Semifree circle actions, Bott towers and quasitoric manifolds
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A Bott tower is the total space of a tower of fibre bundles with base and fibres . 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 the fixed points. We show that a quasitoric manifold over a combinatorial n-cube admitting a semifree action of a 1-dimensional subtorus with 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 Il'inskiĭ, who showed that a smooth compact toric variety admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of Hattori's problem of semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to this product. In the case of Bott towers the homeomorphism is actually a diffeomorphism.Bibliography: 18 titles.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/200510c7ace9a261ada01ac37fda222504e256a8
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
Citations in the corpus
- Rigidity problems in toric topology: A survey (2011, background): “The answer to the strong cohomological rigidity problem with diffeomorphism is affirmative in the following cases : (1) either M or N is ∏m i=1 CP ni (see [34] when ni = 1 for any i and [16] in general), (2) both M and N are Q-trivial Bott manifolds [9],”
- Quasitoric manifolds over a product of simplices (2008, unknown): “A T ] 2 8 N ov 2 00”
- Cohomological rigidity of real Bott manifolds (2008, background): “We also prove that the converse is true, namely a real toric manifold which admits a flat riemannian metric invariant under the action of an elementary abelian 2-group is a real Bott manifold.”
- Toric Topology (2012, background): “Two results were obtained in [210] relating circle actions on Bott towers, their topological structure, and cohomology rings.”