Unstable Adams Operations on p-local compact groups

这个项目的研究对象是所谓的p-局部紧群，更具体地说是这些对象的一族自映射，称为不稳定亚当斯运算。在代数拓扑学中，人们通常试图找到研究拓扑空间的代数模型。p-局部紧群（英语：p-local compact groups）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称为p-完备化的过程隔离编码在这些空间中的p-初等信息。这个概念，由于工作的PI与Broto和奥利弗，准确地概括了什么区别类的p-完全分类空间的紧李群和类似的他们（例如同伦理论的模拟：p-紧群）从一般空间。它将这些空间的同伦理论与产生它的p-初等代数信息联系起来，并使对现象的双向系统研究成为可能。如果G是紧李群，则G的任何自同态诱导其p-完备分类空间的自映射。但如果G是无限的，则存在另一类重要的自映射，称为定义在分类空间上的不稳定亚当斯运算。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核心作用。1992年的定理由于Jackowski，麦克卢尔和奥利弗其中声称，为连接紧凑李群这些地图是确定的同伦由其影响上同调，并为简单的紧凑连接李群这些地图是唯一的自我等价的各自分类空间不诱导同态占据了一个大文件中的两个部分在数学年鉴。p-局部紧群的不稳定亚当斯运算是最近两篇博士论文的主题。Junod（2009）证明在某些限制下，任何p-局部紧群都允许不稳定的亚当斯运算，而Gonzalez分析了这些运算，并能够证明在某些情况下它们的行为与人们通过类比紧李群和p-紧群所期望的非常相似。该项目旨在扩展朱诺和冈萨雷斯的工程，并以全面的方式处理施工作业，研究何时（即，它们存在的程度），如果它们存在，它们是否是唯一的，它们的同伦不动点，以及它们与所谓的p-局部有限群的关系。在这样做，我们将能够捆绑的理论与其他几个工程的主题，特别是推导出一些重要的结果，如所谓的稳定元素定理-声明的模p上同调的分类空间的p-局部紧群可以计算从代数信息，其中定义的对象通过一个封闭的公式。

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.