$\mathbb{Q}$-trivial generalized Bott manifolds

TL;DR

This paper characterizes when a generalized Bott manifold is $ ext{Q}$-trivial by providing necessary and sufficient conditions, linking it to diffeomorphism to certain product bundles over $ ext{Q}$-trivial Bott manifolds.

Contribution

It establishes a complete criterion for $ ext{Q}$-triviality in generalized Bott manifolds and describes their diffeomorphic structure.

Findings

Necessary and sufficient condition for $ ext{Q}$-triviality.

$ ext{Q}$-trivial generalized Bott manifolds are diffeomorphic to specific product bundles.

Connection between $ ext{Q}$-triviality and Bott manifold structures.

Abstract

When the cohomology ring of a generalized Bott manifold with $\mathbb{Q}$-coefficient is isomorphic to that of a product of complex projective spaces $\mathbb{C}P^{n_i}$, the generalized Bott manifold is said to be $\mathbb{Q}$-trivial. We find a necessary and sufficient condition for a generalized Bott manifold to be $\mathbb{Q}$-trivial. In particular, every $\mathbb{Q}$-trivial generalized Bott manifold is diffeomorphic to a $\prod_{n_i>1}\mathbb{C}P^{n_i}$-bundle over a $\mathbb{Q}$-trivial Bott manifold.