## Algebraic & Geometric Topology

### On R L Cohen's $\zeta$–element

Xiugui Liu

#### Abstract

Let $p$ be a prime greater than three. In the $p$–local stable homotopy groups of spheres, R L Cohen constructed the infinite $ζ$–element $ζn−1∈π2pn+1−2pn+2p−5(S)$ of order $p$. In the stable homotopy group $π2pn+1−2pn+2p2−3(V(1))$ of the Smith–Toda spectrum $V(1)$, X Liu constructed an essential element $ϖk$ for $k≥3$. Let $βs∗=j0j1βs∈[V(1),S]2sp2−2s−2p$ denote the Spanier–Whitehead dual of the generator $βs′′=βsi1i0∈π2sp2−2s(V(1))$, which defines the $β$–element $βs$. Let $ξs,k=βs−1∗ϖk$. In this paper, we show that the composite of R L Cohen’s $ζ$–element $ζn−1$ with $ξs,n$ is nontrivial, where $n>4$ and $1. As a corollary, $ξs,n$ is also nontrivial for $1.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1709-1735.

Dates
Revised: 24 February 2011
Accepted: 4 March 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715242

Digital Object Identifier
doi:10.2140/agt.2011.11.1709

Mathematical Reviews number (MathSciNet)
MR2821438

Zentralblatt MATH identifier
1230.55009

Subjects
Primary: 55Q45: Stable homotopy of spheres
Secondary: 55Q10: Stable homotopy groups

#### Citation

Liu, Xiugui. On R L Cohen's $\zeta$–element. Algebr. Geom. Topol. 11 (2011), no. 3, 1709--1735. doi:10.2140/agt.2011.11.1709. https://projecteuclid.org/euclid.agt/1513715242

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