On the Convergence of Products γ̃sh1hn in the Adams Spectral Sequence

Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by (b0hn − h1bn−1) ∈ Ext n+p)q A (Zp,Zp) in the Adams spectral sequence. At the same time, he proved that i∗(h1hn) ∈ Ext +p)q A (H ∗M,Zp) is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn ∈ π(pn+p)q−2M . In this paper, with Lin’s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j̄γsīi′ξn in the stable homotopy groups of spheres. The new one is of degree pq + spq + spq + (s− 2)q + s− 6 and is represented up to a nonzero scalar by h1hnγ̃s in the Es+2,∗ 2 -term of the Adams spectral sequence, where p ≥ 7, q = 2(p− 1), n ≥ 4 and 3 ≤ s < p.