5 Cohomology of toric bundles

Let p : E−→B be a principal bundle with fibre and structure group the torus T ∼ = (C *) n over a topological space B. Let X be a nonsingular projective T-toric variety. One has the X-bundle π : E(X)−→B where E(X) = E × T X, π([e, x]) = p(e). This is a Zariski locally trivial fibre bundle in case p : E−→B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H * (B; Z)-algebra, (ii) the topological K-ring of K * (E(X)) as a K * (B)-algebra when B is compact Hausdorff. When p : E−→B is algebraic over an irreducible, nonsingular, noetherian scheme over C, we describe (iii) the Chow ring of A * (E(X)) as an A * (B)-algebra, and (iv) the Grothendieck ring K 0 (E(X)) of algebraic vector bundles on E(X) as a K 0 (B)-algebra.