On 2-cycles of B Diff (S 1 ) which are represented by foliated S 1 -bundles over T 2
Annales de l'Institut Fourier, Volume 31 (1981) no. 2, p. 1-59

We give several sufficients conditions for a 2-cycle of B Diff(S 1 ) d (resp. B Diff K (R) d ) represented by a foliated S 1 -(resp. R-) bundle over a 2-torus to be homologous to zero. Such a 2-cycle is determined by two commuting diffeomorphisms f, g of S 1 (resp. R). If f, g have fixed points, we construct decompositions: f=πf i , g=πg i , where the interiors of Supp(f i ) Supp(g i ) are disjoint, and f i and g i belong either to {h i n ;nZ} (h i Diff ) or to a one-parameter subgroup generated by a C 1 -vectorfield ξ i . Under some conditions on the norms of f i and g i our theorem says that the 2-cycle determined by f,g( Diff K (R)) is homologous to zero. In particular, if f and g belong to a one-parameter subgroup generated by a smooth vectorfield on R with compact support, our 2-cycle is homologous to zero. As a corollary to our theorem, every topological equivalence class of C 2 -foliated S 1 -bundles over T 2 has a C -foliation which is C -foliated cobordant to zero. To prove our theorem, we show that every element f of Diff K (R) is written as a product of commutators of elements whose supports are contained in Supp(f).

On donne des conditions suffisantes pour qu’un 2-cycle de B Diff(S 1 ) d (resp. B Diff K (R) d ) représenté par un espace S 1 -(resp. R-) fibré feuilleté sur un 2-tore soit homologue à zéro. Un tel cycle est déterminé par deux difféomorphismes f, g commutants de S 1 (resp. R). Si f, g sont des points fixes, ils se décomposent en f=πf i , g=πg i , où les intérieurs des Supp(f i ) Supp(g i ) sont disjoints et où f i et g i appartiennent ou bien à {h i n ;nZ} (h i Diff ) ou bien à un sous-groupe à un paramètre engendré par un champ de vecteurs ξ i de classe C 1 . Sous certaines conditions sur les normes de f i , g i notre théorème montre que le 2-cycle déterminé par f,g( Diff K (R)) est homologue à zéro. En particulier, si f et g appartiennent à un sous-groupe à un paramètre engendré par un champ de vecteurs C sur R de support compact, le 2-cycle est homologue à zéro. Comme corollaire à notre théorème, toute classe d’équivalence topologique d’espace S 1 -fibré C 2 -feuilleté sur T 2 contient un feuilletage C dont la classe de cobordisme C est nulle. Pour démontrer notre théorème, on démontre que tout élément f de Diff K (R) s’écrit comme un produit de commutateurs d’éléments dont les supports sont contenus dans Supp(f).

@article{AIF_1981__31_2_1_0,
     author = {Tsuboi, Takashi},
     title = {On 2-cycles of $B\textasciitilde {\rm Diff}(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {31},
     number = {2},
     year = {1981},
     pages = {1-59},
     doi = {10.5802/aif.828},
     zbl = {0439.57018},
     mrnumber = {84b:57019},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIF_1981__31_2_1_0/

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