germs, Bull. London Math. Soc., 13 (1981), 481-539. | MR | Zbl

[W3] C. T. C. Wall, Determination of the semi-nice dimensions, Math. Proc. Cambridge Philoc. Soc., 97-1 (1983), 12, 79-88. | MR | Zbl

Cité par Sources :