Stable stems

TL;DR

This paper provides a comprehensive analysis of 2-complete stable homotopy groups in both classical and motivic contexts, utilizing spectral sequences to compute and resolve extensions up to high stems, advancing the understanding of stable homotopy theory.

Contribution

It introduces new computations of motivic and classical stable homotopy groups using spectral sequences, including resolving hidden extensions and analyzing the cofiber of tau.

Findings

Computed motivic stable homotopy groups up to the 59-stem.

Resolved all Adams differentials and hidden extensions in the studied range.

Connected motivic and classical homotopy groups through the cofiber of tau.

Abstract

We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, eta, and nu, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences. We also compute the motivic stable homotopy groups of the cofiber of the motivic element tau. This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of tau are the same as the E2-page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.