Nous démontrons que la fibration orientable de fibre ayant même type d’homotopie que l’espace homogène avec rang 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 . Nous démontrons également que le résultat est valable pour les coefficients mod. si ne divise pas l’ordre du groupe de Weyl de .
We show that an orientable fibration whose fiber has a homotopy type of homogeneous space with rank is totally non homologous to zero for rational coefficients. The Jacobian formed by invariant polynomial under the Weyl group of plays a key role in the proof. We also show that it is valid for mod. coefficients if does not divide the order of the Weyl group of .
@article{AIF_1987__37_1_81_0, 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}, address = {Grenoble}, volume = {37}, number = {1}, year = {1987}, doi = {10.5802/aif.1078}, mrnumber = {89g:55019}, zbl = {0608.55006}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1078/} }
TY - JOUR AU - Shiga, H. AU - Tezuka, M. TI - Rational fibrations homogeneous spaces with positive Euler characteristics and Jacobians JO - Annales de l'Institut Fourier PY - 1987 SP - 81 EP - 106 VL - 37 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.1078/ DO - 10.5802/aif.1078 LA - en ID - AIF_1987__37_1_81_0 ER -
%0 Journal Article %A Shiga, H. %A Tezuka, M. %T Rational fibrations homogeneous spaces with positive Euler characteristics and Jacobians %J Annales de l'Institut Fourier %D 1987 %P 81-106 %V 37 %N 1 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.1078/ %R 10.5802/aif.1078 %G en %F AIF_1987__37_1_81_0
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. http://www.numdam.org/articles/10.5802/aif.1078/
[1] 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] Les sous-groupes fermés de rang maximal de Lie clos, Comm. Math. Helv., 23 (1949), 200-221. | EuDML | MR | Zbl
and ,[3] Simple groups of Lie type, John Wilely and Sons, London, 1972. | MR | Zbl
,[4] Invariants of finite groups generated by reflections, Amer. J. Math., 77 (1955), 778-782. | MR | Zbl
,[5] The product of the generators of a finite group generated by reflections, Duke Math. J., 18 (1951), 391-441. | MR | Zbl
,[6] Finitess in the minimal models of Sullivan, Trans. A.M.S., 230 (1977), 173-199. | MR | Zbl
,[7] 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] Commutative Algebra, second edition, Benjamin (1980). | MR | Zbl
,[9] Rational Universal fibration and Flag manifolds, Math. Ann., 258 (1982), 329-340. | EuDML | MR | Zbl
,[10] The mod 2 cohomology rings of extra-special 2-groups and the Spinor groups, Math. Ann., 194 (1971), 197-212. | MR | Zbl
,[11] Deformation theory and rational homotopy type, (to appear).
and ,[12] Lectures on Chevalley groups, Yale Univ. (1967).
,[13] Infinitesimal computations in Topology, Publ. I.H.E.S., 47 (1977), 269-332. | Numdam | MR | Zbl
,[14] Homotopie rationnelle des fibrations de Serre, Ann. Inst. Fourier, 31-3 (1981), 71-90. | Numdam | MR | Zbl
,[15] Quelques questions commentées sur la fibre d'Eilengerg-Moore d'une fibration de Serre, Publ. Lille, 3, no 6 (1981).
,[16] Classifying maps and homogeneous spaces, (preprint).
,[17] A note on the cohomology of a fiber space whose fiber is a homogeneous space, (preprint). | Zbl
, and ,[18] Cohomology automorphisms of some Homogeneous spaces, to appear in Topology and its applications (Singapore conference volume). | Zbl
and ,Cité par Sources :