Computational Chromatic Homotopy Theory

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AbstractAward: DMS 1612020, Principal Investigator: Agnes BeaudryThe project supported by this grant is part of a large scale effort to understand the homotopy groups of spheres, one of the fundamental problems in algebraic topology. The spheres, which are among the simplest geometric objects, are building blocks for more complex entities. The homotopy groups of spheres, which are used to study the connections between these basic components, are collections of continuous functions between spheres considered up to deformations. Many problems in other fields, especially differential topology, have been reduced to the study of these groups, which is notoriously difficult. However, a bridge has been built between algebraic geometry and algebraic topology that allows us to use sophisticated algebraic theory to enable calculations of the homotopy groups of spheres. This bridge is known as chromatic homotopy theory. This project studies two of the most important structural conjectures in this field, the telescope and chromatic splitting conjectures. It has a strong computational component that will provide data to help study these two fundamental problems.Chromatic homotopy theory uses higher analogues of K-theory which give rise to higher periodicity in the stable homotopy groups of spheres. The chromatic splitting conjecture is an attempt at explaining the relationship between different periodicities. Recently, the PI has disproved a special case of this conjecture. In a project with Goerss and Henn, the PI aims to explain this failure and reformulate the conjecture. In a related project with Xu, the PI plans to compute the homotopy groups of the K(2)-local sphere at the prime two. These computations use self-dual resolutions which appear to be related to Brown-Commenetz duality and the structure of the K(2)-local Picard group. The PI plans to study this connection with Bobkova, Goerss and Henn. A parallel part of the project is to study periodicity in relation to the telescope conjecture. In work with Behrens, Bhattacharya, Culver and Xu, the PI plans to detect periodic elements using a resolution of the sphere by topological modular forms. The telescope conjecture gives a connection between periodic elements detected by such methods and those detected by the K(2)-local sphere. Comparing the two computations may shed light on this conjecture.