The Homotopy Theory of Classifying Spaces

DMS-0104318 Jesper Grodal该项目包括发展紧李群(包括有限群)的分类空间的同伦理论及其推广。其目的是解决同伦理论中涉及分类空间的中心问题，以及发现群论和表示论中新的关系问题。另一个目标是在相关领域产生结果，如群作用理论、等变上同调和群上同调。这个项目围绕着发展分类空间BG的同伦理论群论，或者更确切地说，发展它的p-完备性。这涉及到研究归纳理论、p-局部理论、表示理论以及这些对象的更一般的行为。所使用的方法来自于自Sullivan猜想的解以来发展起来的现代不稳定同伦理论，并结合现代群论的思想和技巧。形成一组对称的分类空间BG是对G中的“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中是普遍存在的，因此形成了一类重要的研究空间。此外，与群G相比，分类空间BG的刚性结构较弱，这使得某些在研究前者时不能直接处理后者的技巧是可用的。因此，研究BG的群论为研究原始群G提供了一种有趣而有用的方法。

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.