Characteristic Functions Of P~3and Partial-Quotient Of Moment-Angle Manifolds

The focus of this Ph.D. thesis associated with toric topology is on the following two issues:(1) the existence of characteristic functions of simple convex3-polytopes;(2) The classification on Partial-quotient manifolds of Moment-Angle manifolds over m-gon.In1991, Davis and Januszkiewicz researched in [19] two categories of manifolds denote by M2n and Mn with the local standard Tn-action or Z2n-action and with a simple convex n-polytope P" as the orbit space. These two types of manifolds are known as the quasi-toric manifolds and small covers. We can give a Zn or Z2n color on Pn by using the information of the group action on the manifolds. The Z" or Z2n color on P" is also known as the characteristic functions of P" and is denote by the form (Pnλ). Davis and Januszkiewicz proved that the cohomology ring of quasi-toric manifolds and small covers can be described in terms of (P", λ), and the geometric topology is also completely determined by (P", λ). In other words, the Quasi-toric manifolds or small covers is equivalent to (P",λ).Davis and Januszkiewicz introduced in [19] a Tm-manifold Zp with orbit space P", here m is the number of the faces codimension1of P". This manifold has the following universal property: for every quasi-toric manifold π:M2n→Pn there is a principal Tm-n-bundle Zp→M2n whose composite map with π is the orbit map of Zp-Topology of Zp and their further generalization is very nice itself and at the same time provides an more effective methods for understanding inter-relations of toric manifolds between algebraic and combinatorial objects.In2000, Buchstaber and Panov generalized in [9] the conception of the manifold Zp and called it Moment-Angle manifold. They defined Moment-Angle manifold Zp and Buchstaber-invariant s(P) for any P". By using the new conceptions and results, They have given necessary and sufficient conditions and some equivalent descriptions for the existence of characteristic func-tions of any P". Their research brings new methods and new ideas to toric topology. Owing to Buchstaber and Panov’s conclusions, we can give a new proof about the existence of characteristic functions of P3. By using of new methods, we prove five-color theorem simply. This theorem means that any simple convex3-polytopes can be stained whit5kinds of different colors, such that the adjacent surface has different staining. Furthermore, by using of new methods we given a new method about the proof of four-color theorem, and pointed out the difficulty of the new methods.As subgroup H(1≤dim H≤m-n) of Tm act freely on the Zp, the quotient mapping π:Zp→Zp/H means a principal H-bundle. We call Zp/H the Partial-qotients manifolds of Zp.In particular, when dim H=m-n, Zp/H is a quasi-toric manifold over P. This contact helps us to generalize the conception of characteristic functions from quasi-toric manifold and small covers to Zp and its Partial-quotients manifolds. Firstly, we use the modern language of characteristic functions to give a new proof for Orlik and Raymond’theorem in [54], which is on4-dimensional toric manifold classification and been established in1970. Secondly, by using the theorem and the generalized conception of characteristic functions, we have studied the classifi-cation of Moment-Angle Manifold and the Partial-quotient over m-gon. Finally we consider the contact between other problems of toric topology areas, such as the rigid problem, conjuga-tion space and calculation of complex and real Buchstaber-invariant and the problems we are discussing.