$\mathbb{Q}$-trivial generalized Bott manifolds

Abstract

When the cohomology ring of a generalized Bott manifold with $\mathbb{Q}$-coefficient is isomorphic to that of a product of complex projective spaces $\mathbb{C}P^{n_{i}}$, the generalized Bott manifold is said to be $\mathbb{Q}$-trivial. We find a necessary and sufficient condition for a generalized Bott manifold to be $\mathbb{Q}$-trivial. In particular, every $\mathbb{Q}$-trivial generalized Bott manifold is diffeomorphic to a $\prod_{n_{i}>1}\mathbb{C}P^{n_{i}}$-bundle over a $\mathbb{Q}$-trivial Bott manifold.