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