  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.
