On 2-cycles of $B~{\rm Diff}(S^1)$ which are represented by foliated $S^1$-bundles over $T^2$

Tsuboi, Takashi

On 2-cycles of

B Diff (S^{1})

which are represented by foliated

S^{1}

-bundles over

T^{2}

Tsuboi, Takashi

Annales de l'Institut Fourier, Tome 31 (1981), p. 1-59 / Harvested from Numdam

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Résumé
Abstract

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 $f = π f_{i}$ , $g = π g_{i}$ , où les intérieurs des Supp $(f_{i}) \cup$ Supp $(g_{i})$ sont disjoints et où $f_{i}$ et $g_{i}$ appartiennent ou bien à ${h_{i}^{n}; n \in Z}$ ( $h_{i} \in$ Diff $^{\infty}$ ) 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 $f, g (\in$ Diff $_{K}^{\infty} (R))$ est homologue à zéro. En particulier, si $f$ et $g$ appartiennent à un sous-groupe à un paramètre engendré par un champ de vecteurs $C^{\infty}$ 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 $C^{\infty}$ dont la classe de cobordisme $C^{\infty}$ est nulle. Pour démontrer notre théorème, on démontre que tout élément $f$ de Diff $_{K}^{\infty} (R)$ s’écrit comme un produit de commutateurs d’éléments dont les supports sont contenus dans Supp $(f)$ .

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: $f = π f_{i}$ , $g = π g_{i}$ , where the interiors of Supp $(f_{i}) \cup$ Supp $(g_{i})$ are disjoint, and $f_{i}$ and $g_{i}$ belong either to ${h_{i}^{n}; n \in Z}$ ( $h_{i} \in$ Diff $^{\infty}$ ) 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 $f, g (\in$ Diff $_{K}^{\infty} (R))$ 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 $C^{\infty}$ -foliation which is $C^{\infty}$ -foliated cobordant to zero. To prove our theorem, we show that every element $f$ of Diff $_{K}^{\infty} (R)$ is written as a product of commutators of elements whose supports are contained in Supp $(f)$ .

Publié le : 1981-01-01

MR 84b:57019 | Zbl 0439.57018

DOI : https://doi.org/10.5802/aif.828

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     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},
     volume = {31},
     year = {1981},
     pages = {1-59},
     doi = {10.5802/aif.828},
     mrnumber = {84b:57019},
     zbl = {0439.57018},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 31 (1981) pp. 1-59. doi : 10.5802/aif.828. http://gdmltest.u-ga.fr/item/AIF_1981__31_2_1_0/

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