On the extension problems for three 33-stem homotopy groups

Abstract

This paper tackles the extension problems for three far-unsatble homotopy groups $\ddot{I}{€}_{39}(S^6)$, $\ddot{I}{€}_{40}(S^7)$, and $\ddot{I}{€}_{41}(S^8)$ localized at 2, the puzzles having remained unsolved for forty-five years. By a Toda bracket indexed by 1 included in $\ddot{I}{€}_{39}(S_{(2)}^6)$, 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 $\ddot{I}{€}_{33+n}(S_{(2)}^n)$, ($2\le n\le 9$, $n\ge 27$).

arXiv admin note: text overlap with arXiv:2406.02713