On a family involving R.L Cohen's ξ -element

摘要

In 1981, R.L. Cohen constructed an infinite family of homotopy elements & e 7r?(S) represented by h_0b_k ∈ Ext_A~(3.2(p-1)(p~(k+1)+1))'(Z/p.Z/p) in the Adams spectral sequence, where p ＞ 2 and k ≥1. In this paper, we show that the composite map ξ_(n-1)β1βs+2 is nontrivial in the stable homotopy of spheres π_{(2(p-1)[p~n+(s+3)p+(s+2)]-7(S)}. where p ≥ 5, n ＞ 3 and 0 ≤ s ＜ p-4.