Cohomology and K-theory of generalized Dold manifolds fibred by complex flag manifolds

Let = (n₁, , nₛ), s 2, be a sequence of positive integers and let n=₁ ₉ ₒnⱼ. Let CG () =U (n) / (U (n₁) U (nₛ) ) be the complex flag manifold. Denote by P (m, ) =P (Sᵐ, CG () ) the generalized Dold manifold Sᵐ CG () / where = with: Sᵐ Sᵐ being the antipodal map and: CG () CG (), the complex conjugation. The manifold P (m, ) has the structure of a smooth CG () -bundle over the real projective space RPᵐ. We determine the additive structure of H^* (P (m, ) ;R) when R= Z and its ring structure when R is a commutative ring in which 2 is invertible. As an application, we determine the additive structure of K (P (m, ) ) almost completely and also obtain partial results on its ring structure. The results for the singular homology are obtained for generalized Dold spaces P (S, X) =S X/, where =,: S S is a fixed point free involution and: X X is an involution with Fix (), for a much wider class of spaces S and X.