Semifree circle actions, Bott towers, and quasitoric manifolds

Abstract

A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an n-cube with a semifree circle action and isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This extends a recent result of Ilinskii, who showed that a smooth compact toric variety with a semifree circle action and isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of a problem motivated by Hattori's work on semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold (or Bott tower) is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to the product.

22 pages, LaTEX; substantially revised

References in corpus (1)

• Quasitoric manifolds over a product of simplices

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• Invariance of Pontrjagin classes for Bott manifolds
• The number of orientable small covers over cubes
• Toric Bruhat interval polytopes
• The number of small covers over cubes
• Classification of real Bott manifolds
• Classification problems of toric manifolds via topology
• Quasitoric manifolds over a product of simplices
• Remarks on the classification of quasitoric manifolds up to equivariant homeomorphism
• Cohomological rigidity of real Bott manifolds
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• Some results on the topology of real Bott towers
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• Topological classification of quasitoric manifolds with the second Betti number 2
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• Surprising Examples of Manifolds in Toric Topology!
• Cubes and Generalized Real Bott Manifolds
• The volume polynomial of regular semisimple Hessenberg varieties and the Gelfand-Zetlin polytope
• Strong cohomological rigidity of quasitoric manifolds over the 3-cube
• Universality of actions on $\mathbb{H}{P}^2$