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_{2pⁿ⁺¹−2pⁿ+2p−5}(S) of order p. In the stable homotopy group π_{2pⁿ⁺¹−2pⁿ+2p²−3}(V (1)) of the Smith–Toda spectrum V (1), X Liu constructed an essential element ϖ_k for k ≥ 3. Let β*s = j₀j₁β^s in [V (1),S]_{2sp²−2s−2p} denote the Spanier–Whitehead dual of the generator β′′s = β^si₁i₀ in π_2sp²−2s(V (1)), which defines the β–element β_s. Let ξ_s,k = β*s−1ϖ_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, ζ–element, Adams spectral sequence, May spectral sequence

Mathematical Subject Classification

Primary: 55Q45

Secondary: 55Q10

References

Publication

Received: 13 July 2010
Revised: 24 February 2011
Accepted: 4 March 2011
Published: 3 June 2011

Authors