Foliations and spinnable structures on manifolds

Itiro Tamura

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 2, page 197-214
  • ISSN: 0373-0956

Abstract

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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 M be a compact ( n - 1 ) -connected ( 2 n + 1 ) -dimensional differentiable manifold ( n ≥ 3 ) , then M admits a spinnable structure with axis S 2 n + 1 . Making use of the codimension-one foliation on S 2 n + 1 , this yields that M admits a codimension-foliation.

How to cite

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Tamura, Itiro. "Foliations and spinnable structures on manifolds." Annales de l'institut Fourier 23.2 (1973): 197-214. <http://eudml.org/doc/74123>.

@article{Tamura1973,
abstract = {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 $M$ be a compact $(n-1)$-connected $(2n+1)$-dimensional differentiable manifold $(n\ge 3)$, then $M$ admits a spinnable structure with axis $S^\{2n+1\}$. Making use of the codimension-one foliation on $S^\{2n+1\}$, this yields that $M$ admits a codimension-foliation.},
author = {Tamura, Itiro},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {197-214},
publisher = {Association des Annales de l'Institut Fourier},
title = {Foliations and spinnable structures on manifolds},
url = {http://eudml.org/doc/74123},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Tamura, Itiro
TI - Foliations and spinnable structures on manifolds
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 2
SP - 197
EP - 214
AB - 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 $M$ be a compact $(n-1)$-connected $(2n+1)$-dimensional differentiable manifold $(n\ge 3)$, then $M$ admits a spinnable structure with axis $S^{2n+1}$. Making use of the codimension-one foliation on $S^{2n+1}$, this yields that $M$ admits a codimension-foliation.
LA - eng
UR - http://eudml.org/doc/74123
ER -

References

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  1. [1] N. A'CAMPO, Feuilletages de codimension 1 sur des variétés de dimension 5, C.R. Acad. Sci. Paris, 273 (1971), 603-604. Zbl0221.57009MR44 #4772
  2. [2] J. W. ALEXANDER, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci., 9 (1923), 93-95. JFM49.0408.03
  3. [3] A. H. DURFEE, Foliations of odd-dimensional spheres (to appear). Zbl0231.57016
  4. [4] A. H. DURFEE and H. B. LAWSON, Fibered knots and foliations of highly connected manifolds (to appear). Zbl0231.57015
  5. [5] K. FUKUI, Codimension 1 foliations on simply connected 5-manifolds (to appear). Zbl0271.57007
  6. [6] A. HAEFLIGER, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comm. Math. Helv., 32 (1958), 249-329. Zbl0085.17303MR20 #6702
  7. [7] H. B. LAWSON, Codimension-one foliations of spheres, Ann. of Math., 94 (1971), 494-503. Zbl0236.57014MR44 #4774
  8. [8] J. MILNOR and M. KERVAIRE, Groups of homotopy spheres I, Ann. of Math., 77 (1963), 504-537. Zbl0115.40505MR26 #5584
  9. [9] T. MIZUTANI, Remarks on codimension one foliations of spheres, J. Math. Soc. Japan, 24 (1972), 732-735. Zbl0238.57014MR46 #2689
  10. [10] G. REEB, Sur certaines propriétés topologiques des variétés feuilletées, Actualités Sci. Indust., No. 1183, Hermann, Paris, 1952. Zbl0049.12602MR14,1113a
  11. [11] S. SMALE, On the structure of manifolds, Amer. J. Math., 84 (1962), 387-399. Zbl0109.41103MR27 #2991
  12. [12] I. TAMURA, Every odd dimensional homotopy sphere has a foliation of codimension one, Comm. Math. Helv., 47 (1972), (voir Comm. Math.). Zbl0249.57013MR47 #5887
  13. [13] I. TAMURA, Spinnable structures on differentiable manifolds, Proc. Japan Acad., 48 (1972), 293-296. Zbl0252.57009MR47 #7756
  14. [14] H. E. WINKELNKEMPER, Manifolds as open books (to appear). Zbl0269.57011

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