本项目研究的是具有局部标准的环群作用的紧流形的拓扑性质和几何性质，此类群作用的轨道空间具有好的带角流形的结构。具体研究内容有：. (1) 研究此类流形的一些拓扑不变量以及和底空间组合结构的关系, 特别是研究三维small cover的real moment-angle manifold。. (2) 研究此类流形上几种几何机构的存在问题, 包括正数量曲率或非负数量曲率的黎曼度量，叶状结构和open book分解等。. (3) 如果一个有限维拓扑空间X上存在一个秩为n的环群的自由作用，Carlsson猜想声称X的各个维数的Betti数之和至少为2的n次幂。我们将在已有基础上进一步研究和验证此猜想。.(4) 研究small cover的基本群与轨道空间之间的关系，特别是small cover是aspherical的情形。. (5) 研究其中一类特殊的流形（称为广义实Bott流形）的同胚分类问题。

We study the topological and geometric properties of compact manifolds with locally standard torus actions. The orbit space of such a group action has a nice manifold with corners structure. The main subjects of this research are the following..(1) Study some topological invariants of such manifolds and their relations to the combinatorial structures of the orbit spaces. Especially, study the invariants of 3-dimensional small covers and real moment-angle manifolds..(2) Study the existence of some type of geometric structures on such manifolds, including Riemannian metrics of positive and nonnegative scalar curvature, foliations and open book decompositions etc..(3) If a finite dimensional topological space X admits a free torus group action, Carlsson Conjecture claims that the sum of all the Betti numbers of X is at least 2^n. Based on the known results, we will further study and confirm this conjecture..(4) Study the relations between the fundamental groups of small covers and the orbit spaces, especially when the small cover is aspherical..(5) Study the homeomorphism classification of a special class of such manifolds (called generalized real Bott manifolds), and the classification of a related class of crystallographic groups in Euclidean spaces.