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，我们考虑了纤维范畴和 2-范畴概念之间的关系，并表明由范畴 ε 上的纤维范畴组成的 2-范畴等价于由 ε 的相反范畴到 2-函子组成的 2-范畴。然后，我们使用纤维类别的概念给出了类群表示的一般定义，此外，我们通过构建的表示的各种示例来收集基本事实。我们用 F_p 表示特征 p 的素域，用 TopVect^* 表示 F_p 上的分级拓扑向量空间的类别，其具有由子向量空间 We 组成的 0 邻域的基本系统。观察到，在定义一个空间后，通过给每个 mod p 上同调群一个合适的拓扑，普通 mod p 上同调理论被视为从拓扑空间范畴到 TopVect^* 的函子。适当地考虑线性映射的空间 Hom(V^*, W^*)，我们证明函子 Z^* → Hom(W^*, Z^*) 是函子 V^* → 右伴随的替代V^*【叉积】^^^W^* 由完整的张量积给出并研究了这些函子的性质通过上述准备，我们证明了空间的 mod p 上同调理论被视为仿射群格式的表示。代表为另一方面，为了研究复K-理论的形式群律之间的关系，并试图在Steenrod代数上重建Lanne的不稳定模理论，并提出几个表征Steenrod代数的公理。实K-理论虽然不是复向的，但具有相关的Hopf代数体，我们确定了实K-上同调的KO^*-代数结构复射影空间 CP^l 及其乘积 CP^l × CP^m 此外，我们还确定了无限维复射影空间的实 K-上同调的 KO_*-代数结构。