拓扑学的基本问题是分类问题及不变量的研究，而流形对问题的研究扮演重要的角色。本项目以流形为动态的方式（即等变流形）开展研究，其思想根源可追溯到Klein于1872年发表的极具影响的研究纲领—Erlangen纲领。研究内容主要有：. 1.等变流形的等变协边分类与不变量的研究，以及利用等变流形对标架协边群的计算。重点研究：模2环面群作用下的光滑流形的等变协边分类；任意李群情况下等变酉协边完全不变量，且与Evenness猜想联系起来考虑。. 2.群作用存在性问题。将Halperin-Carlsson猜想以及Kosniowski猜想纳入群作用存在性问题框架下开展工作，探求群作用存在性的阻碍。. 3.广义构形空间和轨道辫子。重点研究：轨道辫子群的基本理论与轨道构形空间基本群的计算；单图Γ的广义构形空间的Hilbert多项式与Γ的染色多项式之关系。.研究还将借助环面拓扑开展。

The fundamental problem in topology contains two respects: classification problem and the study of invariants. Manifolds play an important role on the study of the problem. The project will be carries out from the viewpoint of dynamics. Namely we shall pay our attendtion on equivariant manifolds. The idea of this viewpoint can go back to the deep meaning Erlangen programme posed by Klein in 1872. The project contains the following main contents:. 1. Equivariant bordism and complete invariants of equivariant manifolds, and calculation of framed bordism. We shall pay much more attendtions on the following respects: classification up to equivariant bordism of smooth manifolds with action of mod 2 torus; complete invariants of unitary manifolds under the action of any Lie groups with Evenness conjecture involved.. 2. Existence problem of group actions. We shall carry out our work by putting Halperin-Carlsson conjection and Kosniowski conjecture on the frame of the existence of group actions , from which we will try to find what is obstrctions of existence of group actions.. 3. Generalized configuration spaces and orbit braids. We shall mainly study the basic theory of orbit braid groups and the computation of the fundamental group of the orbit configuration space, and the relation between the Hilbert polynomial of the generalized configuration space over a simple graph Γ and the chromatic polynomial of Γ.. Our research will also be carried out by making use of toric topology.