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

紧李群的分类空间研究是同伦理论的一个重要领域。我们在2004年至2005年期间获得的结果基本上是关于分类空间之间的映射及其应用，以及不变理论。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公报中总结了他们的工作。在p紧群的同伦理论方面取得了进一步的进展。我们在这里陈述我们的主要结果。首先，我们进一步推广了Dror Farjoun和Zabrodsky关于不动点集与同伦不动点集之间关系的结论，该结论与广义Sullivan猜想有关：假设具有有限p群作用的a空间X是p完全的，且其上同维是有限的。当且仅当同伦不动点集为空，则不动点集为空。其次，我们考虑了紧李群G的p完备分类空间的环空间是一组素数的p紧群的条件问题。特别地，我们讨论了当群是单李群的某些子群时，对所有素数都是p紧的分类空间BG。最后，讨论了分类空间的不变量理论和上同调。在Weyl群的作用下，上同调可以表示为一个不变环。所有Weyl小组都是反思小组。我们在某些反射组上得到了一些结果。

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.