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: 9013423
• 负责人: Alejandro Adem
• 金额: $ 3.89万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-15 至 1992-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9013423&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Cohomology; Actions; Finite

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 sufficient 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.

该项目是应用新的上同调和 转换问题的表征理论思想 群与同伦理论 它的具体方面， 主要包括以下几个方面：（1）将研究 某些离散群的，如算术群和映射 类组，基于应用方法应用到现有程序 从模块化表示理论到群体行为 （二） 确定有限群的上同调的一般性质， 以及如何使用等变上同调， 他们扮演的是转化小组 (3)获得必要的和 有限群自由作用于 产品的领域，并分析限制这强加给 当它们具有相同维数时的同调表示。 这项工作的目的是接近一个全球猜想的自由 有限复形的对称秩。 (4)运用无穷论 循环空间和不变理论，以获得精确的上同调 关于对称群的信息，以及在 主意了 它们包括描述某些双上同调 作为有限模型的对称群的覆盖 检测重要特征类的空间。 其他 应用稳定同伦和群上同调也是 计划好了 使用的不变理论被视为一个有趣的， 重要的代数问题本身，编码大部分 对称群的上同调结构。 换句话说，这个项目的各个部分 关注几何物体的对称性。 代数工具， 关于对称性的计算将得到完善，这些 然后将应用工具来回答自然问题。