On R L Cohen’sζ–element

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<s<p−1. As a corollary, ξs,n is also nontrivial for 1<s<p−1.