Smoothability of proper foliations

John Cantwell; Lawrence Conlon

Annales de l'institut Fourier (1988)

  • Volume: 38, Issue: 3, page 219-244
  • ISSN: 0373-0956

Abstract

top
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 .

How to cite

top

Cantwell, John, and Conlon, Lawrence. "Smoothability of proper foliations." Annales de l'institut Fourier 38.3 (1988): 219-244. <http://eudml.org/doc/74809>.

@article{Cantwell1988,
abstract = {Compact, $C^ 2$-foliated manifolds of codimension one, having all leaves proper, are shown to be $C^\{\infty \}$-smoothable. More precisely, such a foliated manifold is homeomorphic to one of class $C^\{\infty \}$. 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 $r$.},
author = {Cantwell, John, Conlon, Lawrence},
journal = {Annales de l'institut Fourier},
keywords = {-foliated manifolds of codimension one having all leaves proper; foliations with nonproper leaves},
language = {eng},
number = {3},
pages = {219-244},
publisher = {Association des Annales de l'Institut Fourier},
title = {Smoothability of proper foliations},
url = {http://eudml.org/doc/74809},
volume = {38},
year = {1988},
}

TY - JOUR
AU - Cantwell, John
AU - Conlon, Lawrence
TI - Smoothability of proper foliations
JO - Annales de l'institut Fourier
PY - 1988
PB - Association des Annales de l'Institut Fourier
VL - 38
IS - 3
SP - 219
EP - 244
AB - Compact, $C^ 2$-foliated manifolds of codimension one, having all leaves proper, are shown to be $C^{\infty }$-smoothable. More precisely, such a foliated manifold is homeomorphic to one of class $C^{\infty }$. 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 $r$.
LA - eng
KW - -foliated manifolds of codimension one having all leaves proper; foliations with nonproper leaves
UR - http://eudml.org/doc/74809
ER -

References

top
  1. [C.C1] J. CANTWELL and L. CONLON, Leaf prescriptions for closed 3-manifolds, Trans. Amer. Math. Soc., 236 (1978), 239-261. Zbl0398.57009MR58 #31105a
  2. [C.C2] J. CANTWELL and L. CONLON, Poincaré-Bendixson theory for leaves of codimension one, Trans. Amer. Math. Soc., 265 (1981), 181-209. Zbl0484.57015MR82f:57019
  3. [C.C3] J. CANTWELL and L. CONLON, Nonexponential leaves at finite level, Trans. Amer. Math. Soc., 269 (1982), 637-661. Zbl0487.57009MR84h:57013
  4. [C.C4] J. CANTWELL and L. CONLON, Smoothing fractional growth, Tôhoku Math. J., 33 (1981), 249-262. Zbl0477.57014MR83e:57022
  5. [De] A. DENJOY, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375. Zbl58.1124.04JFM58.1124.04
  6. [Di] P. DIPPOLITO, Codimension one foliations of closed manifolds, Ann. of Math., 107 (1978), 403-453. Zbl0418.57012MR58 #24288
  7. [E.M.S.] R. EDWARDS, K. MILLETT and D. SULLIVAN, Foliations with all leaves compact, Topology, 16 (1977), 13-32. Zbl0356.57022MR55 #11268
  8. [E] D.B.A. EPSTEIN, Periodic flows on 3-manifolds, Ann. of Math., 95 (1972), 68-92. Zbl0231.58009MR44 #5981
  9. [F] H. FURSTENBERG, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, N.J., 1981. Zbl0459.28023MR82j:28010
  10. [G] C. GODBILLON, Feuilletages, Études Géométriques II, Publ. Inst. de Recherche Math. Avancée, Univ. Louis Pasteur, Strasbourg, 1986. Zbl0724.58002
  11. [Hae] A. HAEFLIGER, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa, 16 (1962), 367-397. Zbl0122.40702MR32 #6487
  12. [Har1] J. HARRISON, Unsmoothable diffeomorphisms, Ann. of Math., 102 (1975), 83-94. Zbl0316.57018MR52 #9294
  13. [Har2] J. HARRISON, Unsmoothable diffeomorphisms on higher dimensional manifolds, Proc. Amer. Math. Soc., 73 (1979), 249-255. Zbl0405.57019MR80g:57045
  14. [H.H] G. HECTOR and U. HIRSCH, Introduction to the Geometry of Foliations, Part B, Vieweg, Braunschweig, 1983. Zbl0552.57001
  15. [I] T. INABA, On stability of proper leaves of codiménsion one foliations, J. Math. Soc. Japan, 29 (1977), 771-778. Zbl0356.57021MR58 #24291
  16. [M] K. MILLETT, Generic properties of proper foliations, I.H.E.S. preprint (1984). Zbl0674.57021
  17. [P] J.F. PLANTE, Foliations with measure preserving holonomy, Ann. of Math., 102 (1975), 327-361. Zbl0314.57018MR52 #11947
  18. [S.S.] R. SACKSTEDER and A.J. SCHWARTZ, Limit sets of foliations, Ann. Inst. Fourier, 15-2 (1965), 201-214. Zbl0136.20904MR32 #6489
  19. [T] T. TSUBOI, Examples of non-smoothable actions on the interval, Preprint (1986). Zbl0671.58018
  20. [W] J. WOOD, Foliations on 3-manifolds, Ann. of Math., 89 (1969), 336-358. Zbl0176.21402MR40 #2123

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.