Stable stems - the computation of stable homotopy groups of spheres

Wayne State University, Detroit MI

Investigators

Abstract

Award: DMS 1606290, Principal Investigator: Daniel C. Isaksen High-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.