Some nontrivial secondary Adams differentials on the fourth line (Q6108710)

For studying the \(p\)-component of the stable homotopy groups \(\pi_*(S)\) of spheres, the Adams spectral sequence \(E_2^{s,t}=\mathrm{Ext}_{A_*}^{s,t}(\mathbb F_p,\mathbb F_p)\Rightarrow \pi_{t-s}(S)\) is one of the useful tools. Here, \(p\) denotes a prime number. In this paper, the authors introduce a procedure to compute the secondary differential \(d_2\) of the Adams spectral sequence, and show some non-trivial actions of the secondary differential \(d_2\colon E_2^{4,t}\to E_2^{6,t+1}\) of the Adams spectral sequence on elements in the fourth line \(E_2^{4,t}\). Previously, all of the non-trivial actions of the secondary differential on \(E_2^{s,t}\) with \(s\le 3\) were determined by a different procedure. It is well known that the secondary differential of the Adams spectral sequence is given by the secondary differential of the algebraic Novikov spectral sequence converging to the \(E_2\)-page of the Adams-Novikov spectral sequence converging to \(\pi_*(S)\). The above procedures are based on this fact. The authors' procedure is to compute the secondary differentials of the algebraic Novikov spectral sequence by the differentials of a cobar complex, while the other is to compute them by using matrix Massey products. By their procedure, we may compute the differentials without worrying about the indeterminacies of Massey products.