Journal of Symplectic Geometry

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

Suyoung Choi and Mikiya Masuda

Full-text: Open access

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


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