Classification of $Q$-trivial Bott manifolds

Abstract

A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $CP^1$-bundle starting with a point, where each $CP^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A $Q-trivial Bott manifold$ of dimension $2_n$ is a Bott manifold whose cohomology ring is isomorphic to that of $(CP^1)^n$ 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, the number of diffeomorphism classes of $Q$-trivial Bott manifolds of dimension $2_n$ 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.