On 2-cycles of which are represented by foliated -bundles over
Annales de l'institut Fourier (1981)
- Volume: 31, Issue: 2, page 1-59
- ISSN: 0373-0956
Access Full Article
topAbstract
topHow to cite
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 -
References
top- [1] K. FUKUI, A remark on the foliated cobordisms of codimension-one foliated 3-manifolds, J. Math. Kyoto Univ., 18-1 (1978), 189-197. Zbl0375.57010MR58 #7652
- [2] C. GODBILLON et J. VEY, Un invariant des feuilletages de codimension 1, C. R. Acad. Sci., Paris, 273 (1971), 92-95. Zbl0215.24604MR44 #1046
- [3] M. R. HERMAN, Sur le groupe des difféomorphismes du tore, Ann. de l'Inst. Fourier, 23, 2 (1973), 75-86. Zbl0269.58004MR52 #11988
- [4] M. R. HERMAN, The Godbillon-Vey invariant of foliations by planes of T3, Geometry and Topology, Rio de Janeiro, Springer Lecture Notes, 597 (1976), 294-307. Zbl0366.57006MR56 #9548
- [5] N. KOPELL, Commuting diffeomorphisms, Global Analysis, Symp. Pure Math., vol. XIV, A.M.S., (1970), 165-184. Zbl0225.57020MR42 #5285
- [6] S. MACLANE, Homology, Springer Verlag, New York (1963). Zbl0133.26502
- [7] J. N. MATHER, The vanishing of the homology of certain groups of homeomorphisms, Topology, vol. 10 (1971), 297-298. Zbl0207.21903MR44 #5973
- [8] J. N. MATHER, On Haefliger's classifying space. I, Bull. A.M.S., 77 (1971), 1111-1115. Zbl0224.55022MR44 #1047
- [9] J. N. MATHER, Integrability in codimension 1, Comment, Math. Helv., 48 (1973), 195-233. Zbl0284.57016MR50 #8556
- [10] J. N. MATHER, Commutators of diffeomorphisms, Comment, Math. Helv., 49 (1974), 512-528. Zbl0289.57014MR50 #8600
- [11] T. MIZUTANI, Foliated cobordisms of S3 and examples of foliated 4-manifolds, Topology, vol. 13 (1974), 353-362. Zbl0295.57012MR50 #11267
- [12] T. MIZUTANI, S. MORITA and T. TSUBOI, Foliated J-bundle and the Godbillon-Vey classes of codimension one foliations, to appear. Zbl0465.57010
- [13] T. MIZUTANI and T. TSUBOI, Foliations without holonomy and foliated bundles, Sci. Reports of the Saitama Univ., 9, 1 (1979), 45-55. Zbl0443.57019MR81a:57027
- [14] S. MORITA and T. TSUBOI, The Godbillon-Vey class of codimension one foliations without holonomy, Topology, 19 (1980), 43-49. Zbl0435.57006MR82d:57015
- [15] R. MOUSSU et R. ROUSSARIE, Relations de conjugaison et de cobordisme entre certains feuilletages, I.H.E.S. Publ. Math., 43 (1974), 143-168. Zbl0356.57018MR50 #11269
- [16] T. NISHIMORI, Compact leaves with abelian holonomy, Tôhoku Math. J., 27 (1975), 259-272. Zbl0324.57019MR51 #14097
- [17] T. NISHIMORI, SRH-decompositions of codimension-one foliations and the Godbillon-Vey class, Tôhoku Math. J., 32 (1980), 9-34. Zbl0413.57020MR81e:57030
- [18] G. OSHIKIRI, The surgery of codimension-one foliations, Tôhoku Math. J., 31 (1979), 63-70. Zbl0407.57022MR81c:57027
- [19] F. SERGERAERT, Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications, Ann. Sc. Ec. Norm. Sup., 4e série, t. 5 (1972), 599-660. Zbl0246.58006MR54 #6182
- [20] F. SERGERAERT, Feuilletages et difféomorphismes infiniment tangents à l'identité, Inventiones math., 39 (1977), 253-275. Zbl0327.58004MR57 #13973
- [21] F. SERGERAERT, BГ [d'après MATHER et THURSTON], Séminaire Bourbaki 30e année, 1977/1978, n° 524, Springer Lecture Notes 710. Zbl0434.57020
- [22] S. STERNBERG, Local Cn transformations of real line, Duke Math. J., 24 (1957), 97-102. Zbl0077.06201MR21 #1371
- [23] F. TAKENS, Normal forms for certain singularities of vector-fields, Ann. Inst. Fourier, 23, 2 (1973), 163-195. Zbl0266.34046MR51 #1872
- [24] W. THURSTON, Non-cobordant foliations of S3, Bull. A.M.S., 78 (1972), 511-514. Zbl0266.57004MR45 #7741
- [25] W. THURSTON, Foliations and groups of diffeomorphisms, Bull. A.M.S., 80 (1974), 304-307. Zbl0295.57014MR49 #4027
- [26] W. THURSTON, A local construction of foliations for three-manifolds, Proc. Symp. Pure Math., vol. 27 (1975), 315-319. Zbl0323.57014MR52 #1725
- [27] W. THURSTON, Existence of codimension-one foliations, Ann. of Math., 104 (1976), 249-268. Zbl0347.57014MR54 #13934
- [28] G. WALLET, Nullité de l'invariant de Godbillon-Vey d'un tore, C. R. Acad. Sc., Paris, t. 283 (1976), 821-823. Zbl0341.57018MR54 #11353
- [29] J. WOOD, Bundles with totally disconnected structure group, Comment, Math. Helv., 46 (1971), 257-273. Zbl0217.49202MR45 #2732
- [30] P. E. CONNER and E. E. FLOYD, Differentiable periodic maps, Springer-Verlag (1964). Zbl0125.40103MR31 #750
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.