Topological classification of quasitoric manifolds

Abstract

A $\emph{quasitoric manifold}$ is a 2n-dimensional closed smooth manifold with a locally standard $n$-dimensional torus action whose orbit space is a simple polytope. This is a topological analogue of a projective toric manifold in algebraic geometry. One of the most interesting examples of quasitoric manifold is a \emph{generalized Bott manifold} which is constructed by a sequence of complex projective bundles and whose orbit space is a product of simplices. In fact, a generalized Bott manifold is not only a quasitoric manifold but also a projective toric manifold. The purpose of this thesis is to classify quasitoric manifolds topologically. The integral cohomology ring plays an important role in this classification. In the first part, we study whether a quasitoric manifold is equivalent to a generalized Bott manifold or not when its integral cohomology ring is isomorphic to that of a generalized Bott manifold. In some cases, the integral cohomology ring determines the generalized Bott manifold structure. But there is an example of quasitoric manifold which is not equivalent but homeomorphic to a generalized Bott manifold. In the second part, we classify quasitoric manifolds with the second Betti number $\beta_2=2$. Interestingly, they are distinguished by their integral cohomology rings up to homeomorphism. Moreover, we can count quasitoric manifolds with $\beta_2=2$ not homeomorphic to a generalized Bott manifold in each dimension. In the last part, we consider quasitoric manifolds whose rational cohomology rings are isomorphic to that of a trivial generalized Bott manifold, that is, the product of complex projective spaces. We say that such a quasitoric manifold is $\emph{\mathbb{Q}-trivial}$. We find a necessary and sufficient condition for a generalized Bott manifold to be $\mathbb{Q}$ -trivial. In particular, every $\mathbb{Q}$ -trivial generalized Bott manifold is diffeomorphic to a product of complex projective spaces provided that there is no $\mathbb...