We give several sufficients conditions for a 2-cycle of Diff (resp. Diff) represented by a foliated -(resp. -) bundle over a 2-torus to be homologous to zero. Such a 2-cycle is determined by two commuting diffeomorphisms , of (resp. ). If , have fixed points, we construct decompositions: , , where the interiors of Supp Supp are disjoint, and and belong either to ( Diff) or to a one-parameter subgroup generated by a -vectorfield . Under some conditions on the norms of and our theorem says that the 2-cycle determined by Diff is homologous to zero. In particular, if and belong to a one-parameter subgroup generated by a smooth vectorfield on with compact support, our 2-cycle is homologous to zero. As a corollary to our theorem, every topological equivalence class of -foliated -bundles over has a -foliation which is -foliated cobordant to zero. To prove our theorem, we show that every element of Diff is written as a product of commutators of elements whose supports are contained in Supp.
On donne des conditions suffisantes pour qu’un 2-cycle de Diff (resp. Diff) représenté par un espace -(resp. -) fibré feuilleté sur un 2-tore soit homologue à zéro. Un tel cycle est déterminé par deux difféomorphismes , commutants de (resp. ). Si , sont des points fixes, ils se décomposent en , , où les intérieurs des Supp Supp sont disjoints et où et appartiennent ou bien à ( Diff) ou bien à un sous-groupe à un paramètre engendré par un champ de vecteurs de classe . Sous certaines conditions sur les normes de , notre théorème montre que le 2-cycle déterminé par Diff est homologue à zéro. En particulier, si et appartiennent à un sous-groupe à un paramètre engendré par un champ de vecteurs sur de support compact, le 2-cycle est homologue à zéro. Comme corollaire à notre théorème, toute classe d’équivalence topologique d’espace -fibré -feuilleté sur contient un feuilletage dont la classe de cobordisme est nulle. Pour démontrer notre théorème, on démontre que tout élément de Diff s’écrit comme un produit de commutateurs d’éléments dont les supports sont contenus dans Supp.
@article{AIF_1981__31_2_1_0, author = {Tsuboi, Takashi}, title = {On 2-cycles of $B~{\rm Diff}(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$}, journal = {Annales de l'Institut Fourier}, pages = {1--59}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {31}, number = {2}, year = {1981}, doi = {10.5802/aif.828}, mrnumber = {84b:57019}, zbl = {0439.57018}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.828/} }
TY - JOUR AU - Tsuboi, Takashi TI - On 2-cycles of $B~{\rm Diff}(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$ JO - Annales de l'Institut Fourier PY - 1981 SP - 1 EP - 59 VL - 31 IS - 2 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.828/ DO - 10.5802/aif.828 LA - en ID - AIF_1981__31_2_1_0 ER -
%0 Journal Article %A Tsuboi, Takashi %T On 2-cycles of $B~{\rm Diff}(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$ %J Annales de l'Institut Fourier %D 1981 %P 1-59 %V 31 %N 2 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.828/ %R 10.5802/aif.828 %G en %F AIF_1981__31_2_1_0
Tsuboi, Takashi. On 2-cycles of $B~{\rm Diff}(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$. Annales de l'Institut Fourier, Volume 31 (1981) no. 2, pp. 1-59. doi : 10.5802/aif.828. http://www.numdam.org/articles/10.5802/aif.828/
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