Stable homotopy methods in p-local group theory

p-局域群论中的稳定同伦方法

• 批准号: 1007619
• 负责人: Kari Ragnarsson
• 金额: $ 7.67万
• 依托单位: DePaul University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007619&HistoricalAwards=false
• 关键词: Stable; homotopy; methods; local; group

In this project, the Principal Investigator will employ methods from stable homotopy theory to provide insight into the surprising relationship between p-local group theory, modular representation theory and stable homotopy theory. Specifically, the PI will first give a much simplified description of the p-local finite group model for the homotopy theory of fusion systems. This shows that a p-local finite group on a finite p-group is equivalent to map from the classifying space into a p-complete, nilpotent space with a stable retract satisfying Frobenius reciprocity up to homotopy. In recent work, joint with Radu Stancu, the PI has shown that saturated fusion systems on a finite p-group are in bijective correspondence with stable idempotents of the classifying space that satisfy Frobenius reciprocity. The PI will extend this result to a bijection between p-local finite groups and idempotents that satisfy Frobenius reciprocity up to structured homotopies. The crucial question in the field is on the existence and uniqueness of classifying spaces for saturated fusion systems (in the form of p-local finite groups). Using results from this project, the question can be approached by refining the current construction of classifying spectra from being a construction up to homotopy to producing structured classifying spectra, and steps will be taken in this direction.Groups are fundamental objects in mathematics used to keep track of the symmetries of an object (e.g. a set or a space). A group is said to act on an object if that object exhibits the symmetries encoded by the group. Among such objects the classifying space of a group is of special importance, as its homotopical properties contain information about all possible actions of the group on topological spaces. A finite group can be characterized as the fundamental group of its classifying space --the set of paths in the space beginning and ending in a fixed point where two paths are identified if one can be continuously deformed into the other. For a given prime p, the p-local structure of a group can be thought of as the system of symmetries of the group that can be detected by actions on sets whose cardinality is a power of p. This notion is made precise by fusion systems as introduced by Puig and developed by Broto-Levi-Oliver. As conjectured by Martino-Priddy, and proved by Oliver, p-localizing the classifying space mirrors this construction in topology. More generally one can consider abstract fusion systems that do not necessarily come from groups and p-local finite groups are a model for the corresponding classifying space suggested by Broto-Levi-Oliver. In this project the PI will elucidate this correspondence by providing a simpler model for classifying spaces of abstract fusion systems and take important steps toward proving the existence of classifying spaces for arbitrary fusion systems which is a fundamental question in the field.

在这个项目中，主要研究者将采用稳定同伦理论的方法，以深入了解p-局部群理论，模表示理论和稳定同伦理论之间令人惊讶的关系。 具体来说，PI将首先给出一个非常简化的描述的p-局部有限群模型的同伦理论的融合系统。 这表明有限p-群上的p-局部有限群等价于从分类空间映射到p-完备幂零空间，其稳定收缩满足Frobenius互易性直到同伦。 在最近的工作中，与Radu Stancu联合，PI已经证明了有限p群上的饱和融合系统与满足Frobenius互易性的分类空间的稳定幂等元是双射对应的。 PI将这个结果扩展到p-局部有限群和幂等元之间的双射，这些幂等元满足Frobenius互反直到结构同伦。 该领域的关键问题是饱和融合系统（以p-局部有限群的形式）分类空间的存在性和唯一性。 利用这个项目的结果，这个问题可以通过改进当前的分类光谱的结构来解决，从同伦结构到产生结构化的分类光谱，并将朝着这个方向采取步骤。群是数学中的基本对象，用于跟踪对象（例如集合或空间）的对称性。 如果一个对象表现出由群编码的对称性，则称该群作用于该对象。在这些对象中，群的分类空间具有特殊的重要性，因为它的同伦性质包含了群在拓扑空间中所有可能作用的信息。 一个有限群可以被描述为它的分类空间的基本群--空间中开始和结束于一个固定点的路径集合，如果一个路径可以连续变形为另一个路径，则可以识别两个路径。对于一个给定的素数p，群的p-局部结构可以被认为是群的对称性系统，可以通过对集合的作用来检测，集合的基数是p的幂。正如Martino-Priddy所指出的，并由奥利弗所证明的，p-局部化分类空间在拓扑学上反映了这种构造。 更一般地，可以考虑不一定来自群的抽象融合系统，并且p-局部有限群是Broto-Levi-奥利弗建议的相应分类空间的模型。 在这个项目中，PI将通过为抽象融合系统的分类空间提供一个更简单的模型来阐明这种对应关系，并采取重要步骤来证明任意融合系统的分类空间的存在性，这是该领域的一个基本问题。