On R L Cohen’s ζ–element

Liu, Xiugui

doi:10.2140/agt.2011.11.1709

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On R L Cohen's $\zeta$–element

Xiugui Liu

Algebraic & Geometric Topology 11 (2011) 1709–1735

DOI: 10.2140/agt.2011.11.1709

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 $ζ_{n - 1} \in π_{2 p^{n + 1} - 2 p^{n} + 2 p - 5} (S)$ of order $p$ . In the stable homotopy group $π_{2 p^{n + 1} - 2 p^{n} + 2 p^{2} - 3} (V (1))$ of the Smith–Toda spectrum $V (1)$ , X Liu constructed an essential element $ϖ_{k}$ for $k \geq 3$ . Let $β_{s}^{*} = j_{0} j_{1} β^{s} \in {[V (1), S]}_{2 s p^{2} - 2 s - 2 p}$ denote the Spanier–Whitehead dual of the generator $β_{s}^{''} = β^{s} i_{1} i_{0} \in π_{2 s p^{2} - 2 s} (V (1))$ , which defines the $β$ –element $β_{s}$ . Let $ξ_{s, k} = β_{s - 1}^{*} ϖ_{k}$ . In this paper, we show that the composite of R L Cohen’s $ζ$ –element $ζ_{n - 1}$ with $ξ_{s, n}$ is nontrivial, where $n > 4$ and $1 < s < p - 1$ . As a corollary, $ξ_{s, n}$ is also nontrivial for $1 < s < p - 1$ .

Keywords

stable homotopy groups of spheres, $\zeta$–element, Adams spectral sequence, May spectral sequence

Mathematical Subject Classification 2000

Primary: 55Q45

Secondary: 55Q10

References

Publication

Received: 13 July 2010

Revised: 24 February 2011

Accepted: 4 March 2011

Published: 3 June 2011

Authors

Xiugui Liu
School of Mathematical Sciences and LPMC Nankai University Tianjin 300071 China

Volume 11, issue 3 (2011)