Classification of $\mathbb{Q}$-trivial Bott manifolds

A Bott manifold is a closed smooth manifold obtained as the total space of an iterated CP -bundle starting with a point, where each CP -bundle is the projectivization of a Whitney sum of two complex line bundles. A Q-trivial Bott manifold of dimension 2n is a Bott manifold whose cohomology ring is isomorphic to that of (CP ) with Q-coefficients. We find all diffeomorphism types of Q-trivial Bott manifolds and show that they are distinguished by their cohomology rings with Z-coefficients. As a consequence, we see that the number of diffeomorphism classes in Q-trivial Bott manifolds of dimension 2n is equal to the number of partitions of n. We even show that any cohomology ring isomorphism between two Q-trivial Bott manifolds is induced by a diffeomorphism.