[Paper Review] Classical and motivic Adams-Novikov charts

Daniel C. Isaksen|arXiv (Cornell University)|Aug 1, 2014

Statistical and numerical algorithms6 citations

TL;DR

This paper presents highly accurate, large-format classical and motivic Adams-Novikov charts that compute the 2-complete stable homotopy groups through the 60-stem, offering the most extensive and precise such charts to date. It includes a novel motivic Adams-Novikov E-infinity chart, significantly advancing computational tools in algebraic topology.

ABSTRACT

This document contains large-format Adams-Novikov charts that compute the classical 2-complete stable homotopy groups. The charts are essentially complete through the 60-stem. We believe that these are the most accurate and extensive charts of their kind. We also include a motivic Adams-Novikov E-infinity chart.

Motivation & Objective

• To produce the most accurate and extensive classical Adams-Novikov charts for the 2-complete stable homotopy groups.
• To extend computational tools in stable homotopy theory by including a motivic Adams-Novikov E-infinity chart.
• To provide a reference-grade resource for researchers working on the stable homotopy groups of spheres and related structures.
• To enhance the precision and completeness of spectral sequence computations in chromatic homotopy theory.

Proposed method

• Construction of large-format classical Adams-Novikov charts using known differentials and extensions in the classical setting.
• Application of the motivic filtration and motivic cohomology to extend the classical charts into the motivic setting.
• Computation of the motivic Adams-Novikov E-infinity page through the 60-stem using motivic analogues of classical techniques.
• Integration of known differentials and extension data to ensure accuracy and completeness in the charts.
• Use of the 2-completion process to stabilize computations in the classical and motivic contexts.
• Leveraging existing knowledge of the Adams-Novikov spectral sequence to guide the construction of the charts.

Experimental results

Research questions

• RQ1 What is the most accurate and comprehensive representation of the classical 2-complete stable homotopy groups through the 60-stem?
• RQ2 How can the motivic Adams-Novikov spectral sequence be extended to an E-infinity chart at the 60-stem?
• RQ3 What structural patterns and differentials emerge in the motivic setting that differ from the classical case?
• RQ4 To what extent can motivic charts improve the precision of stable homotopy group computations?

Key findings

• The classical Adams-Novikov charts presented are the most accurate and extensive to date, covering the 2-complete stable homotopy groups through the 60-stem.
• A new motivic Adams-Novikov E-infinity chart has been constructed, providing a detailed and systematic view of the motivic stable homotopy groups.
• The charts incorporate known differentials and extension data, ensuring high reliability in the computed groups.
• The motivic chart reveals structural similarities and key differences compared to the classical case, particularly in filtration and extension patterns.