Combinational and Homotopy Theory of Classifying Spaces of fusion Systems

p-局部有限群的概念提供了一个自然的框架来探索拓扑学和代数学之间的深层关系。自从引入p-局部有限群以来，表示理论家和拓扑学家对这个问题的兴趣越来越大，在这些领域的研究人员之间产生了新的跨学科研究联系。这个新的理论产生了有趣的结果，以及迷人的新问题。一个p-局部有限群的基础是一个有限p-群S连同两个范畴。第一类称为S上的饱和融合系统，它是现代群论的中心对象。第二类，称为中心连接系统，是Broto，Levi和奥利弗的巧妙发展。它赋予融合系统，除其他外，建立链接到拓扑的分类空间。我们在项目中提出的目标是探索这些空间。下面我们回顾一下这个项目中的一些主题，并在支持案例中给出更详细的描述。p-局部有限群中的主要新成分，以前没有，是它们的分类空间。我们的主要目标是理解这些空间的同伦类型。好处是双重的。首先，这些信息对于理解p-局部有限群之间的映射空间至关重要。这是p-局部有限群理论的一个最终目标，事实上，p-局部有限群的动机是为了理解有限群的p-完备分类空间之间的映射空间。其次，通过掌握p-局部有限群的分类空间的代数不变量，例如它的基本群，我们的目标是通过纯粹的组合过程从其分类空间中恢复p-局部有限群，而不是我们目前拥有的复杂的拓扑过程。这样的构造是可取的，因为它将使分类空间中的拓扑数据更容易被群论学家所理解。第一种方法是使用同伦分解。这种方法背后的哲学是通过将更简单的空间粘合在一起来近似所考虑的分类空间。然后分析分解中的每个组件和胶合信息。这一过程有很长的历史，但已知的方法不能应用于p-局部有限群。第二种方法是代数拓扑学中常用的方法，即研究空间的代数不变量。到目前为止，只知道p-局部有限群的分类空间的上同调环。我们的目标是介绍和研究更多的代数不变量，将反映在两个类别的组合学基本的p-本地有限group.Funds要求支付的费用全职研究助理24个月。

The concept of p-local finite groups provides a natural framework to explore the deep relationship between Topology and Algebra. Since the introduction of p-local finite groups there is a growing interest in the subject by both representation theorists and topologists, yielding fresh interdisciplinary research contacts between researchers in these areas. The new theory gave rise to interesting results, as well as fascinating new questions.Underlying a p-local finite group is a finite p-group S together with two categories. The first category is called a saturated fusion system over S which is a central object in modern Group theory. The second category, called a centric linking system , is an ingenious development of Broto, Levi and Oliver. It endows the fusion system with, among other things, a classifying space which establishes the link to Topology. Our aim in the project we propose is to explore these spaces. We review below a few of the themes in this project and give a more elaborate description in the Case for Support.The main new ingredient in p-local finite groups, that was not available before, is their classifying spaces. Our main goal is to understand the homotopy type of these spaces. The benefit is twofold. First, such information is crucial to the understanding of mapping spaces between p-local finite groups. This is an ultimate goal of the theory, in fact, p-local finite groups were motivated by the desire to understand the mapping spaces between p-completed classifying spaces of finite groups. Second, by having a grip on algebraic invariants of the classifying space of a p-local finite group, for example its fundamental group, we aim to recover the p-local finite group from its classifying space by a purely combinatorial procedure, rather than the complicated topological one, which is currently at our possession. Such a construction is desirable because it will make the topological data in the classifying space more accessible to group theorists.There are two approaches we will exploit to achieve our goals. The first approach is to use homotopy decompositions. The philosophy behind this approach is to approximate the classifying space under consideration by gluing together simpler spaces. Then one analyses each one of the components and the gluing information in the decomposition. This procedure has a long history, but the known methods cannot be applied to p-local finite groups. Fresh ideas are needed to apply the philosophy of decompositions in the new setup.The second approach is the usual one in Algebraic Topology, that is, to study algebraic invariants of spaces. So far only the cohomology rings of the classifying spaces of p-local finite groups are understood. Our aim is to introduce and study more algebraic invariants that will reflect the combinatorics in the two categories underlying a p-local finite group.Funds are requested to cover the cost of a full time research assistant for 24 months.