Representation theory of groupoids and its applications to complex cobordism theory

群形表示论及其在复杂共边理论中的应用

基本信息

批准号：
11640092
负责人：
YAMAGUCHI Atsushi
金额：
$ 1.22万
依托单位：
Osaka Prefecture University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

项目摘要

First, we considered the relations between the notions of fibered categories and 2-categories and showed that the 2-category consisting of fibered categories over a category ε is equivalent to the 2-category consisting of 2-functors from the opposite category of ε to the 2-category cat consisting of categories. Then, we gave a general definition of the representation of groupoid using the notion of fibered category. Moreover, we collected basic facts by constructing various examples of representations of groupoids and defining the trivial representations and regular representations.We denote by F_p the prime field of characteristic p and by TopVect^* the category of graded topological vector spaces over F_p which have fundamental systems of the neighborhood of 0 consisting of sub vector spaces. We observed that the ordinary mod p cohomology theory is regaded as a functor from the category of topological spaces to TopVect^* by giving a suitable topology to each mod p cohomology group of … More a space. After defining the space Hom(V^*, W^*) of linear maps suitably, we showed that a functor Z^* → Hom(W^*, Z^*) is a substitute for the right adjoint of the functor V^* → V^*【cross product】^^^W^* given by the completed tensor product and investigated properties of these functors. With the preparations above, we showed that the mod p cohomology theory of spaces are regarded as representations of the affine group scheme represented by the dual Steenrod algebra and tried to reconstruct the Lanne's theory of unstable modules over the Steenrod algebras and submit several axioms which characterize the Steenrod algebras.On the other hand, in order to investigate the relations between the formal group law of the complex K-theory and the real K-theory which has the associated Hopf algebroid though it is not complex oriented, we determined the KO^*-algebra structures of the real K-cohomology of complex projective spaces CP^l and their product CP^l × CP^m. Moreover, we determined the KO_*-algebra structure of the real K-cohomology of infinite dimensional complex projective space. Less

首先，我们考虑了纤维类别的注释与2类别之间的关系，并表明由类别ε的纤维类别组成的2类类别等于2类别，该2类别由来自相反类别的2类funktors组成的2类funktors to相反的类别与类别组成的2类猫。然后，我们使用纤维类别的通知给出了组素的表示的一般定义。此外，我们通过构建了群体素的表示的各种示例并定义了微不足道的表示和常规表示来收集基本事实。我们用f_p表示特征p的素数和topVect^**等级拓扑矢量空间的类别，而f_p的类别具有由子矢量空间组成的0个基础系统的基本系统。我们观察到，普通的mod p共同体学理论是从拓扑空间的类别中引起的，通过为每个空间的每个mod p共同体学组提供合适的拓扑^*。在适当地定义线性映射的空间HOM（V^*，W^*）之后，我们表明函数Z^*→HOM（W^*，Z^*）是替代函数v^*→V^*【cross产品的正确伴随，由完整的Tensor tensor产品给出，并由这些函数的构图给出。借助上面的准备，我们表明，空间的模式同步理论被认为是由双重steenrod代数代表的仿射组方案的表示，并试图重建兰妮在steenrod代数上的不稳定模块的理论，并提交了几个符合steenrod alggebras的范围，以实现其他关系，以实现其他关系，以实现其他关系。 K理论具有相关的HOPF代数，尽管它并不复杂，但我们确定了复杂射击空间CP^l的真实K-cohomology的KO^* - 代数结构及其产物CP^l×Cp^m。此外，我们确定了KO _* - 无限尺寸复杂射击空间的真实K-合理学的代数结构。较少的