Stable stems - the computation of stable homotopy groups of spheres

稳定茎 - 球体稳定同伦群的计算

• 批准号: 1606290
• 负责人: Daniel Isaksen
• 金额: $ 16.68万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1606290&HistoricalAwards=false
• 关键词: Stable; stems; computation; stable; homotopy

Award: DMS 1606290, Principal Investigator: Daniel C. IsaksenHigh-dimensional spheres are the basic building blocks of all geometric objects. It turns out that spheres of different dimensions can fit together in only certain combinations to create more complicated geometric objects. The computation of stable homotopy groups is essentially the same as counting these combinations. This computation has been a major topic of research since the middle of the 20th century. This work belongs to the field of homotopy theory, which is a technique for studying geometric objects up to certain kinds of deformations. Motivic homotopy theory is a version of homotopy theory that applies to problems in algebraic geometry. This project exploits the similarities and differences between classical homotopy theory and motivic homotopy theory to compute stable homotopy groups.The computation of stable homotopy groups of the sphere spectrum is among the most fundamental problems in homotopy theory. The projects goals are to apply Adams spectral sequences at the prime 2 to compute: (1) classical stable homotopy groups; (2) motivic stable homotopy groups over C; (3) motivic stable homotopy groups over R; and (4) C2-equivariant stable homotopy groups. These four computations are closely interrelated. Their connections reveal structure that is not apparent within just one type of stable homotopy group. At face value, the computation of the C2-equivariant Adams spectral sequence presents unmanageable technical complexities. There is a path to C2-equivariant computations that proceeds through intermediate stages of C-motivic and R-motivic calculations. At each stage, new complexities arise, but they are manageable when taken one at a time. The first step is to compute algebraic Ext groups that serve as the E2-page of the Adams spectral sequence. These Ext groups are in themselves quite complicated, and typically are obtained with an auxiliary spectral sequence. Since this part of the problem is entirely algebraic, computers can be used to great effect here. The second step is to compute Adams differentials and hidden extensions. This process usually requires subtle work with Toda brackets and is no longer algebraic. Several techniques will be employed: (a) brute force computation in a range of dimensions; (b) machine-assisted computation to produce algebraic data and to organize the many individual computational facts into a consistent whole; (c) description of the global structure of stable homotopy groups by means of periodicity operators; and (d) comparison between the Adams-Novikov and Adams spectral sequences.

奖项：DMS 1606290，首席研究员：Daniel C. Isaksenhigh维球是所有几何对象的基本基础。 事实证明，不同尺寸的球体只能在某些组合中融合在一起，以创建更复杂的几何对象。 稳定同质组的计算基本与计算这些组合相同。 自20世纪中叶以来，该计算一直是研究的主要主题。 这项工作属于同质理论领域，这是一种研究几何对象，直到某些变形的技术。 动机同义理论是一种同义理论的版本，适用于代数几何学中的问题。 该项目利用了经典同质性理论和动机同义理论之间的相似性和差异来计算稳定的同型组。球形谱的稳定同型组的计算是同型理论中最根本的问题之一。 项目目标是在Prime 2上应用ADAMS光谱序列来计算：（1）经典稳定同型组； （2）c上的动机稳定同性恋组； （3）r上的动机稳定同质组； （4）C2均衡稳定同型组。 这四个计算是密切相关的。 它们的连接揭示了仅在一种稳定的同型组中不明显的结构。 从表面上看，C2-Equivariant Adams光谱序列的计算呈现出难以管理的技术复杂性。 C2等级计算有一条途径，该计算通过C-动力和R-动力计算的中间阶段进行。 在每个阶段，都会出现新的复杂性，但是一次一次服用时，它们是可以控制的。第一步是计算用作ADAMS光谱序列的E2页的代数ext基团。 这些EXT组本身很复杂，通常是通过辅助光谱序列获得的。 由于该问题的这一部分完全是代数，因此可以在此处使用计算机来实现巨大效果。 第二步是计算Adams差异和隐藏的扩展。这个过程通常需要与TODA支架的微妙合作，并且不再是代数。 将采用几种技术：（a）在一定范围内的蛮力计算； （b）机器辅助计算以产生代数数据并将许多单个计算事实组织成一致的整体； （c）通过周期性操作员描述稳定同质组的全球结构； （d）Adams-Novikov和Adams光谱序列之间的比较。