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

Abstract

A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\mathbb{C}P^1$-bundle starting with a point, where each $\mathbb{C}P^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A $\mathbb{Q}$-trivial Bott manifold of dimension $2n$ is a Bott manifold whose cohomology ring is isomorphic to that of $(\mathbb{C}P^1)^n$ with $\mathbb{Q}$-coefficients. We find all diffeomorphism types of $\mathbb{Q}$-trivial Bott manifolds and show that they are distinguished by their cohomology rings with $\mathbb{Z}$-coefficients. As a consequence, the number of diffeomorphism classes of $\mathbb{Q}$-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 $\mathbb{Q}$-trivial Bott manifolds is induced by a diffeomorphism.