Homotopy Methods in a Generalization of Lie Group Theory

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2002到2003年间得到的结果基本上是关于分类空间之间的映射及其应用，Dwyer-Wilkerson定义了p-紧群并研究了它的性质。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，我们推广了Dror Farjoun和Zabrodsky关于不动点集和同伦不动点集之间关系的一个结果，该结果与推广的Sullivan猜想有关：对于有限p-群Π，设Π-空间X是F_p-完备的，CD_p(X)是有限的。则X^π为空集当且仅当同伦不动点集X^&lt；hπ&gt；为空。作为应用，我们讨论了p-紧群的齐次空间上考虑作用的延拓问题。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。得到了BG对所有素数都是p紧的充要条件。当H是单李群G的子群时，我们问BH对于素数集是否p紧，并得到了某些结果。

The research on the classifying spaces of compact Lie groups is one of the major area in Homotopy Theory. Our results obtained during 2002 through 2003 are basically concerned with maps between classifying spaces and their applications Dwyer-Wilkerson defined a 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 generalize 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 : For a finite p-group Π, suppose a Π -space X is F_p-complete and cd_p(X) is finite. Then X^πis an empty set if and only if the homotopy fixed point set X^<hπ> is empty. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups. 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. A necessary and sufficient condition that BG be p-compact toral for all primes has been obtained. We ask if BH is p-compact for a set of primes when H is a subgroup of a simple Lie group G and obtaine certain results.