Properties of Bott manifolds and cohomological rigidity

Abstract

The cohomological rigidity problem for toric manifolds asks whether the integral cohomology ring of a toric manifold determines the topological type of the manifold. In this paper, we consider the problem with the class of one-twist Bott manifolds to get an affirmative answer to the problem. We also generalize the result to quasitoric manifolds. In doing so, we show that the twist number of a Bott manifold is well-defined and is equal to the cohomological complexity of the cohomology ring of the manifold. We also show that any cohomology Bott manifold is homeomorphic to a Bott manifold. All these results are also generalized to the case with ${\mathbb{Z}}_{\left(2\right)}$–coefficients, where ${\mathbb{Z}}_{\left(2\right)}$ is the localized ring at 2.