On the extension problems for the 33-stem homotopy groups of the 6-, 7- and 8-spheres

Abstract

This paper tackles the extension problems for the 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. We introduce a tool for the theory of determinations of unstable homotopy groups, namely, the $\mathcal{Z}$-shape Toda bracket, by which we are able to solve the extension problems with respect to these three homotopy groups.