Abstract
It is shown that for any smooth foliated manifold the identity component of the group of all leaf preserving diffeomorphisms is perfect. This result generalizes in a sense a well-known theorem of Thurston.
Similar content being viewed by others
References
Abe, K., Fukui, K.: On commutators of equivariant diffeomorphisms. Proc. Japan Acad.54, Ser. A 52–54 (1978).
Banyaga, A.: Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. Comment. Math. Helv.53, 174–227 (1978).
Banyaga, A., de la Llave, R., Wayne, C. E.: Cohomology equations and commutators of germs of contact diffeomorphisms. Trans. Amer. Math. Soc.312, 755–778 (1989).
Epstein, D. B. A.: The simplicity of certain groups of homeomorphisms. Comp. Math.22, 165–173 (1970).
Epstein, D. B. A.: Commutators ofC ∞-diffeomorphisms. Appendix to [12]. Comment. Math. Helv.59, 111–122 (1984).
Fukui, K.: Homologies of the group Diff∞ (R n, 0) and its subgroups. J. Math. Kyoto Univ.20, 475–487 (1980).
Herman, M. R.: Sur le groupe des difféomorphismes du tore. Ann. Inst. Fourier (Grenoble)23, 75–86 (1973).
Kan, D.: A combinatorial definition of homotopy groups. Ann. Math.67, 282–317 (1958).
Lichnerowicz, A.: Remarques sur deux théorièmes de Banyaga. C. R. Acad. Sci. Paris, Série A287, 1121–1124 (1978).
May, J. P.: Simplicial Objects in Algebraic Topology. Princeton: Van Nostrand. 1967.
Mather, J. N.: Commutators of diffeomorphisms. Comment. Math. Helv. I49, 512–528 (1974); II50, 33–40 (1975); III60, 122–124 (1985).
Mather, J. N.: A curious remark concerning the geometric transfer map. Comment. Math. Helv.59, 86–110 (1984).
Masson, A.: Sur la perfection du groupe des difféomorphisms d'une variété à bord infinitement tangents à l'identité su le bord. C. R. Acad. Sci. Paris, Série A285, 837–839 (1977).
Moser, J.: A rapidly convergent iteration method and non-linear differential equations, II. Ann. Scuola Norm. Sup. Pisa20, 499–535 (1966).
Palis, J., Smale, S.: Structural stability theorems. Proc. Symp. in Pure math.14, 223–231 Amer. Math. Soc. 1970.
Rybicki, T.: Isomorphisms between groups of diffeomorphisms. Proc. Amer. Math. Soc.123, 1, 303–310 (1995).
Sadowski, M.: The proof of theorem of Thurston. University of Gdańsk, August 1980 (in Polish). Preprint.
Sergeraert, F.: Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications. Ann. Scient. Ec. Norm. Sup. 4 série5, 599–660 (1972).
Sergeraert, F.: Feuilletages et difféomorphismes infiniment tangents à l'identité. Invent. Math.39, 253–275 (1977).
Sternberg, S.: On the structure of local homeomorphisms, III. Amer. J. Math.81, 578–604 (1959).
Thurston, W.: On the structure of volume preserving diffeomorphisms. Unpublished Ms. (1973).
Thurston, W.: Foliations and groups of diffeomorphisms. Bull. Amer. Math. Soc.80, 304–307 (1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rybicki, T. The identity component of the leaf preserving diffeomorphism group is perfect. Monatshefte für Mathematik 120, 289–305 (1995). https://doi.org/10.1007/BF01294862
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01294862