Computations in Equivariant Homotopy and Algebraic K-Theory

等变同伦和代数 K 理论中的计算

• 批准号: 1207774
• 负责人: Michael Hill
• 金额: $ 29.3万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207774&HistoricalAwards=false
• 关键词: Computations; Equivariant; Homotopy; Algebraic; Theory

AbstractAward: DMS 1207774, Principal Investigator: Michael A. HillThis project seeks to broaden our knowledge of equivariant computations and, by universality of certain maps, of the homotopy groups of spheres. Equivariant homotopy calculations are notoriously difficult, hybridizing representation theory and classical stable homotopy theory in beautiful, and sometimes mysterious, ways. The recent solution by the principal investigator, Hopkins, and Ravenel to the Kervaire Invariant One problem introduced several new tools and techniques to equivariant homotopy, rigidifying earlier homotopical work and generalizing naturally occurring filtrations. In this project, the PI intends to use these new, exciting tools to continue making computational inroads in equivariant homotopy. There are two main approaches: computing equivariant homotopy groups of the families of spectra introduced by the PI, Hopkins, and Ravenel, and reconceptualizing parts of the cyclotomic trace approaches to algebraic K-theory. The former uses slice filtration techniques to tackle spectra like the Hopkins-Miller spectra, allowing a direct attack on the stable homotopy groups of spheres from a non-traditional approach. The latter builds on the norm machinery to rewrite the classical constructions in the equivariant approaches to algebraic K-theory, recasting them in computationally more amenable forms.The project addresses directly the heart of algebraic topology: computing numbers and invariants to understand spaces. The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and geometric objects like spaces. Equivariant algebraic topology remembers a collection of symmetries inherent in a space as part of the data, systematically grouping spaces with the same symmetries, and the numbers and invariants produced must reflect this. Remembering the extra structure makes richer, but more complicated, computations, and it allows us to tease apart otherwise interconnected problems. For example, using equivariant methods, the principal investigator, Hopkins, and Ravenel solved the Kervaire Invariant One problem, the oldest outstanding problem in algebraic topology with roots dating back to the 1930s. This in turn gave information about how we can build spaces out of simpler ones like spheres. This project aims to build on the techniques developed in the solution, tackling other computational problems in algebra and topology.

摘要奖：DMS 1207774，首席研究员：迈克尔·A·希尔这个项目旨在扩大我们对等变计算的认识，并通过某些映射的普适性来扩大球面同伦群的知识。等变同伦计算是出了名的困难，以美丽的、有时甚至是神秘的方式将表示理论和经典的稳定同伦理论混合在一起。最近，主要研究者Hopkins和Ravenel对Kervaire不变量一问题的解决方案引入了几个新的工具和技术来等变同伦，僵化了早期的同伦工作，并推广了自然发生的过滤。在这个项目中，PI打算使用这些新的、令人兴奋的工具来继续在等变同伦中取得计算进展。主要有两种方法：计算PI、Hopkins和Ravenel引入的谱族的等变同伦群，以及将部分割圆迹方法重新概念化到代数K-理论。前者使用切片过滤技术来处理像霍普金斯-米勒光谱这样的光谱，允许从非传统方法直接攻击球体的稳定同伦群。后者建立在范数机制的基础上，以代数K理论的等变方法重写经典结构，以计算上更易于接受的形式重新塑造它们。该项目直接解决了代数拓扑的核心：计算数字和不变量来理解空间。代数拓扑学的目标是系统地在像数这样的代数对象和像空间这样的几何对象之间建立联系。等变代数拓扑学将空间中固有的对称性集合作为数据的一部分，系统地将具有相同对称性的空间分组，产生的数字和不变量必须反映这一点。记住额外的结构会让计算变得更丰富，但也更复杂，它让我们能够梳理出原本相互关联的问题。例如，主要研究者Hopkins和Ravenel使用等变方法解决了Kervaire不变一问题，这是代数拓扑学中最古老的悬而未决的问题，其根可以追溯到20世纪30年代。这反过来提供了关于我们如何从球体等更简单的空间中建造空间的信息。这个项目旨在建立在解决方案中开发的技术的基础上，解决代数和拓扑学中的其他计算问题。

项目成果

Detecting exotic spheres in low dimensions using coker J

• DOI: 10.1112/jlms.12301
• 发表时间: 2017-08
• 期刊: Journal of the London Mathematical Society
• 作者: Mark Behrens;Michael Hill;Michael J. Hopkins;M. Mahowald