no. 3
Smoothability of proper foliations
Annales de l'Institut Fourier, Tome 38 (1988) no. 3, pp. 219-244.

Il a été prouvé que toutes les variétés feuilletées compactes de classe C 2 , de codimension 1, dont toutes les feuilles sont propres, sont de classe C ∞ . Plus précisément, une telle variété feuilletée est homéomorphe à une variété de classe C ∞ . En d’autres termes, le résultat n’est pas vrai pour un feuilletage à feuilles non-propres. Dans ce cas précis, il y a une différence du point de vue topologique entre les classes C r et C r+1 , pour tout entier naturel r.

Compact, C 2 -foliated manifolds of codimension one, having all leaves proper, are shown to be C ∞ -smoothable. More precisely, such a foliated manifold is homeomorphic to one of class C ∞ . The corresponding statement is false for foliations with nonproper leaves. In that case, there are topological distinctions between smoothness of class C r and of class C r+1 for every nonnegative integer r.

@article{AIF_1988__38_3_219_0,
     author = {Cantwell, John and Conlon, Lawrence},
     title = {Smoothability of proper foliations},
     journal = {Annales de l'Institut Fourier},
     pages = {219--244},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {3},
     year = {1988},
     doi = {10.5802/aif.1146},
     mrnumber = {90f:57034},
     zbl = {0644.57013},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1146/}
}
TY  - JOUR
AU  - Cantwell, John
AU  - Conlon, Lawrence
TI  - Smoothability of proper foliations
JO  - Annales de l'Institut Fourier
PY  - 1988
SP  - 219
EP  - 244
VL  - 38
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1146/
DO  - 10.5802/aif.1146
LA  - en
ID  - AIF_1988__38_3_219_0
ER  - 
%0 Journal Article
%A Cantwell, John
%A Conlon, Lawrence
%T Smoothability of proper foliations
%J Annales de l'Institut Fourier
%D 1988
%P 219-244
%V 38
%N 3
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1146/
%R 10.5802/aif.1146
%G en
%F AIF_1988__38_3_219_0
Cantwell, John; Conlon, Lawrence. Smoothability of proper foliations. Annales de l'Institut Fourier, Tome 38 (1988) no. 3, pp. 219-244. doi : 10.5802/aif.1146. http://www.numdam.org/articles/10.5802/aif.1146/

[C.C1] J. Cantwell and L. Conlon, Leaf prescriptions for closed 3-manifolds, Trans. Amer. Math. Soc., 236 (1978), 239-261. | MR | Zbl

[C.C2] J. Cantwell and L. Conlon, Poincaré-Bendixson theory for leaves of codimension one, Trans. Amer. Math. Soc., 265 (1981), 181-209. | MR | Zbl

[C.C3] J. Cantwell and L. Conlon, Nonexponential leaves at finite level, Trans. Amer. Math. Soc., 269 (1982), 637-661. | MR | Zbl

[C.C4] J. Cantwell and L. Conlon, Smoothing fractional growth, Tôhoku Math. J., 33 (1981), 249-262. | MR | Zbl

[De] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375. | JFM | Numdam

[Di] P. Dippolito, Codimension one foliations of closed manifolds, Ann. of Math., 107 (1978), 403-453. | MR | Zbl

[E.M.S.] R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 13-32. | MR | Zbl

[E] D.B.A. Epstein, Periodic flows on 3-manifolds, Ann. of Math., 95 (1972), 68-92. | MR | Zbl

[F] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, N.J., 1981. | MR | Zbl

[G] C. Godbillon, Feuilletages, Études Géométriques II, Publ. Inst. de Recherche Math. Avancée, Univ. Louis Pasteur, Strasbourg, 1986. | Zbl

[Hae] A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa, 16 (1962), 367-397. | Numdam | MR | Zbl

[Har1] J. Harrison, Unsmoothable diffeomorphisms, Ann. of Math., 102 (1975), 83-94. | MR | Zbl

[Har2] J. Harrison, Unsmoothable diffeomorphisms on higher dimensional manifolds, Proc. Amer. Math. Soc., 73 (1979), 249-255. | MR | Zbl

[H.H] G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Part B, Vieweg, Braunschweig, 1983. | Zbl

[I] T. Inaba, On stability of proper leaves of codiménsion one foliations, J. Math. Soc. Japan, 29 (1977), 771-778. | MR | Zbl

[M] K. Millett, Generic properties of proper foliations, I.H.E.S. preprint (1984).

[P] J.F. Plante, Foliations with measure preserving holonomy, Ann. of Math., 102 (1975), 327-361. | MR | Zbl

[S.S.] R. Sacksteder and A.J. Schwartz, Limit sets of foliations, Ann. Inst. Fourier, 15-2 (1965), 201-214. | Numdam | MR | Zbl

[T] T. Tsuboi, Examples of non-smoothable actions on the interval, Preprint (1986). | Zbl

[W] J. Wood, Foliations on 3-manifolds, Ann. of Math., 89 (1969), 336-358. | MR | Zbl

Cité par Sources :