Rational fibrations homogeneous spaces with positive Euler characteristics and Jacobians

Shiga, H.; Tezuka, M.

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 $G=\mathrm{rang}U$ 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 $G=\mathrm{rang}U$ 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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