Topological classification of generalized Bott towers

Abstract

If $B$ is a toric manifold and $E$ is a Whitney sum of complex line bundles over $B$, then the projectivization $P(E)$ of $E$ is again a toric manifold. Starting with $B$ as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives a supporting evidence to what we call cohomological rigidity problem for toric manifolds "Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?" We provide two more results which support the cohomological rigidity problem.