对有限群G，本项目在一般拓扑空间范畴上建立层理论观点的由正交表示环RO(G)分次的等变Bredon上同调理论，并应用于实代数簇的上同调研究。主要研究内容包括：（1）在G-CW复形范畴和更一般的具有某种良好可缩开覆盖性质的仿紧Hausdorff空间范畴上建立层论视角的RO(G)分次等变Bredon上同调理论；（2）研究RO(G)分次等变Bredon上同调与等变Cech 超上同调的同构关系；（3）利用上述结果在一般实代数簇上建立实版本的Deligne-Beilinson上同调理论。.本项目的研究意义在于：（1）建立了层论观点的RO(G)分次等变Bredon上同调理论，把研究对象的几何和组合性质纳入到等变上同调研究；（2）通过建立实版本Deligne-Beilinson上同调，把等变拓扑理论和Deligne上同调理论引入到实代数簇的研究中，打开从代数、拓扑、几何综合研究的思路。

The project constructs the equivariant Bredon cohomology on a general category of topological spaces from the sheaf theoretic viewpoint. This cohomology theory is graded by the real orthogonal representation ring RO(G) for a finite group G. Furthermore, we apply the theory to the study of the cohomology theories on real algebraic varieties. The main contents of this project are: (1) Construction of the sheaf version of RO(G)-graded equivariant Bredon cohomology on the category of G-CW complexes, and on the category of paracompact Hausdorff spaces with some "good and contractible open cover" property; (2) The isomorphism between RO(G)-graded equivariant Bredon cohomology and equivariant Cech hypercohomology; (3) Applying the above results to real algebraic varieties and constructing a "real" version of Deligne-Beilinson cohomology theory..The research significance of this project is: (1) The sheaf theoretical version of RO(G)-graded equivariant Bredon cohomology is built, through which the geometric and combinatorial properties of the research objects are brought into the equivariant cohomology theory; (2) By creating a real version of Deligne-Beilinson cohomology theory on the real algebraic varieties, we introduce the equivariant topology theory and Deligne cohomology theory to the study of real algebraic varieties. This leads to a comprehensive research thread to real algebraic varieties, which combines the viewpoints from algebra, topology and geometry.