Volume 11, issue 3 (2011)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 ζ–element ζn1 π2pn+12pn+2p5(S) of order p. In the stable homotopy group π2pn+12pn+2p23(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]2sp22s2p denote the Spanier–Whitehead dual of the generator βs = βsi1i0 π2sp22s(V (1)), which defines the β–element βs. Let ξs,k = βs1ϖk. In this paper, we show that the composite of R L Cohen’s ζ–element ζn1 with ξs,n is nontrivial, where n > 4 and 1 < s < p 1. As a corollary, ξs,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