Mathematical Sciences: Lie Groups up to Homotopy

数学科学：李群到同伦

• 批准号: 9505006
• 负责人: Clarence Wilkerson
• 金额: $ 9.95万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505006&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Lie; Groups; up

9505006 Wilkerson Ever since work of Serre in the 1950's, algebraic topology has often been studied one prime at a time. This idea, borrowed from algebra and number theory, has resulted in new questions being raised. For example, what does it mean for a topological space to be a Lie group at a particular prime p? One can answer with a reformulation that combines the group structure on G and the topological structure on G into a single classifying space, BG. Further abstraction of the properties true for all BG leads to the definition of a p-compact group which is a suitable homotopical analogue of a Lie group at the prime p. Remarkably, these p-compact groups possess analogues of standard Lie theoretical constructions such as maximal tori, Weyl groups, and root systems. The goal is to use these structures to produce a classification that is similar in spirit to the classification of compact connected Lie groups, but different in detail. A by-product would be the complete solution to the Steenrod question (1961) of which spaces have mod-p cohomology algebras that are finitely generated polynomial rings. Algebraic topology studies geometric objects by abstracting more tractable algebraic information. One rich source of such data is the group of symmetries of a geometric configuration. These so- called Lie groups, after the 19th century analyst and geometer, Sophus Lie, have had profound applications in algebra, physics, and number theory. One advantage of algebraic topology and homotopy theory is that its natural tools, such as homology and homotopy groups, are impervious to small deformations or changes in the geometric structure under study. The goal of this work is to create a classification of the homotopical analogues of Lie groups by the study of the algebraic topology of their classifying spaces. ***

小行星9505006 自从工作塞尔在20世纪50年代，代数拓扑往往是研究一个总理的时间。 这个从代数和数论中借来的想法导致了新的问题的提出。 例如，一个拓扑空间在一个特定的素数p上是一个李群意味着什么？ 我们可以用一种重新表述的方法来回答这个问题，它把G上的群结构和G上的拓扑结构结合到一个单一的分类空间BG中。 进一步抽象的属性真正的所有BG导致定义的p-紧群，这是一个合适的同伦类似物的李群在总理p.值得注意的是，这些p-紧群拥有类似物的标准李理论建设，如最大环面，外尔群，和根系。 我们的目标是使用这些结构来产生一种分类，这种分类在精神上类似于紧致连通李群的分类，但在细节上不同。 一个副产品是Steenrod问题（1961）的完整解，其中空间具有mod-p上同调代数，这些代数是n-生成的多项式环。 代数拓扑学通过抽象更易处理的代数信息来研究几何对象。 这种数据的一个丰富来源是几何构型的对称群。 这些所谓的李群，在19世纪世纪的分析家和几何学家索菲斯·李之后，在代数、物理和数论中有着深远的应用。 代数拓扑和同伦理论的一个优点是，它的自然工具，如同调和同伦群，是不受所研究的几何结构的小变形或变化。 本工作的目标是通过研究李群的分类空间的代数拓扑来建立李群的同伦类似物的分类。 ***