Semifree circle actions, Bott towers, and quasitoric manifolds

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.Comment: 22 pages, LaTEX; substantially revise

Moment-angle complexes from simplicial posets

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T^m-space Z_S to an arbitrary simplicial poset S on m vertices. Face rings Z[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen--Macaulay rings. Our primary motivation is to study the face rings Z[S] by topological methods. The space Z_S has many important topological properties of the original moment-angle complex Z_K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z_S is isomorphic to the Tor-algebra of the face ring Z[S]. This leads directly to a generalisation of Hochster's theorem, expressing the algebraic Betti numbers of the ring Z[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z_S from below by proving the toral rank conjecture for the moment-angle complexes Z_S.Comment: 17 pages, 2 pictures; v2 revised, v3 minor correction