The E_3 page of the Adams spectral sequence

Abstract

In [Bau06], Baues computed the secondary Steenrod algebra, the algebra of all secondary cohomology operations. Together with Jibladze, they showed that this gives an algorithm that computes all Adams d_2 differentials for the sphere [BJ04]. The goal of this paper is to reinterpret their results in the language of synthetic spectra in order to achieve stronger computational results. Using this, we obtain an algorithm that computes hidden extensions on the E_3 page that jump by one filtration, in addition to the d_2 differentials of Baues–Jibladze. We then implement and run this algorithm for the sphere up to the 140th stem. Combined with a generalized version of the Leibniz rule, these hidden extensions allow us to compute many longer differentials with ease. In particular, we resolve all remaining unknown d_2, d_3, d_4 and d_5 differentials of the sphere up to the 95th stem.