Algebraic & Geometric Topology

On R L Cohen's $\zeta$–element

Xiugui Liu

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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 ζn1π2pn+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 k3. Let βs=j0j1βs[V(1),S]2sp22s2p denote the Spanier–Whitehead dual of the generator βs=βsi1i0π2sp22s(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<p1. As a corollary, ξs,n is also nontrivial for 1<s<p1.

Article information

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

Received: 13 July 2010
Revised: 24 February 2011
Accepted: 4 March 2011
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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