In this paper we study a new structure, called a spinnable structure, on a differentiable manifold. Roughly speaking, a differentiable manifold is spinnable if it can spin around a codimension 2 submanifold, called the axis, as if the top spins.
The main result is the following: let be a compact -connected -dimensional differentiable manifold , then admits a spinnable structure with axis . Making use of the codimension-one foliation on , this yields that admits a codimension-foliation.
On étudie une structure nouvelle, dite spinnable, sur des variétés différentielles. On dit qu’une variété différentielle est spinnable si elle peut tourner autour d’une sous-variété de codimension 2 qui s’appelle l’axe, comme une toupie.
Le résultat principal de cet article est le suivant : soit une variété différentielle compacte, -connexe de dimension , du feuilletage de codimension 1 sur , on en déduit que admet une feuilletage de codimension 1.
@article{AIF_1973__23_2_197_0, author = {Tamura, Itiro}, title = {Foliations and spinnable structures on manifolds}, journal = {Annales de l'Institut Fourier}, pages = {197--214}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {23}, number = {2}, year = {1973}, doi = {10.5802/aif.468}, mrnumber = {50 #14788}, zbl = {0269.57012}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.468/} }
TY - JOUR AU - Tamura, Itiro TI - Foliations and spinnable structures on manifolds JO - Annales de l'Institut Fourier PY - 1973 SP - 197 EP - 214 VL - 23 IS - 2 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.468/ DO - 10.5802/aif.468 LA - en ID - AIF_1973__23_2_197_0 ER -
Tamura, Itiro. Foliations and spinnable structures on manifolds. Annales de l'Institut Fourier, Volume 23 (1973) no. 2, pp. 197-214. doi : 10.5802/aif.468. http://www.numdam.org/articles/10.5802/aif.468/
[1] Feuilletages de codimension 1 sur des variétés de dimension 5, C.R. Acad. Sci. Paris, 273 (1971), 603-604. | MR | Zbl
,[2] A lemma on systems of knotted curves, Proc. Nat. Acad. Sci., 9 (1923), 93-95. | JFM
,[3] Foliations of odd-dimensional spheres (to appear). | Zbl
,[4] Fibered knots and foliations of highly connected manifolds (to appear). | Zbl
and ,[5] Codimension 1 foliations on simply connected 5-manifolds (to appear). | Zbl
,[6] Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comm. Math. Helv., 32 (1958), 249-329. | MR | Zbl
,[7] Codimension-one foliations of spheres, Ann. of Math., 94 (1971), 494-503. | MR | Zbl
,[8] Groups of homotopy spheres I, Ann. of Math., 77 (1963), 504-537. | MR | Zbl
and ,[9] Remarks on codimension one foliations of spheres, J. Math. Soc. Japan, 24 (1972), 732-735. | MR | Zbl
,[10] Sur certaines propriétés topologiques des variétés feuilletées, Actualités Sci. Indust., No. 1183, Hermann, Paris, 1952. | MR | Zbl
,[11] On the structure of manifolds, Amer. J. Math., 84 (1962), 387-399. | MR | Zbl
,[12] Every odd dimensional homotopy sphere has a foliation of codimension one, Comm. Math. Helv., 47 (1972), (voir Comm. Math.). | MR | Zbl
,[13] Spinnable structures on differentiable manifolds, Proc. Japan Acad., 48 (1972), 293-296. | MR | Zbl
,[14] Manifolds as open books (to appear). | Zbl
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