Homotopy Methods in a Generalization of Lie Group Theory

紧李群的分类空间研究是同伦理论的一个重要领域。Dwyer-Wilkerson定义了一个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一个与广义Sullivan猜想有关的结果：对于有限p群Π，假设Π -空间X是f_p -完全的，cd_p(X)是有限的。则当且仅当同伦不动点集X^<hπ>为空时，X^π是空集。作为应用，讨论了考虑p紧群在齐次空间上作用的可拓问题。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。得到了对所有素数BG是p紧集合的充分必要条件。当H是单李群G的子群时，我们讨论了H对一组素数是否p紧，并得到了一些结果。

The research on the classifying spaces of compact Lie groups is one of the major area in Homotopy Theory. Our results obtained during 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.