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Classifying spaces of compact Lie groups topology of p-compact groups

p-紧群拓扑的紧李群空间分类

基本信息

批准号：
16540088
负责人：
ISHIGURO Kenshi
金额：
$ 2.24万
依托单位：
Fukuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540088/
关键词：
Classifying spaces Homotopy p-compact groups Compact Lie groups Homotopy fixed point set Invariant theory Weyl群コンパクトLie群 p-local finite

项目摘要

The research on the classifying spaces of compact Lie groups is one of the major area in Homotopy Theory. Our results obtained during 2004 through 2005 are basically concerned with maps between classifying spaces and their applications, as well as the invariant theory. Dwyer--Wilkerson introduced the notion of p-compact group and studied its properties. The purely homotopy theoretic object appears to be a good generalization of a compact Lie group. A p-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note written by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of p-compact groups are being made.We state here our main results. First, we considered a further generalization of a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture : Suppose a-space X with an action of a finite p-group is p-complete and the cohomological dimension is finite. Then the fixed point set is an empty set if and only if the homotopy fixed point set is empty. Next, we consider a problem on the conditions of a compact Lie group G that its loop space of the p-completed classifying space be a p-compact group for a set of primes. In particular, we discuss the classifying spaces BG that are p-compact for all primes when the groups are certain subgroups of simple Lie groups. Finally, we discuss the invariant theory and the cohomology of classifying spaces. The cohomology can be expressed as an invariant ring under the action of the Weyl group. All Weyl groups are reflection groups. We obtained some results on certain reflection groups.

紧李群的分类空间的研究是同伦理论的主要领域之一。我们在2004到2005年间得到的结果基本上是关于分类空间及其应用之间的映射，以及不变量理论。Dwyer-Wilkerson引入了p-紧群的概念，并研究了它的性质。纯同伦理论对象似乎是紧李群的一个很好的推广。P-紧群具有丰富的结构，如极大环面、Weyl群等。Moeller在AMS公报上发表的一篇文章总结了他们的工作。P-紧群的同伦理论正在取得进一步的进展。我们在这里陈述了我们的主要结果。首先，我们考虑了Dror Farjoun和Zabrodsky关于不动点集与同伦不动点集之间关系的一个结果的进一步推广，该结果与推广的Sullivan猜想有关：设a-空间X具有有限p-群作用，X是p-完备的，上同调维度是有限的。则不动点集是空集当且仅当同伦不动点集为空。接下来，我们考虑了紧李群G的一个条件，即它的p-完备分类空间的环空间对于素数集是p-紧群。特别地，我们讨论了当群是单李群的某些子群时，对所有素数是p紧的分类空间BG。最后，我们讨论了分类空间的不变理论和上同调。上同调可表示为Weyl群作用下的不变环。所有的Weyl群都是反射群。我们得到了关于某些反射群的一些结果。