A Toda bracket in the stable homotopy groups of spheres

Abstract

Let $p$ be a prime number greater than five. In the $p$–local stable homotopy groups of spheres, H Toda and J Lin, respectively, constructed the elements

$$\begin{array}{cc}{\gamma }_s\in {\pi }_{2s{p}^3-2p^2-2p-2s+1}\left(S\right),& \\ {\omega }_{m,n}\in {\pi }_{2p^{n+1}-2p^n+2p^{m+1}-2p^m+2p-6}\left(S\right)& \end{array}$$

of order $p$. In this paper, we show the nontriviality of the Toda bracket $\langle {\gamma }_s,p,{\omega }_{m,n}\rangle$ in the stable homotopy groups of spheres, where $n\ge m+2>6$, $3\le s<p$.