## Journal of Symplectic Geometry

- J. Symplectic Geom.
- Volume 10, Number 3 (2012), 447-461.

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

Suyoung Choi and Mikiya Masuda

#### 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.

#### Article information

**Source**

J. Symplectic Geom., Volume 10, Number 3 (2012), 447-461.

**Dates**

First available in Project Euclid: 16 October 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.jsg/1350392493

**Mathematical Reviews number (MathSciNet)**

MR2983437

**Zentralblatt MATH identifier**

1261.57028

#### Citation

Choi, Suyoung; Masuda, Mikiya. Classification of $\mathbb{Q}$-trivial Bott manifolds. J. Symplectic Geom. 10 (2012), no. 3, 447--461. https://projecteuclid.org/euclid.jsg/1350392493