# 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 (1981)

- Volume: 31, Issue: 2, page 1-59
- ISSN: 0373-0956

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topTsuboi, 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 31.2 (1981): 1-59. <http://eudml.org/doc/74497>.

@article{Tsuboi1981,

abstract = {We give several sufficients conditions for a 2-cycle of $B$ Diff$(S^1)_d$ (resp. $B$ Diff$_K(\{\bf R\})_d$) represented by a foliated $S^1$-(resp. $\{\bf R\}$-) 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. $\{\bf R\}$). If $f$, $g$ have fixed points, we construct decompositions: $f=\pi f_i$, $g = \pi g_i$, where the interiors of Supp$(f_i) \cup $ Supp$(g_i)$ are disjoint, and $f_i$ and $g_i$ belong either to $\lbrace h^n_i; n\in \{\bf Z\}\rbrace $ ($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 (\in $ Diff$^\infty _K(\{\bf R\}))$ is homologous to zero. In particular, if $f$ and $g$ belong to a one-parameter subgroup generated by a smooth vectorfield on $\{\bf R\}$ 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$^\infty _K(\{\bf R\})$ is written as a product of commutators of elements whose supports are contained in Supp$(f)$.},

author = {Tsuboi, Takashi},

journal = {Annales de l'institut Fourier},

keywords = {classifying space of the discrete group of diffeomorphisms of the circle; suspended foliations over the 2-torus; cobordance classes of foliations},

language = {eng},

number = {2},

pages = {1-59},

publisher = {Association des Annales de l'Institut Fourier},

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

url = {http://eudml.org/doc/74497},

volume = {31},

year = {1981},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 31

IS - 2

SP - 1

EP - 59

AB - We give several sufficients conditions for a 2-cycle of $B$ Diff$(S^1)_d$ (resp. $B$ Diff$_K({\bf R})_d$) represented by a foliated $S^1$-(resp. ${\bf R}$-) 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. ${\bf R}$). If $f$, $g$ have fixed points, we construct decompositions: $f=\pi f_i$, $g = \pi g_i$, where the interiors of Supp$(f_i) \cup $ Supp$(g_i)$ are disjoint, and $f_i$ and $g_i$ belong either to $\lbrace h^n_i; n\in {\bf Z}\rbrace $ ($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 (\in $ Diff$^\infty _K({\bf R}))$ is homologous to zero. In particular, if $f$ and $g$ belong to a one-parameter subgroup generated by a smooth vectorfield on ${\bf R}$ 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$^\infty _K({\bf R})$ is written as a product of commutators of elements whose supports are contained in Supp$(f)$.

LA - eng

KW - classifying space of the discrete group of diffeomorphisms of the circle; suspended foliations over the 2-torus; cobordance classes of foliations

UR - http://eudml.org/doc/74497

ER -

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