受代数几何中环面簇理论的影响, 近几十年来，拓扑学中对变换群理论的研究，尤其是对环面作用的研究得以迅速发展。这一研究使拓扑学、组合数学、辛几何、交换代数和同调代数等学科相互交叉,形成了一个新的研究方向- - -环面拓扑。环面拓扑研究带环面作用的空间的拓扑性质,该作用的轨道空间有比较好的组合结构，这些组合信息可以帮助我们理解原来空间的拓扑和计算一些拓扑不变量。从这一角度出发，本项目拟开展以下两类问题的研究：（1）给定一个单纯复形或一个单纯偏序集，可以联系带环面（或2-环面）作用的矩角复形，从单纯复形或单纯偏序集的组合性质、代数性质或染色性质来研究与矩角复形相关的空间的性质。（2）对于带2-环面作用且有特殊几何特点的一些光滑闭流形，尤其是轨道空间组合结构比较复杂的小覆盖，讨论它们在一些等价关系（如等变协边、等变同胚、D-J等价等）下的分类及类的个数计算问题等。

Toric varieties appeared in algebraic geometry in the early 1970s. With influence of the applications of topological and combinatorial methods in the study of toric varieties, the study on topological transformation groups, especially on torus actions, has been developed rapidly during the last several decades. This approach is presented as a bridge connecting topology with combinatorics, symplectic geometry, commutative and homological algebra. As a result, it provides a new direction called toric topology. In toric topology, it is concerned with the combinatorics of orbit spaces of actions and understanding the topology of the space with the actions through the combinatorial information. From that point of view, this project is devoted to the study of the following problems: (1) Given a simplicial complex or a simplicial poset, a moment-angle complex is assigned to it. We will study properties of some spaces associated to the moment-angle complex with torus actions(2-torus actions) through combinatorics, coloring， and algebraic aspects of the simplicial complex or the simplicial poset; (2) For some closed manifolds with 2-torus actions and specified geometric properties, especially small covers over simple polytopes with complicated combinatorics, we focus on the classification of the manifolds (up to equivariant cobordism, equivariant homeomorphism, D-J equivalence etc), calculation of the number of the equivalence classes, and other related properties.