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Homotoy Theory of Classifying Spaces

空间分类同伦理论

基本信息

批准号：
09640138
负责人：
ISHIGURO Kenshi
金额：
$ 1.47万
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：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640138/
关键词：
Classifying spaces Homotopy p-compact groups Compact Lie groups Pairing Admissible maps Whitehead積 Coxter群

项目摘要

The research on the classifying spaces of compact Lie groups has been one of the major area in Homotoy Theory. Our results obtained during 1997 through 1998 are basically concerned with maps between classifying spaces and their applications. Dwyer-Wilkerson defined ap-compact group and studied its properties. The purely homotopy theoritic 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 wrtten by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of the classifying spaces of p-compact groups are being made.We state here our main results. First, we consider the maps of p-compact groups of the form BX * BY*BZ.The main theorem shows that if the restriction map on BY is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the p-compact group Z.Next, for G =S^3 * .. * S^3, let X be a genus of BG.We investigate the monoid of rational equivalences of X, denoted by epsilon(X). It is shown that a submonoid of epsilon_0(X), denoted by delta_00(X), determines the decomposability of the space X.We also show converses to some known results for the classifying spaces of p-toral groups or p-compact toral group. Suppose G is a compact Lie group. The following results are obtained. If there is a positive integer k such that the n-th homotopy groups of the p-completion of BG are zero for all n <greater than or equal> k then the loop space of this space is a p-compact toral group. If the canonical map Rep(G, K)*[BG, BK] is bijective for any compact connected Lie group K, then G is a p-toral group. in addition, our work containesa research on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group.

紧李群的分类空间的研究一直是同伦理论的主要领域之一。我们在1997到1998年间得到的结果基本上是关于分类空间和它们的应用之间的映射。Dwyer-Wilkerson定义了AP-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》上的一篇笔记总结了他们的工作。关于p-紧群的分类空间的同伦理论正在取得进一步的进展。我们在这里陈述我们的主要结果。首先，我们考虑形式为BX*by*BZ的p-紧群的映射.主要定理表明，如果By上的限制映射是弱满同态，则对BX的限制应该通过p-紧群Z的中心的分类空间来分解.其次，G=S^3*.*S^3，设X是BG的亏格，我们研究X的有理等价么半群，记为epsilon(X)。证明了E_0(X)的一个子么半群决定了空间X的可分解性，并证明了关于p-型群或p-紧型群的分类空间的一些已知结果。设G是紧李群。得到了以下结果。如果存在正整数k，使得BG的p-完成的n阶同伦群对所有大于或等于&gt；k的n&lt；k都为零，则该空间的圈空间是p-紧的。如果标准映射Rep(G，K)*[BG，BK]对任意紧连通李群K是双射的，则G是p-群。此外，我们还研究了紧Lie群的p-完备分类空间的环空间是p-紧的条件。