COHOMOLOGY OF TORUS MANIFOLD BUNDLES

Let X be a 2n-dimensional torus manifold with a locally standard T congruent to (S-1)(n) action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p: E -> B and let pi : E(X) -> B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H*(B)-algebra and the topological K-ring of E(X) as a K*(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.