On the group of real analytic diffeomorphisms
[Sur le groupe des difféomorphismes analytiques réels]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 4, pp. 601-651.

Le groupe des difféomorphismes analytiques réels d’une variété analytique réelle est un groupe riche. Il est dense dans le groupe des difféomorphismes lisses. Herman a montré que, pour le tore de dimension n, sa composante connexe de l’identité est un groupe simple. Pour les variétés U(1) fibrées, pour les variétés admettant une action semi-libre spéciale de U(1), et pour les variétés de dimension 2 ou 3 admettant une action non-triviale de U(1), on montre que la composante de l’identité du groupe des difféomorphismes analytiques réels est un groupe parfait.

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the n-dimensional torus, its identity component is a simple group. For U(1) fibered manifolds, for manifolds admitting special semi-free U(1) actions and for 2- or 3-dimensional manifolds with nontrivial U(1) actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

DOI : 10.24033/asens.2104
Classification : 57R52, 57R50, 58A07, 58F18, 57R30, 53C12, 58C15, 37C05
Keywords: diffeomorphism groups, foliations, real analytic, rotations, $U(1)$ action, circle bundles
Mot clés : groupes de difféomorphismes, feuilletages, analytique réel, rotations, action de $U(1)$, fibrés en cercle
@article{ASENS_2009_4_42_4_601_0,
     author = {Tsuboi, Takashi},
     title = {On the group of real analytic diffeomorphisms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {601--651},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {4},
     year = {2009},
     doi = {10.24033/asens.2104},
     zbl = {1181.58009},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2104/}
}
TY  - JOUR
AU  - Tsuboi, Takashi
TI  - On the group of real analytic diffeomorphisms
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2009
SP  - 601
EP  - 651
VL  - 42
IS  - 4
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2104/
DO  - 10.24033/asens.2104
LA  - en
ID  - ASENS_2009_4_42_4_601_0
ER  - 
%0 Journal Article
%A Tsuboi, Takashi
%T On the group of real analytic diffeomorphisms
%J Annales scientifiques de l'École Normale Supérieure
%D 2009
%P 601-651
%V 42
%N 4
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/asens.2104/
%R 10.24033/asens.2104
%G en
%F ASENS_2009_4_42_4_601_0
Tsuboi, Takashi. On the group of real analytic diffeomorphisms. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 4, pp. 601-651. doi : 10.24033/asens.2104. http://www.numdam.org/articles/10.24033/asens.2104/

[1] V. I. ArnolʼD, Small denominators. I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 21-86, (= Amer. Math. Soc. Translations 46 (1965), 213-284). | MR | Zbl

[2] A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and its Applications 400, Kluwer Academic Publishers Group, 1997. | MR | Zbl

[3] H. Cartan, Idéaux de fonctions analytiques de n variables complexes, Ann. Sci. École Norm. Sup. 61 (1944), 149-197. | Numdam | MR | Zbl

[4] H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77-99. | Numdam | MR | Zbl

[5] D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math. 22 (1970), 165-173. | Numdam | MR | Zbl

[6] H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958), 460-472. | MR | Zbl

[7] H. Grauert & R. Remmert, Coherent analytic sheaves, Grund. Math. Wiss. 265, Springer, 1984. | MR | Zbl

[8] R. C. Gunning, Introduction to holomorphic functions of several variables. Vol. II, The Wadsworth & Brooks/Cole Mathematics Series, 1990. | MR | Zbl

[9] S. Haller & J. Teichmann, Smooth perfectness through decomposition of diffeomorphisms into fiber preserving ones, Ann. Global Anal. Geom. 23 (2003), 53-63. | MR | Zbl

[10] M.-R. Herman, Simplicité du groupe des difféomorphismes de classe C , isotopes à l’identité, du tore de dimension n, C. R. Acad. Sci. Paris 273 (1971), 232-234. | MR | Zbl

[11] M.-R. Herman, Sur le groupe des difféomorphismes 𝐑-analytiques du tore, in Differential topology and geometry (Proc. Colloq., Dijon, 1974), Springer Lecture Notes in Math. 484, 1975, 36-42. | MR | Zbl

[12] U. Hirsch, Some remarks on analytic foliations and analytic branched coverings, Math. Ann. 248 (1980), 139-152. | MR | Zbl

[13] J. Mather, On the homology of Haefliger's classifying space, C.I.M.E. Differential Topology (1976), 71-116. | Zbl

[14] C. B. Morrey Jr., The analytic embedding of abstract real-analytic manifolds, Ann. of Math. 68 (1958), 159-201. | MR | Zbl

[15] J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I and II, Ann. Scuola Norm. Sup. Pisa 20 (1966), 265-315 and 499-535. | Numdam | MR | Zbl

[16] P. Orlik & F. Raymond, Actions of SO (2) on 3-manifolds, in Proc. Conf. on Transformation Groups (New Orleans, La, 1967), Springer, 1968, 297-318. | MR | Zbl

[17] F. Raymond, Classification of the actions of the circle on 3-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51-78. | MR | Zbl

[18] H. L. Royden, The analytic approximation of differentiable mappings, Math. Ann. 139 (1960), 171-179. | MR | Zbl

[19] C. L. Siegel, Iteration of analytic functions, Ann. of Math. 43 (1942), 607-612. | MR | Zbl

[20] W. Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304-307. | MR | Zbl

Cité par Sources :