Mathematical Sciences: Lie Groups up to Homotopy

[50505006] Wilkerson自从Serre在20世纪50年代的工作以来，代数拓扑常常一次只研究一个素数。这个从代数和数论中借鉴来的观点引发了新的问题。例如，一个拓扑空间在特定素数p处是李群是什么意思？我们可以用一种重新表述来回答这个问题，即把G上的群结构和G上的拓扑结构组合成一个单独的分类空间BG。对所有BG的性质的进一步抽象导致了p紧群的定义，它是在素数p上的李群的合适的同邻类似物。值得注意的是，这些p紧群具有标准李理论结构的类似物，如极大环面、Weyl群和根系统。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连通李群的分类，但在细节上有所不同。副产物将是Steenrod问题（1961）的完全解，该问题的空间具有模p上同代数，这些代数是有限生成的多项式环。代数拓扑通过抽象更易于处理的代数信息来研究几何对象。这类数据的一个丰富来源是几何构型的对称组。这些所谓的李群，以19世纪的分析学家和几何学家索夫斯·李命名，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，如同伦和同伦群，不受所研究的几何结构的微小变形或变化的影响。本工作的目的是通过研究李群的同调类似物的分类空间的代数拓扑来建立一类李群的同调类似物。＊＊＊

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. ***