On 2-cycles of $B\phantom{\rule{3.33333pt}{0ex}}\mathrm{Diff}\left({S}^{1}\right)$ which are represented by foliated ${S}^{1}$-bundles over ${T}^{2}$
Annales de l'Institut Fourier, Volume 31 (1981) no. 2, pp. 1-59.

We give several sufficients conditions for a 2-cycle of $B$ Diff$\left({S}^{1}{\right)}_{d}$ (resp. $B$ Diff${}_{K}\left(\mathbf{R}{\right)}_{d}$) represented by a foliated ${S}^{1}$-(resp. $\mathbf{R}$-) 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. $\mathbf{R}$). If $f$, $g$ have fixed points, we construct decompositions: $f=\pi {f}_{i}$, $g=\pi {g}_{i}$, where the interiors of Supp$\left({f}_{i}\right)\cup$ Supp$\left({g}_{i}\right)$ are disjoint, and ${f}_{i}$ and ${g}_{i}$ belong either to $\left\{{h}_{i}^{n};n\in \mathbf{Z}\right\}$ (${h}_{i}\in$ Diff${}^{\infty }$) or to a one-parameter subgroup generated by a ${C}^{1}$-vectorfield ${\xi }_{i}$. Under some conditions on the norms of ${f}_{i}$ and ${g}_{i}$ our theorem says that the 2-cycle determined by $f,g\left(\in$ Diff${}_{K}^{\infty }\left(\mathbf{R}\right)\right)$ is homologous to zero. In particular, if $f$ and $g$ belong to a one-parameter subgroup generated by a smooth vectorfield on $\mathbf{R}$ 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 ${C}^{\infty }$-foliation which is ${C}^{\infty }$-foliated cobordant to zero. To prove our theorem, we show that every element $f$ of Diff${}_{K}^{\infty }\left(\mathbf{R}\right)$ is written as a product of commutators of elements whose supports are contained in Supp$\left(f\right)$.

On donne des conditions suffisantes pour qu’un 2-cycle de $B$ Diff$\left({S}^{1}{\right)}_{d}$ (resp. $B$ Diff${}_{K}\left(\mathbf{R}{\right)}_{d}$) représenté par un espace ${S}^{1}$-(resp. $\mathbf{R}$-) 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. $\mathbf{R}$). Si $f$, $g$ sont des points fixes, ils se décomposent en $f=\pi {f}_{i}$, $g=\pi {g}_{i}$, où les intérieurs des Supp$\left({f}_{i}\right)\cup$ Supp$\left({g}_{i}\right)$ sont disjoints et où ${f}_{i}$ et ${g}_{i}$ appartiennent ou bien à $\left\{{h}_{i}^{n};n\in \mathbf{Z}\right\}$ (${h}_{i}\in$ Diff${}^{\infty }$) ou bien à un sous-groupe à un paramètre engendré par un champ de vecteurs ${\xi }_{i}$ de classe ${C}^{1}$. Sous certaines conditions sur les normes de ${f}_{i}$, ${g}_{i}$ notre théorème montre que le 2-cycle déterminé par $f,g\phantom{\rule{4pt}{0ex}}\left(\in$ Diff${}_{K}^{\infty }\left(\mathbf{R}\right)\right)$ est homologue à zéro. En particulier, si $f$ et $g$ appartiennent à un sous-groupe à un paramètre engendré par un champ de vecteurs ${C}^{\infty }$ sur $\mathbf{R}$ 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 ${C}^{\infty }$ dont la classe de cobordisme ${C}^{\infty }$ est nulle. Pour démontrer notre théorème, on démontre que tout élément $f$ de Diff${}_{K}^{\infty }\left(\mathbf{R}\right)$ s’écrit comme un produit de commutateurs d’éléments dont les supports sont contenus dans Supp$\left(f\right)$.

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title = {On 2-cycles of $B~{\rm Diff}(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$},
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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://archive.numdam.org/articles/10.5802/aif.828/

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