Properties determined by cohomology ring in toric topology

Abstract

The purpose of this thesis is to research the relations between manifolds which admit certain well-behaved actions of the torus and their cohomology rings. If $\textsl{B}$ is a toric manifold and if $\textsl{E}$ is a Whitney sum of complex line bundles over $\textsl{B}$, then the projectivization $\textsl{P(E)}$ of $\textsl{E}$ is again a toric manifold. Starting with $\textsl{B}$ as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a $\emph{generalized Bott tower}$. In fact, the top manifold in tower, called a $\textit{generalized Bott manifold}$, is indeed a toric manifold and its orbit space can be identified with a product of simplices. The generalized Bott tower is one of the most interesting objects in the present thesis, as well as toric topology. The first part of this thesis is the study of classifications of toric manifolds via topology. Of special interest is the following problem which is now called the $\textit{cohomological rigidity problem; Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic as graded rings}$? In general, a cohomology ring invariant is too weak to determine the topological type. It can be a strong invariant, however, for some specific classes of manifolds. In this thesis, we give partial affirmative solutions to this problem. The second part is the study of combinatorial polytopes as orbit spaces of toric manifolds. A simple polytope $\textsl{P}$ is $\textit{(toric) cohomologically rigid}$ if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold $\textsl{M}$ over $\textsl{P}$, i.e., there exists a quasitoric manifold $\textsl{M}$ over $\textsl{P}$, and whenever there exists a quasitoric manifold $N$ over another polytope $\textsl{Q}$ with $H^\ast (M) = H^\ast (N)$ there is a combinatorial equivalence $P \approx Q$. Although $H^\ast (M)$ contains some information of $\textsl{P}$, not every si...