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. Isaksen 高维球体是所有几何对象的基本构建块。 事实证明，不同尺寸的球体只能以某些组合方式组合在一起，以创建更复杂的几何对象。 稳定同伦群的计算本质上与计算这些组合相同。 自 20 世纪中叶以来，这种计算一直是研究的主要课题。 这项工作属于同伦理论领域，这是一种研究几何对象到某些变形的技术。 动机同伦理论是同伦理论的一个版本，适用于代数几何问题。 该项目利用经典同伦理论和动机同伦理论之间的异同来计算稳定同伦群。球谱稳定同伦群的计算是同伦理论中最基本的问题之一。 该项目的目标是应用素数 2 处的 Adams 谱序列来计算： (1) 经典稳定同伦群； (2) C 上的动机稳定同伦群； (3) R 上的基元稳定同伦群； (4) C2 等变稳定同伦群。 这四种计算密切相关。 它们的联系揭示了仅在一种类型的稳定同伦群中不明显的结构。 从表面上看，C2 等变 Adams 谱序列的计算呈现出难以管理的技术复杂性。 有一条通往 C2 等变计算的路径，该路径通过 C-动机和 R-动机计算的中间阶段进行。 在每个阶段，都会出现新的复杂性，但如果一次解决一个问题，它们是可以控制的。第一步是计算充当 Adams 谱序列的 E2 页的代数 Ext 群。 这些 Ext 组本身相当复杂，并且通常是通过辅助谱序列获得的。 由于这部分问题完全是代数问题，因此计算机在这里可以发挥很大的作用。 第二步是计算 Adams 微分和隐藏扩展。这个过程通常需要对 Toda 括号进行微妙的处理，并且不再是代数的。 将采用多种技术：（a）在一定维度范围内进行强力计算； (b) 机器辅助计算，产生代数数据并将许多单独的计算事实组织成一个一致的整体； (c) 通过周期性算子描述稳定同伦群的全局结构； (d) Adams-Novikov 和 Adams 谱序列之间的比较。