Rational homotopy of Serre fibrations

Jean-Claude Thomas

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 3, page 71-90
  • ISSN: 0373-0956

Abstract

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In rational homotopy theory, we show how the homotopy notion of pure fibration arises in a natural way. It can be proved that some fibrations, with homogeneous spaces as fibre are pure fibrations. Consequences of these results on the operation of a Lie group and the existence of Serre fibrations are given. We also examine various measures of rational triviality for a fibration and compare them with and whithout the hypothesis of pure fibration.

How to cite

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Thomas, Jean-Claude. "Rational homotopy of Serre fibrations." Annales de l'institut Fourier 31.3 (1981): 71-90. <http://eudml.org/doc/74508>.

@article{Thomas1981,
abstract = {In rational homotopy theory, we show how the homotopy notion of pure fibration arises in a natural way. It can be proved that some fibrations, with homogeneous spaces as fibre are pure fibrations. Consequences of these results on the operation of a Lie group and the existence of Serre fibrations are given. We also examine various measures of rational triviality for a fibration and compare them with and whithout the hypothesis of pure fibration.},
author = {Thomas, Jean-Claude},
journal = {Annales de l'institut Fourier},
keywords = {rational homotopy theory; pure fibration; fibrations with homogeneous spaces as fibre; existence of Serre fibrations; rational triviality for a fibration; minimal model},
language = {eng},
number = {3},
pages = {71-90},
publisher = {Association des Annales de l'Institut Fourier},
title = {Rational homotopy of Serre fibrations},
url = {http://eudml.org/doc/74508},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Thomas, Jean-Claude
TI - Rational homotopy of Serre fibrations
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 3
SP - 71
EP - 90
AB - In rational homotopy theory, we show how the homotopy notion of pure fibration arises in a natural way. It can be proved that some fibrations, with homogeneous spaces as fibre are pure fibrations. Consequences of these results on the operation of a Lie group and the existence of Serre fibrations are given. We also examine various measures of rational triviality for a fibration and compare them with and whithout the hypothesis of pure fibration.
LA - eng
KW - rational homotopy theory; pure fibration; fibrations with homogeneous spaces as fibre; existence of Serre fibrations; rational triviality for a fibration; minimal model
UR - http://eudml.org/doc/74508
ER -

References

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  1. [1] A. BOREL, Sur la cohomologie des espaces fibrés..., Ann. of Math., vol. 57, n° 1 (1953), 115-207. Zbl0052.40001MR14,490e
  2. [2] A. BOREL and J.P. SERRE, Impossibilité de fibrer... C.R.A.S., (1950), 2258 et 943-945. Zbl0039.19201
  3. [3] H.J. BAUES, J.M. LEMAIRE, Minimal model in homotopy theory, Math. Ann., 225 (1977), 219-242. Zbl0322.55019MR55 #4174
  4. [4] H. CARTAN, La transgression dans un groupe de Lie... Colloque de Topologie, Masson, Paris, (1951) 51-71. Zbl0045.30701
  5. [5] W. GREUB et al., Connection curvative and cohomology, Vol. III, Academic Press (1976). 
  6. [6] Y. FELIX, Classification homotopique des espaces rationnels à cohomogie fixée, Nouveaux Mémoires de la S.M.F., (1980). Zbl0447.55010
  7. [7] J.B. FRIEDLANDER and S. HALPERIN. — An arithmetic Characterization of the rational homotopy groups of certain spaces, Invent. Math., 53 (1979), 117-133. Zbl0396.55010MR81f:55006b
  8. [8] S. HALPERIN, Lecture notes on minimal models, Preprint 111, Université de Lille I, (1977). 
  9. [9] S. HALPERIN, Rational fibration, minimal models..., Trans of the A.M.S., vol. 244 (1978), 199-223. Zbl0387.55010MR58 #24264
  10. [10] S. HALPERIN, Finiteness in the minimal model of Sullivan, Trans. of the A.M.S., vol. 230 (1977), 173-199. Zbl0364.55014MR57 #1493
  11. [11] S. HALPERIN and J. STASHEFF, Obstruction to homotopy equivalences, Advances in Math., 32 (1979), 233-279. Zbl0408.55009MR80j:55016
  12. [12] J.L. KOSZUL, Homologie et cohomologie des algèbres de Lie, Bull. S.M.F., 78 (1950). Zbl0039.02901MR12,120g
  13. [13] J.M. LEMAIRE, "Autopsie d'un meurtre"..., Ann. Sc. E.N.S. 4ème série, t. 1.1 (1978), 93-100. Zbl0382.55011MR58 #18423
  14. [14] J. NEISENDORFER, Formal and coformal spaces, Illinois Journal of Mathematics, vol. 22, Number 4 (1978), 565-580. Zbl0396.55011MR58 #18429
  15. [15] J.P. SERRE, Homologie singulière des espaces fibrés, Ann. of Math., Vol. 54 (1951), 425-505. Zbl0045.26003MR13,574g
  16. [16] D. SULLIVAN, Infinitesimal computation in topology, Publi. de l'I.H.E.S., n° 47 (1977). Zbl0374.57002MR58 #31119
  17. [17] J.C. THOMAS, Homotopie rationnelle des fibrés de Serre, Thèse Université de Lille I (1980) et C.R.A.S., n° 290 (1980), 811-813. 

Citations in EuDML Documents

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  1. Aleksy Tralle, On compact symplectic and Kählerian solvmanifolds which are not completely solvable
  2. Toshihiro Yamaguchi, Formality of the function space of free maps into an elliptic space
  3. Anna Dumańska-Małyszko, Zofia Stępień, Aleksy Tralle, Generalized symmetric spaces and minimal models
  4. Wojciech Andrzejewski, Aleksy Tralle, Fat bundles and formality
  5. Gregory Lupton, Variations on a conjecture of Halperin
  6. H. Shiga, M. Tezuka, Rational fibrations homogeneous spaces with positive Euler characteristics and Jacobians

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