Partial groups and maps between p-completed classifying spaces

在任何数学学科中，人们可以问的最基本的问题之一是，给定两个性质相似的对象，它们之间的所有映射的集合是什么，至少达到某种等价关系。例如，在同伦理论中，人们可以明确地问，给定两个拓扑空间，它们之间的所有同伦映射类的集合是什么。类似地，在群论中，问题变成，给定两个群G和H，它们之间的所有同态或表示的集合是什么。p-局部有限群的理论正好处于同伦理论和群论的交叉点上。它是由Broto，奥利弗和PI在2000年初引入的，目的是创建一个统一的框架，在这个框架中，可以研究和推广有限群空间的p-局部同伦理论。他们取得了一些基本成果，并创造了一个研究领域，吸引了显着的注意数学家之间的工作都在组和代表性理论，并在代数拓扑。特别是设置的p-本地群体使他们能够获得代数分类的所有同伦类的自同伦等价的一个分类空间的p-本地有限群。这一点，沿着许多其他成功的主题，提出了期望，p-本地组理论将成为有助于理解同伦类更一般的地图之间的分类空间。然而，迄今为止，为实现这一目标所做的一切努力都失败了。一个p-局部有限群由三个数据位、一个有限p-群S和两个范畴F和L组成。范畴F称为S上的饱和融合系统，它以S的子群为对象，以子群之间的同态为态射，使得满足某些公理。这样的范畴在群论和模表示论中自然出现。范畴L，称为与F相关联的中心联结系统，在精确意义上是F的一种充实，它允许人们将一个拓扑上有意义的分类空间与F相关联。Broto、奥利弗和PI发展了一种阻碍理论，证明与给定融合系统F相关的中心连接系统的存在性和唯一性，但无法证明这样的连接系统确实存在，而且如果它们存在的话，它们确实是唯一的。Chermak（VR）最近引入了链接系统概念的进一步抽象，他将其命名为部分群，或者更具体地说是局部性。仔细研究这些对象使他能够提供一个壮观的积极解决的存在/唯一性问题。部分群体，不像链接系统，配备了一个非常自然的概念，它们之间的态射。另一方面，进一步的细化-局部性-仍然没有一个很好的概念，它们之间的态射，但有几个自然的想法，值得认真考虑。局部性与p-局部有限群之间的密切关系表明，从这个角度重新审视该理论可能会给出p-局部群的分类空间之间的映射空间的信息。该项目旨在仔细研究局部性，它们之间的态射，以及与链接系统/p-局部群的相互作用，以深入了解如何在其分类空间之间对映射或映射的同伦类进行分类的问题。

One of the most fundamental questions one can ask in any mathematical discipline is, given two objects of a similar nature, what is the set of all maps between them, at least up to some equivalence relation. In homotopy theory, for instance, once can ask specifically, given two topological spaces, what is the set of all homotopy classes of maps between them. Similarly, in group theory the question becomes, given two groups G and H, what is the set of all homomorphisms, or representations between them. The theory of p-local finite groups sits squarely in the intersection between homotopy theory and group theory. It was introduced by Broto, Oliver and the PI in the early 2000's with the aim to create a unified framework in which the p-local homotopy theory of classifying spaces of finite group can be studied and generalised. They obtained a number of fundamental results, and created a field of study which attracted significant attention among mathematicians working both in group and representation theory, and in algebraic topology. In particular the setup of p-local groups enabled them to obtain an algebraic classification of the group of all homotopy classes of self homotopy equivalences of a classifying space of a p-local finite group. This, along with numerous other successes in the subject, raised the expectation that p-local group theory will become instrumental in understanding homotopy classes of more general maps between classifying spaces. To date however, all attempts to reach this goal have failed. A p-local finite group consists of three bits of data, a finite p-group S, and two categories, F and L. The category F called a saturated fusion system over S, has subgroups of S as its objects and homomorphisms between them as morphisms, such that certain axioms are satisfied. Such categories arise naturally in group theory and modular representation theory. The category L, called a centric linking systems associated to F, is in a precise sense an enrichment of F which allows one to associate a topologically meaningful classifying space with F. Broto, Oliver and the PI developed an obstruction theory to the existence and uniqueness of a centric linking system associated to a given fusion system F, but were not able to show that such linking systems do exist, and indeed are unique if they do. Chermak (the VR) recently introduced a further abstraction of the concept of a linking system which he named partial groups, or more specifically localities. A careful study of these objects allowed him to deliver a spectacular positive solution to the existence/uniqueness problem.Partial groups, unlike linking systems, come equipped with a very natural concept of morphisms between them. The further refinement - localities - on the other hand still do not possess a good concept of morphisms between them, but there are several natural ideas which deserve serious consideration. It is the close relationship between localities and p-local finite groups which suggest that revisiting the theory from this perspective may give information about mapping spaces between classifying spaces of p-local groups. This project is aimed at a careful study of localities, morphisms between them, and the interaction with linking systems/p-local groups, in order to gain insight into the question of how to classify maps or homotopy classes of maps between their classifying spaces.