Invariance of Pontrjagin classes for Bott manifolds

Abstract

A Bott manifold is the total space of some iterated $ℂ{\mathbb{P}}^1$–bundles over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are $\mathbb{Z}∕2$–trivial, where a Bott manifold is called $\mathbb{Z}∕2$–trivial if its cohomology ring with $\mathbb{Z}∕2$–coefficients is isomorphic to that of a product of copies of $ℂ{\mathbb{P}}^1$.