On R L Cohen's $\zeta$–element

Abstract

Let $p$ be a prime greater than three. In the $p$–local stable homotopy groups of spheres, R L Cohen constructed the infinite $\zeta$–element ${\zeta }_{n-1}\in {\pi }_{2p^{n+1}-2p^n+2p-5}\left(S\right)$ of order $p$. In the stable homotopy group ${\pi }_{2p^{n+1}-2p^n+2p^2-3}\left(V\left(1\right)\right)$ of the Smith–Toda spectrum $V\left(1\right)$, X Liu constructed an essential element ${\varpi }_k$ for $k\ge 3$. Let ${\beta }_s^{\ast }=j_0{j}_1{\beta }^s\in {\left[V\left(1\right),S\right]}_{2s{p}^2-2s-2p}$ denote the Spanier–Whitehead dual of the generator ${\beta }_s^{\prime \prime }={\beta }^s{i}_1{i}_0\in {\pi }_{2s{p}^2-2s}\left(V\left(1\right)\right)$, which defines the $\beta$–element ${\beta }_s$. Let ${\xi }_{s,k}={\beta }_{s-1}^{\ast }{\varpi }_k$. In this paper, we show that the composite of R L Cohen’s $\zeta$–element ${\zeta }_{n-1}$ with ${\xi }_{s,n}$ is nontrivial, where $n>4$ and $1<s<p-1$. As a corollary, ${\xi }_{s,n}$ is also nontrivial for $1<s<p-1$.