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September 2012 Classification of $\mathbb{Q}$-trivial Bott manifolds
Suyoung Choi, Mikiya Masuda
J. Symplectic Geom. 10(3): 447-461 (September 2012).

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.

Citation

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Suyoung Choi. Mikiya Masuda. "Classification of $\mathbb{Q}$-trivial Bott manifolds." J. Symplectic Geom. 10 (3) 447 - 461, September 2012.

Information

Published: September 2012
First available in Project Euclid: 16 October 2012

zbMATH: 1261.57028
MathSciNet: MR2983437

Rights: Copyright © 2012 International Press of Boston

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Vol.10 • No. 3 • September 2012
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