Properties of Bott manifolds and cohomological rigidity

Abstract

The cohomological rigidity problem for toric manifolds asks whether the cohomology ring of a toric manifold determines the topological type of the manifold. In this paper, we consider the problem with the class of one-twist Bott manifolds to get an affirmative answer to the problem. We also generalize the result to quasitoric manifolds. In doing so, we show that the twist number of a Bott manifold is well-defined and is equal to the cohomological complexity of the cohomology ring of the manifold. We also show that any cohomology Bott manifold is homeomorphic to a Bott manifold. All these results are also generalized to the case with ${\mathbb{Z}}_{(2)}$-coefficients, where ${\mathbb{Z}}_{(2)}$ is the localized ring at 2.

22 pages

References in corpus (2)

• Semifree circle actions, Bott towers, and quasitoric manifolds
• Toric cohomological rigidity of simple convex polytopes

Cited by in corpus (3)

• The Kähler geometry of Bott manifolds
• Invariance of Pontrjagin classes for Bott manifolds
• Classification of rationally elliptic toric orbifolds