Invariance of Pontrjagin classes for Bott manifolds

Abstract

A Bott manifold is the total space of some iterated $\mathbb{C}{P}^1$-bundle 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 $\mathbb{Z}/2$-coefficient is isomorphic to that of a product of $\mathbb{C}{P}^1$'s.

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