Rational fibrations homogeneous spaces with positive Euler characteristics and Jacobians
Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 81-106.

Nous démontrons que la fibration orientable de fibre ayant même type d’homotopie que l’espace homogène G/U avec rang G=rangU est totalement non homologue à zéro pour les coefficients rationnels. Nous utilisons le jacobien formé par des poloynômes invariants pour le groupe de Weyl de G. Nous démontrons également que le résultat est valable pour les coefficients mod.p si p ne divise pas l’ordre du groupe de Weyl de G.

We show that an orientable fibration whose fiber has a homotopy type of homogeneous space G/U with rank G=rangU is totally non homologous to zero for rational coefficients. The Jacobian formed by invariant polynomial under the Weyl group of G plays a key role in the proof. We also show that it is valid for mod.p coefficients if p does not divide the order of the Weyl group of G.

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     author = {Shiga, H. and Tezuka, M.},
     title = {Rational fibrations homogeneous spaces with positive {Euler} characteristics and {Jacobians}},
     journal = {Annales de l'Institut Fourier},
     pages = {81--106},
     publisher = {Institut Fourier},
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Shiga, H.; Tezuka, M. Rational fibrations homogeneous spaces with positive Euler characteristics and Jacobians. Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 81-106. doi : 10.5802/aif.1078. https://www.numdam.org/articles/10.5802/aif.1078/

[1] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math., 57 (1953), 115-207. | MR | Zbl

[2] A. Borel and J.D. Siebenthal, Les sous-groupes fermés de rang maximal de Lie clos, Comm. Math. Helv., 23 (1949), 200-221. | EuDML | MR | Zbl

[3] R.W. Cater, Simple groups of Lie type, John Wilely and Sons, London, 1972. | MR | Zbl

[4] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math., 77 (1955), 778-782. | MR | Zbl

[5] H. Coxeter, The product of the generators of a finite group generated by reflections, Duke Math. J., 18 (1951), 391-441. | MR | Zbl

[6] S. Halperin, Finitess in the minimal models of Sullivan, Trans. A.M.S., 230 (1977), 173-199. | MR | Zbl

[7] B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., 81 (1959), 997-1032. | MR | Zbl

[8] H. Matsumura, Commutative Algebra, second edition, Benjamin (1980). | MR | Zbl

[9] W. Meier, Rational Universal fibration and Flag manifolds, Math. Ann., 258 (1982), 329-340. | EuDML | MR | Zbl

[10] D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the Spinor groups, Math. Ann., 194 (1971), 197-212. | MR | Zbl

[11] M. Schlessinger and J. Stascheff, Deformation theory and rational homotopy type, (to appear).

[12] R. Steinberg, Lectures on Chevalley groups, Yale Univ. (1967).

[13] D. Sullivan, Infinitesimal computations in Topology, Publ. I.H.E.S., 47 (1977), 269-332. | Numdam | MR | Zbl

[14] J.C. Thomas, Homotopie rationnelle des fibrations de Serre, Ann. Inst. Fourier, 31-3 (1981), 71-90. | Numdam | MR | Zbl

[15] J.C. Thomas, Quelques questions commentées sur la fibre d'Eilengerg-Moore d'une fibration de Serre, Publ. Lille, 3, no 6 (1981).

[16] H. Shiga, Classifying maps and homogeneous spaces, (preprint).

[17] A. Kono, H. Shiga and M. Tezuka, A note on the cohomology of a fiber space whose fiber is a homogeneous space, (preprint). | Zbl

[18] H. Shiga and M. Tezuka, Cohomology automorphisms of some Homogeneous spaces, to appear in Topology and its applications (Singapore conference volume). | Zbl

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