#### Volume 11, issue 3 (2011)

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On R L Cohen's $\zeta$–element

### Xiugui Liu

Algebraic & Geometric Topology 11 (2011) 1709–1735
##### 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 ${\zeta }_{n-1}\in {\pi }_{2{p}^{n+1}-2{p}^{n}+2p-5}\left(S\right)$ of order $p$. In the stable homotopy group ${\pi }_{2{p}^{n+1}-2{p}^{n}+2{p}^{2}-3}\left(V\left(1\right)\right)$ 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. As a corollary, ${\xi }_{s,n}$ is also nontrivial for $1.

##### Keywords
stable homotopy groups of spheres, $\zeta$–element, Adams spectral sequence, May spectral sequence
Primary: 55Q45
Secondary: 55Q10