Mathematical Sciences: Cohomology and Actions of Finite Groups

这个项目是将新的上同调和表示论思想应用于变换群和同伦理论中的问题的一次尝试。它涉及的具体方面可以描述如下：(1)将对某些离散群的研究，如算术群和映射类群，应用从模表示理论到群行动的方法，纳入现有的程序。(2)确定有限群的上同调的一般性质，以及利用等变上同调，这如何影响它们作为变换群所起的作用。(3)得到了有限群在球面乘积上自由作用的充要条件，并分析了当同调表示具有相同的维度时，这对同调表示的限制。这项工作的目的是接近一个关于有限复数的自由对称秩的整体猜想。(4)利用无限循环空间理论和不变量理论得到关于对称群的精确上同调信息，并考虑几个应用。它们包括描述作为检测重要特征类的空间的有限模型的对称群的某些双覆盖的上同调。我们还计划将其应用于稳定同伦和群上同调。所使用的不变量理论本身就被视为一个有趣而重要的代数问题，它编码了对称群的大部分上同调结构。换句话说，这个项目的各个部分都与几何对象的对称性有关。关于对称性的计算的代数工具将被完善，然后这些工具将被应用于回答自然问题。

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.