The Homotopy Theory of Classifying Spaces

空间分类的同伦理论

• 批准号: 0104318
• 负责人: Jesper Grodal
• 金额: $ 7.25万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104318&HistoricalAwards=false
• 关键词: Homotopy; Theory; Classifying; Spaces

DMS-0104318Jesper GrodalThe project consists of developing the homotopy theory of classifying spaces of compact Lie groups (including finite groups) and their generalizations. The goal of this is to solve central problems in homotopy theory involving classifying spaces, as well as to discover new relationsto problems in group theory and representation theory. A further goal is to produce results in related fields such as in the theory of group actions, equivariant cohomology, and group cohomology. The project centersaround developing the homotopy theoretic group theory of the classifying space BG, or rather of its p-completion. This involves examining the induction theory, p-local theory, representation theory as well as the more general actions of these objects. The methods used come from modern unstable homotopy theory, developed since the solution of the Sullivan conjecture, combined with ideas and techniques from modern group theory.Forming the classifying space BG of a group of symmetries G is a way of geometrically or topologically encoding the "symmetries" present in G. Geometric spaces of this form and generalizations thereof are ubiquitous in topology, and hence form an important class of spaces to study. Furthermore, the less rigid structure of the classifying space BG, compared to that ofthe group G, makes certain techniques available in the study of the former which were not available when dealing with the later directly. Hence, studying the grouptheory of BG provides an interesting and useful way of studying the original group G.

DMS-0104318 Jesper Grodal该项目包括发展紧李群（包括有限群）及其推广的分类空间的同伦理论。其目标是解决同伦理论中涉及分类空间的中心问题，以及发现与群论和表示论中问题的新关系。另一个目标是在相关领域产生结果，例如在群作用理论，等变上同调和群上同调。该项目centersaround发展同伦理论群理论的分类空间BG，或更确切地说，其p-完成。这涉及到研究归纳理论，p-局部理论，表征理论以及这些对象的更一般的行为。所使用的方法来自于现代不稳定同伦理论，该理论是在解决沙利文猜想之后发展起来的，并结合了现代群论的思想和技术。形成一组对称G的分类空间BG是对G中存在的“对称”进行几何或拓扑编码的一种方式。这种形式的几何空间及其推广在拓扑学中无处不在，因此形成了一类重要的空间来研究。 此外，由于分类空间BG的结构比群G的结构刚性小，使得直接研究分类空间BG时所不具备的某些技术在分类空间BG的研究中成为可能。因此，研究BG的群论为研究原群G提供了一种有趣而有用的方法。