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

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.

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.

@article{AIF_1989__39_2_459_0,
     author = {Nakai, Isao},
     title = {Topological stability theorem for composite mappings},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {39},
     number = {2},
     year = {1989},
     pages = {459-500},
     doi = {10.5802/aif.1174},
     zbl = {0673.58025},
     mrnumber = {91e:58020},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_2_459_0}
}
Nakai, Isao. Topological stability theorem for composite mappings. Annales de l'Institut Fourier, Volume 39 (1989) no. 2, pp. 459-500. doi : 10.5802/aif.1174. http://www.numdam.org/item/AIF_1989__39_2_459_0/

[A] V. I. Arnold, Evolution of wave fronts and equivariant Morse lemma, Comm. Pure Appl. Math., 29 (1976), 557-582. | MR 55 #9148 | Zbl 0343.58003

[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 58 #2866 | Zbl 0365.58010

[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 85h:58023 | Zbl 0525.58008

[Da] J. Damon, Topological stability in the nice dimensions, Topology, 18 (1979), 129-142. | MR 80h:58015 | Zbl 0454.58003

[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 0354.58011

[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 83j:58019 | Zbl 0486.58005

[Du3] J.-P. Dufour, Familles de courbes planes différentiables, Topology, 22-4 (1983), 449-474. | MR 84k:58034 | Zbl 0521.58012

[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 84e:58010 | Zbl 0499.58008

[GG] M. Golubitsky, V. Guillemin, Stable mappings and their singularities, Graduate Text in Math. 14, Springer-Verlag. | MR 49 #6269 | Zbl 0294.58004

[Gi] C. Gibson et al., Topological stability of smooth mappings, Lecture notes in Math. 552, Springer, Berlin, 1976. | MR 55 #9151 | Zbl 0377.58006

[L-T] D. T. Lê, B. Teissier, Report on the problem session, Proceedings of Symposia in Pure Math., Vol. 40 (1983), Part. 2. | MR 84k:32002 | Zbl 0514.14001

[M1] J. N. Mather, Stability of C∞-mappings : II. Infinitesimal stability implies stability, Ann. of Math., 89 (1969), 259-291. | MR 41 #4582 | Zbl 0177.26002

[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 45 #2747 | Zbl 0211.56105

[M4] J. N. Mather, Stratification and mappings. Proc. Conference on Dynamical Systems (e.g. M. M. Peixoto, Academic Press, 1973), pp. 195-232. | MR 51 #4306 | Zbl 0286.58003

[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 0721.58008

[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 85c:58016 | Zbl 0523.58009

[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 55 #13472 | Zbl 0251.58001

[Th] R. Thom, Sur la théorie des enveloppes, J. Math. Pure et Appl., t. XL, fasc. 2 (1962). | MR 25 #4454 | Zbl 0105.16102

[W1] C. T. C. Wall, Stability, Pencils and Polytopes, Bull. London Math. Soc., 12 (1980), 401-421. | MR 82h:58009 | Zbl 0433.58006

[W2] C. T. C. Wall, Finite determinacy of smooth map germs, Bull. London Math. Soc., 13 (1981), 481-539. | MR 83i:58020 | Zbl 0451.58009

[W3] C. T. C. Wall, Determination of the semi-nice dimensions, Math. Proc. Cambridge Philoc. Soc., 97-1 (1983), 12, 79-88. | MR 86c:58011 | Zbl 0568.58008