An unstable approach to the May-Lawrence matrix Toda bracket and the 2nd James-Hopf invariant

In this paper, we give an unstable approach of the May-Lawrence matrix Toda bracket, which becomes a useful tool to detect the unstable phenomena. Then, we give a generalization of the classical isomorphisms between homotopy groups of $(J{S}^m,S^m)$ and $(J{S}^{2m},⁎)$ localized at 2. After that we provide a generalized H-formula for matrix Toda brackets. As an application, we show a new construction of ${\overline{\kappa }}^{\prime }\in {\pi }_{26}\left(S^6\right)$ localized at 2 which improves the construction of ${\overline{\kappa }}^{\prime }$ given by [4].