Stable stems - the computation of stable homotopy groups of spheres

奖项：DMS 1606290，首席研究员：丹尼尔·C·伊萨克森高维球体是所有几何物体的基本构件。事实证明，不同维度的球体只能以特定的组合组合在一起，以创建更复杂的几何对象。稳定同伦群的计算实质上与计算这些组合是相同的。自20世纪中叶以来，这种计算一直是一个主要的研究课题。这项工作属于同伦理论领域，同伦理论是一种研究直到某种变形的几何对象的技术。动机同伦理论是应用于代数几何问题的同伦理论的一种版本。本课题利用经典同伦理论和动机同伦理论的异同来计算稳定同伦群。球谱的稳定同伦群的计算是同伦理论中最基本的问题之一。项目的目标是应用素数为2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上的动机稳定同伦群；(4)C2-等变稳定同伦群。这四种计算是密切相关的。它们的联系揭示了仅在一种稳定同伦群中并不明显的结构。从表面上看，C2等变亚当斯谱序列的计算呈现出难以处理的技术复杂性。有一条通往C2等变计算的路径，它通过C-Motivic和R-Motivic计算的中间阶段进行。在每个阶段，都会出现新的复杂性，但如果一次只处理一个，这些问题是可以控制的。第一步是计算作为亚当斯谱序列的E2-PAGE的代数Ext群。这些Ext基团本身相当复杂，并且通常利用辅助光谱序列来获得。由于这部分问题完全是代数问题，计算机在这里可以发挥很大的作用。第二步是计算亚当斯微分和隐延拓。这个过程通常需要对Toda括号进行微妙的处理，不再是代数的。将采用几种技术：(A)在一定维度范围内的蛮力计算；(B)产生代数数据并将许多单独的计算事实组织成一致整体的机器辅助计算；(C)利用周期算子描述稳定同伦群的全局结构；以及(D)Adams-Novikov和Adams谱序列之间的比较。

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.