A Product Involving the β-Family in Stable Homotopy Theory

In the stable homotopy groups πq(pⁿ+pᵐ+1)-3(S) of the sphere spectrum S localized at the prime p greater than three, J. Lin constructed an essential family ξm,n for n ≥ m + 2 5. In this paper, the authors show that the composite ξm,nβs in πq(pⁿ+pᵐ+sp+s)-5(S) for 2 ≤ s p is non-trivial, where q = 2(p - 1) and βs in πq(sp+s-1)-2(S) is the known β-family. We show our result by explicit combinatorial analysis of the (modified) May spectral sequence.