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On the genus of elliptic fibrations

Author: J.-B. Gatsinzi
Journal: Proc. Amer. Math. Soc. 132 (2004), 597-606
MSC (2000): Primary 55P62; Secondary 55M30
Published electronically: August 20, 2003
MathSciNet review: 2022386
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Abstract: A simply connected topological space is called elliptic if both $\pi_*(X, \mathbb{Q})$ and $H^*(X, \mathbb{Q})$ are finite-dimensional $\mathbb{Q}$-vector spaces. In this paper, we consider fibrations for which the fibre $X$ is elliptic and $ H^*(X, \mathbb{Q}) $ is evenly graded. We show that in the generic cases, the genus of such a fibration is completely determined by generalized Chern classes of the fibration.

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  • 1. A. Dold, Halbexakte Homotopiefunktoren, Lecture Notes in Math., no. 12, Springer-Verlag, New York, 1966. MR 33:6622
  • 2. A. Dold and R. Lashoff, Principal quasi-fibrations and fibre homotopy equivalence of bundles, Illinois J. Math. 3 ($1959$), $285-305$. MR 21:331
  • 3. J.-P. Doeraene, L.S.-category in a model category, J. Pure and Appl. Algebra 84 ($1993$), $215-261$. MR 94b:55017
  • 4. E. Dror and A. Zabrodsky, Unipotency and nilpotency in homotopy equivalences, Topology 18 ($1979$), $187-197$. MR 81g:55008
  • 5. Y. Félix, La dichotomie elliptique-hyperbolique en homotopie rationelle, Astérisque 176, Société Mathématique de France, $1989$. MR 91c:55016
  • 6. Y. Félix and S. Halperin, Rational LS category and its applications, Trans. Amer. Math. Soc. 273 ($1982$), $1-37$. MR 84h:55011
  • 7. T. Ganea, Lusternik-Schnirelmann category and strong category, Illinois J. Math. 11 ($1967$), $417-427$. MR 37:4814
  • 8. S. Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 ($1977$), $ 173-199$. MR 57:1493
  • 9. S. Halperin, Lectures on minimal models, Mémoire de la Société Mathématique de France, $9-10$, $1983$. MR 85i:55009
  • 10. I. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 ($1978$), $331-348$. MR 80i:55001
  • 11. G. Lupton, Note on a conjecture of Stephen Halperin, Lecture Notes in Math., no. 1440, Springer-Verlag, New York, 1990, $148-163$.MR 92a:55012
  • 12. W. Meier, Rational universal fibrations and flag manifolds, Math. Ann. 258 (1981/82), $329-340$. MR 83g:55009
  • 13. M. Schlessinger and J. Stasheff, Deformations theory and rational homotopy type, preprint.
  • 14. H. Shiga and M. Tezuka, Rational fibrations, homogeneous spaces with positive Euler characteristics and Jacobians, Ann. Inst. Fourier 37 ($1987$), $81 - 106$.MR 89g:55019
  • 15. D. Stanley, The sectional category of spherical fibrations, Proc. Amer. Math. Soc. 128 ($2000$), $3137-3143$. MR 2001a:55004
  • 16. D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 ($1977$), $ 269-331$. MR 58:31119
  • 17. D. Tanré, Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Math., no. 1025, Springer-Verlag, Berlin, 1983. MR 86b:55010
  • 18. D. Tanré, Fibrations et Classifiants, In Homotopie algébrique et algèbre locale, Astérisque $113-114$, $1984$.MR 85h:55018
  • 19. J. C. Thomas, Rational homotopy of Serre fibrations, Ann. Inst. Fourier (Grenoble) 31 (1981), $71-90$. MR 83c:55016

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Additional Information

J.-B. Gatsinzi
Affiliation: University of Botswana, Private Bag 0022, Gaborone, Botswana

Keywords: Rational homotopy, Lusternik-Schnirelmann category, genus, sectional category
Received by editor(s): October 6, 2001
Received by editor(s) in revised form: September 19, 2002
Published electronically: August 20, 2003
Additional Notes: Supported by a grant from Université Catholique de Louvain
Communicated by: Paul Goerss
Article copyright: © Copyright 2003 American Mathematical Society

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