Mathematical Sciences: Cohomology and Actions of Finite Groups

数学科学：有限群的上同调和作用

• 批准号: 8901414
• 负责人: Alejandro Adem
• 金额: $ 2.15万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901414&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Cohomology; Actions; Finite

8901414 Adem This project is an attempt to apply new cohomological and representation-theoretic ideas to problems in transformation groups and homotopy theory. The particular aspects which it involves can be described as follows: (1) Incorporate the study of certain discrete groups, such as arithmetic groups and mapping class groups, to an existing program based on applying methods from modular representation theory to group actions. (2) Determine general properties of the cohomology of finite groups, and how, using equivariant cohomology, this influences the role they play as transformation groups. (3) Obtain necessary and sufficeint conditions for a finite group to act freely on a product of spheres, and analyze the restrictions this imposes on the homology representations when they are of the same dimension. This work is aimed at approaching a global conjecture on the free rank of symmetry of a finite complex. (4) Use theory of infinite loop spaces and invariant theory to obtain precise cohomological information on the symmetric groups, with several applications in mind. They include describing the cohomology of certain double covers of the symmetric groups which serve as finite models for a space which detects important characteristic classes. Other applications to stable homotopy and group cohomology are also planned. The invariant theory used is seen as an interesting and important algebraic problem in its own right, encoding most of the cohomological structure of the symmetric groups. In other words, the various parts of this project all concern the symmetry of geometric objects. Algebraic tools for making calculations about symmetry will be perfected, and these tools will then be applied to answer natural questions.

8901414 Adem本课题是将新的上同调和表示论思想应用于变换群和同伦理论问题的尝试。它涉及的具体方面可以描述如下：(1)将某些离散群的研究，如算术群和映射类群，结合到一个现有的程序中，该程序基于将模块化表示理论的方法应用于群行为。(2)确定有限群上同调的一般性质，以及如何利用等变上同调影响它们作为变换群的作用。(3)得到有限群在球的积上自由作用的充分必要条件，并分析了同维同调表示所受的限制。本文的目的是探讨有限复合体的自由对称秩的一个全局猜想。(4)利用无限循环空间理论和不变量理论获得了对称群上的精确上同信息，并给出了一些应用。它们包括描述对称群的某些双盖的上同调，这些对称群作为空间的有限模型，可以检测到重要的特征类。本文还讨论了稳定同伦和群上同伦的其他应用。所使用的不变量理论本身被视为一个有趣而重要的代数问题，它编码了对称群的大多数上同调结构。换句话说，这个项目的各个部分都与几何物体的对称性有关。用于计算对称性的代数工具将得到完善，然后这些工具将被应用于回答自然问题。