Semifree circle actions, Bott towers and quasitoric manifolds

M. Masuda ; T. E. Panov

Sbornik. Mathematics, Tome 199 (2008) no. 8, pp. 1201-1223

Résumé

A Bott tower is the total space of a tower of fibre bundles with base $\mathbb C P^1$ and fibres $\mathbb C P^1$. 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 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'inskiǐ, 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.

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