On the extension problems for three 33-stem homotopy groups

TL;DR

This paper solves longstanding extension problems in specific unstable homotopy groups of spheres at the prime 2 by employing Toda brackets and advanced homotopy theory techniques, providing new detailed group computations.

Contribution

It introduces a novel approach using Toda brackets and deuspension properties to resolve 45-year-old extension problems in 33-stem homotopy groups of spheres at 2.

Findings

Resolved extension problems for $oldsymbol{\pi_{39}(S^{6})}$, $oldsymbol{\pi_{40}(S^{7})}$, and $oldsymbol{\pi_{41}(S^{8})}$.

Produced a comprehensive table of $oldsymbol{\pi_{33+n}(S^{n}_{(2)})}$ for specified n.

Enhanced understanding of unstable homotopy groups at the prime 2.

Abstract

This paper tackles the extension problems for three far-unsatble homotopy groups $\pi_{39}(S^{6})$, $\pi_{40}(S^{7})$, and $\pi_{41}(S^{8})$ localized at 2, the puzzles having remained unsolved for forty-five years. By a Toda bracket indexed by 1 included in $\pi_{39}(S^{6}_{(2)})$, which makes better use of the deuspension property of homotopy classes, we address the problems. As a corollary, through Thomeier's 8-step backward theorem of the metastable homotopy theory, together with the results of Oda, Mukai and Miyauchi, we show a table of the 33-stem homotopy groups $\pi_{33+n}(S^{n}_{(2)})$, ($2\leq n\leq 9$, $n\geq27$).