Characteristic classes of subfoliations

Luis A. Cordero; X. Masa

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 2, page 61-86
  • ISSN: 0373-0956

Abstract

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This paper is devoted to define a characteristic homomorphism for a subfoliation ( F 1 , F 2 ) and to study its relation with the usual characteristic homomorphism for each foliation (as defined by Bott). Moreover, two applications are given: 1) the Yamato’s 2-codimensional foliation is shown to be no homotopic to F 2 in a (1,2)-codimensional subfoliation; 2) an obstruction to the existence of d everywhere independent and transverse infinitesimal transformations of a foliation F 2 is obtained, when F 2 and these transformations generated a new foliation F 1 .

How to cite

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Cordero, Luis A., and Masa, X.. "Characteristic classes of subfoliations." Annales de l'institut Fourier 31.2 (1981): 61-86. <http://eudml.org/doc/74498>.

@article{Cordero1981,
abstract = {This paper is devoted to define a characteristic homomorphism for a subfoliation $(F_1,F_2)$ and to study its relation with the usual characteristic homomorphism for each foliation (as defined by Bott). Moreover, two applications are given: 1) the Yamato’s 2-codimensional foliation is shown to be no homotopic to $F_2$ in a (1,2)-codimensional subfoliation; 2) an obstruction to the existence of $d$ everywhere independent and transverse infinitesimal transformations of a foliation $F_2$ is obtained, when $F_2$ and these transformations generated a new foliation $F_1$.},
author = {Cordero, Luis A., Masa, X.},
journal = {Annales de l'institut Fourier},
keywords = {characteristic classes of subfoliations; obstruction to the existence of d everywhere independent and transverse infinitesimal transformations of a foliation},
language = {eng},
number = {2},
pages = {61-86},
publisher = {Association des Annales de l'Institut Fourier},
title = {Characteristic classes of subfoliations},
url = {http://eudml.org/doc/74498},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Cordero, Luis A.
AU - Masa, X.
TI - Characteristic classes of subfoliations
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 2
SP - 61
EP - 86
AB - This paper is devoted to define a characteristic homomorphism for a subfoliation $(F_1,F_2)$ and to study its relation with the usual characteristic homomorphism for each foliation (as defined by Bott). Moreover, two applications are given: 1) the Yamato’s 2-codimensional foliation is shown to be no homotopic to $F_2$ in a (1,2)-codimensional subfoliation; 2) an obstruction to the existence of $d$ everywhere independent and transverse infinitesimal transformations of a foliation $F_2$ is obtained, when $F_2$ and these transformations generated a new foliation $F_1$.
LA - eng
KW - characteristic classes of subfoliations; obstruction to the existence of d everywhere independent and transverse infinitesimal transformations of a foliation
UR - http://eudml.org/doc/74498
ER -

References

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  1. [1] R. BOTT, Lectures on characteristic classes and foliations, Lectures on Algebraic and Differential Topology, Lecture Notes in Math., vol. 279, Springer-Verlag, Berlin and New York, 1972, pp. 1-94. Zbl0241.57010MR50 #14777
  2. [2] R. BOTT and A. HAEFLIGER, On characteristic classes of Г-foliations, Bull. Amer. Math. Soc., 78 (1972), 1039-1044. Zbl0262.57010MR46 #6370
  3. [3] L. A. CORDERO and P. M. GADEA, Exotic characteristic classes and subfoliations, Ann. Inst. Fourier, Grenoble, 26 (1976), 225-237; errata, ibid. 27, fasc. 4 (1977). Zbl0313.57010MR53 #6584
  4. [4] B. L. FEIGIN, Characteristic classes of flags of foliations. Functional Anal. Appl., 9 (1975), 312-317. Zbl0328.57008MR53 #6585
  5. [5] C. GODBILLON, Cohomologies d'algèbres de Lie de champs de vecteurs, Séminaire Bourbaki, 25e année (1971/1972), n° 421. Zbl0296.17010
  6. [6] A. HAEFLIGER, Sur les classes caractéristiques des feuilletages, Séminaire Bourbaki, 24e année (1971/1972), n° 412. Zbl0257.57011
  7. [7] G. HOCHSCHILD and J. P. SERRE, Cohomology of Lie algebras, Ann. of Math., 57 (1953), 591-603. Zbl0053.01402MR14,943c
  8. [8] F. KAMBER and Ph. TONDEUR, Foliated bundles and characteristic classes, Lecture Notes in Math., vol. 493, Springer-Verlag, Berlin and New York, 1975. Zbl0308.57011MR53 #6587
  9. [9] C. LAZAROV and H. SHULMAN, Obstructions to foliation-preserving Lie group actions, Topology, 18 (1979), 255-256. Zbl0413.57022MR80m:57020
  10. [10] C. LAZAROV and H. SHULMAN, Obstructions to foliation-preserving vector fields, To appear in J. Pure and Appl. Algebra. Zbl0523.57014
  11. [11] D. LEHMANN, Classes caractéristiques exotiques et J-connexité des espaces de connexions, Ann. Inst. Fourier, Grenoble, 24, 3 (1974), 267-306. Zbl0268.57009MR50 #14784
  12. [12] R. MOUSSU, Sur les classes exotiques des feuilletages. Géométrie Differentielle, Colloque, Santiago de Compostela, Espagne 1972, Lecture Notes in Math., vol. 392, Springer-Verlag, Berlin and New York, 1974, pp. 37-42. Zbl0292.57021MR50 #14785
  13. [13] H. SUZUKI, A property of a characteristic class of an orbit foliation, Transform. Groups, Proc. Conf. Univ. Newcastle 1976, London Math. Soc. Lecture Notes Series, vol. 26 (1977), Cambridge Univ. Press, Cambridge, pp. 190-203. Zbl0356.57013MR57 #13974
  14. [14] I. VAISMAN, Almost-multifoliate Riemannian manifolds, An. St. Univ. Iasi, 16 (1970), 97-103. Zbl0202.53003MR43 #1075
  15. [15] K. YAMATO, Examples of foliations with non-trivial exotic characteristic classes, Osaka J. Math., 12 (1975), 401-417. Zbl0318.57026MR53 #1607

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