环分次等变Bredon上同调与实代数簇Deligne-Beilinson上同调

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.

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。

对有限群G，本项目在一类拓扑空间范畴上建立了基于层理论的，由正交表示环RO(G)分次的等变Bredon上同调理论，此理论适合把研究对象的几何和组合性质纳入到等变上同调理论。本项目同时研究了RO(G)分次等变Bredon上同调与等变Cech 超上同调的同构关系。.本项目同时把上述结果应用于实代数簇的上同调研究，在一般实代数簇上定义了Bredon复形，证明了Bredon复形与复形Z(V)之间的拟同构关系，从而导出实代数簇上Cech超上同调与整数双分次的Bredon上同调之间的同构定理。此同构定理是在实代数簇上建立实版本的Deligne-Beilinson上同调理论的一个理论基础。相关成果已在《Journal of Algebra》等期刊上发表。.通过本项目获得的上述研究成果，本项目负责人的后续研究思路是研究实代数簇上的双分次上同调理论和motivic理论，后续深入研究已获得国家自然科学基金地区基金项目（编号：11461047）资助，现正密集开展相关研究工作。