Combinational and Homotopy Theory of Classifying Spaces of fusion Systems

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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.