Unstable Adams Operations on p-local compact groups

这个项目研究的对象是所谓的p局部紧群，更具体地说是这些对象的一组自映射，称为不稳定亚当斯操作。在代数拓扑学中，人们通常试图找到用于研究拓扑空间的代数模型。p局部紧群是与紧李群相关的空间的同伦理论行为的代数模型，或者更准确地说，它们的分类空间被修改为通过一个称为p补全的过程来隔离在这些空间中编码的单个素数p的p初级信息。这个概念，由于PI与Broto和Oliver的工作，精确地概括了紧李群的p完备分类空间及其同类（例如同伦理论类比：p紧群）与一般空间的区别。它将这些空间的同伦理论与产生它的p-初级代数信息联系起来，并使现象的双向系统研究成为可能。如果G是紧李群，则G的任何自同态都能导出其p完备分类空间的自映射。但是如果G是无限的，那么在分类空间上定义了另一类重要的自映射称为不稳定亚当斯操作。这些映射在紧李群和更一般的p紧群的分类空间的同伦理论中起着中心作用。1992年由Jackowski， McClure和Oliver提出的定理：对于连通紧连李群，这些映射通过它们对上同调的影响被确定到同伦；对于简单紧连李群，这些映射是各自分类空间中唯一不由同态诱导的自等价。这一定理在《数学年报》上占了两大部分。p局部紧群的不稳定Adams运算是最近两篇博士论文的主题。Junod（2009）表明，在某些限制条件下，任何p局部紧群都承认不稳定的Adams操作，而Gonzalez分析了这些操作，并能够证明在某些情况下，它们的行为非常类似于人们通过类比紧Lie和p紧群所期望的。本项目旨在推广Junod和Gonzalez的工作，全面地处理运算的构造，研究它们何时（即在多大程度上）存在，如果存在是否唯一，它们的同伦不动点，以及它们与所谓p局部有限群的关系。这样一来，我们就可以把这个理论与本学科的其他几部著作联系起来，特别是可以推导出一些重要的结果，比如所谓的稳定元定理——即p局部紧群的分类空间的模p上同调可以从代数信息中计算出来，这些代数信息通过一个封闭公式来定义对象。

The object under investigation in this project are the so called p-local compact groups, and more particularly a family of self maps of these objects called unstable Adams operations. In algebraic topology one generally attempts to find algebraic models for the study of topological spaces. p-local compact groups are an algebraic model for the homotopy theoretic behaviour of spaces associated with compact Lie groups, or more precisely, their classifying spaces modified as to isolate the p-primary information encoded in these spaces for a single prime number p, by a process called p-completion. This concept, due to the work of the PI with Broto and Oliver, encapsulates precisely what distinguishes the class of p-completed classifying spaces of compact Lie groups and the likes of them (e.g. the homotopy theoretic analog: p-compact groups) from general spaces. It connects the homotopy theory of these spaces to the p-primary algebraic information which gives rise to it, and enables a two-way systematic study of phenomena. If G is a compact Lie group then any endomorphism of G induces a self map of its p-completed classifying space. But if G is infinite, then there is an additional important class of self maps called unstable Adams operations defined on the classifying space. These maps play a central role in the homotopy theory of classifying spaces of compact Lie groups, and more generally p-compact groups. The 1992 theorems due to Jackowski, McClure and Oliver which claims that for connected compact Lie groups these maps are determined up to homotopy by their effect on cohomology, and that for simple compact connected Lie groups these maps are the only self equivalences of the respective classifying space not induced by homomorphism occupy a large paper in two parts in the Annals of Mathematics. Unstable Adams operations for p-local compact groups are the subject of two recent PhD theses. Junod (2009) showed that under some restrictions, any p-local compact group admits unstable Adams operations, while Gonzalez analysed these operations and was able to show that in some cases their behaviour is very similar to what one would expect by analogy to compact Lie and p-compact groups. This project aims to extend the works of Junod and Gonzalez and deal in a comprehensive way with the construction of the operations, study when (i.e., for what degree) they exist, if they exist whether they are unique, their homotopy fixed point, and their relation to the so called p-local finite groups. In doing so we will be able to tie up the theory with several other works in the subject, and in particular to deduce a number of important results, such as the so called Stable Elements Theorem - the statement that the mod p cohomology of classifying space of a p-local compact group can be computed from algebraic information which defines the object by means of a closed formula.