On the extension problems for three 33-stem homotopy groups

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 $\pi_{33+n}(S^{n}_{(2)})$, ($2\leq n\leq 9$, $n\geq27$).