Rational fibrations, homogeneous spaces with positive Euler characteristics and Jacobians (Q1085832)

We show that an orientable fibration whose fiber has homotopy type of a homogeneous space G/U with rank G\(=rank U\) is totally nonhomologous to zero for rational coefficients. The Jacobian formed by invariant polynomials 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.