Stable homotopy groups of spheres

Area of Science:

• Algebraic Topology
• Differential Geometry
• Computational Mathematics

Background:

• Stable homotopy groups of spheres are fundamental in algebraic topology, yet notoriously difficult to compute.
• Previous computational methods were complex and prone to human error.
• Motivic homotopy theory offers a novel framework for studying these groups.

Purpose of the Study:

• To present a streamlined computational method for stable homotopy groups of spheres.
• To apply this method to determine these groups and classify smooth structures on spheres.
• To leverage motivic homotopy theory as a deformation of classical homotopy theory.

Main Methods:

• Utilizing motivic homotopy theory as a computational tool.
• Employing the Adams spectral sequence as the primary mathematical framework.
• Integrating significant machine computation to enhance accuracy and efficiency.

Main Results:

• Successfully computed the first 61 stable homotopy groups of spheres.
• Provided information on stable homotopy groups in dimensions 62 through 90.
• Determined the groups of homotopy spheres classifying smooth structures on spheres up to dimension 90 (excluding dimension 4).

Conclusions:

• The developed method offers a more efficient and less error-prone approach to computing stable homotopy groups.
• This work advances the understanding of the topology of spheres and their smooth structures.
• Motivic homotopy theory proves to be a powerful tool in algebraic topology.