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 supporting evidence to what we call the 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.

A simple convex polytope P is cohomologically rigid if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over P . Not every P has this property, but some important polytopes such as simplices or cubes are known to be cohomologically rigid. In this article we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes including products of simplices. Cohomological rigidity of P is related to the bigraded Betti numbers of its Stanley-Reisner ring, another important invariant coming from combinatorial commutative algebra.

Abstract. We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.

A quasitoric manifold is a 2n-dimensional compact smooth manifold with a locally standard action of an n-dimensional torus whose orbit space is a simple polytope. We classify quasitoric manifolds with second Betti number β 2 = 2 topologically. Interestingly, they are distinguished by their cohomology rings up to homeomorphism.

Abstract. Several rigidity problems in toric topology are addressed in [35]. In this paper, we survey results on those problems including recent development.