On the Convergence of Products $$ \ifmmode\expandafter\tilde\else\expandafter\~\fi{\gamma }_{s} h_{1} h_{n} $$ in the Adams Spectral Sequence

Abstract

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 \( {\left( {b_{0} h_{n} - h_{1} b_{{n - 1}} } \right)} \in {\text{Ext}}^{{3,{\left( {p^{n} + p} \right)}q}}_{A} {\left( {\mathbb{Z}_{p} ,\mathbb{Z}_{p} } \right)} \) in the Adams spectral sequence. At the same time, he proved that \( i_{ * } {\left( {h_{1} h_{{n}} } \right)} \in {\text{Ext}}^{{2,{\left( {p^{n} + p} \right)}q}}_{A} {\left( {H^{ * } M,\mathbb{Z}_{p} } \right)} \) is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element \( \xi _{n} \in \pi _{{{\left( {p^{n} + p} \right)}q - 2}} M \). 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 \( j{j}\ifmmode{'}\else$'$\fi\overline{j} \gamma ^{s} \overline{i} {i}\ifmmode{'}\else$'$\fi\xi _{n} \) in the stable homotopy groups of spheres. The new one is of degree pnq + sp2q + spq + (s − 2)q + s − 6 and is represented up to a nonzero scalar by \( h_{1} h_{n} \ifmmode\expandafter\tilde\else\expandafter\~\fi{\gamma }_{s} \) in the \( E^{{s + 2, * }}_{2} - {\text{term}} \) of the Adams spectral sequence, where p ≥ 7, q = 2(p − 1), n ≥ 4 and 3 ≤ s < p.