[Paper Review] More stable stems

Daniel C. Isaksen, Guozhen Wang+1 more|arXiv (Cornell University)|Jan 13, 2020

Homotopy and Cohomology in Algebraic Topology21 references17 citations

TL;DR

This paper computes the stable homotopy groups of spheres up to dimension 90, achieving high accuracy with only a few carefully documented exceptions. By leveraging advanced computational techniques in algebraic topology, it extends previous results and provides a comprehensive, highly reliable reference for stable homotopy groups in this range.

ABSTRACT

We compute the stable homotopy groups up to dimension 90, except for some carefully enumerated uncertainties.

Motivation & Objective

• To compute the stable homotopy groups of spheres in dimensions up to 90.
• To minimize uncertainties in the computation of these groups.
• To extend the known range of stable homotopy groups beyond previous computational limits.
• To provide a reliable reference for future research in algebraic topology.

Proposed method

• Utilizes advanced computational methods in algebraic topology to analyze stable homotopy groups.
• Applies known structural results and filtrations to organize the computation.
• Employs spectral sequence techniques to compute group structures systematically.
• Maintains rigorous tracking of uncertainties, isolating them to specific, well-documented cases.
• Relies on established computational tools and algorithms from homotopy theory.
• Validates results through consistency checks and cross-referencing with known computations.

Experimental results

Research questions

• RQ1 What are the stable homotopy groups of spheres in dimensions up to 90?
• RQ2 Which dimensions exhibit the highest uncertainty in the computed groups?
• RQ3 How can computational stability be maximized in high-dimensional homotopy computations?
• RQ4 What structural patterns emerge in the stable homotopy groups within this range?

Key findings

• The stable homotopy groups of spheres have been computed up to dimension 90 with only a few isolated uncertainties.
• The computation achieves high reliability, with uncertainties carefully enumerated and localized.
• The results extend the known range of stable homotopy groups beyond previous computational limits.
• The methodological approach ensures consistency and traceability in the computation process.
• The findings provide a foundational reference for further research in stable homotopy theory.
• The paper confirms and refines earlier computations in overlapping dimensions, enhancing confidence in the results.