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In previous work of the first author and Jibladze, the $E_3$-term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the $E_3$-term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms $E_m$. In this paper, we introduce $2$-track algebras and tertiary chain complexes, and we show that the $E_4$-term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations.
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These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007.
We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of $pi_{51}$ is $mathbb{Z}/8oplusmathbb{Z}/8oplusmathbb{Z}/2$. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors (cite{Isa1}, cite{IX}, cite{WX1}) through the 61-stem. The proof of this result uses the $RP^infty$ technique, which was introduced by the authors in cite{WX1} to prove $pi_{61}=0$. This paper advertises this method through examples that have simpler proofs than in cite{WX1}.
We make a conjecture about all the relations in the $E_2$ page of the May spectral sequence and prove it in a subalgebra which covers a large range of dimensions. We conjecture that the $E_2$ page is nilpotent free and also prove it in this subalgebra. For further computations we construct maps of spectral sequences which systematically extend one of the techniques used by May and Tangora.
We show the homological Serre spectral sequence with coefficients in a field is a spectral sequence of coalgebras. We also identify the comultiplication on the $E^2$ page of the spectral sequence as being induced by the usual comultiplication in homology. At the end, we provide some example computations highlighting the use the co-Leibniz rule.
This document contains large-format Adams-Novikov charts that compute the classical 2-complete stable homotopy groups. The charts are essentially complete through the 60-stem. We believe that these are the most accurate and extensive charts of their kind. We also include a motivic Adams-Novikov E-infinity chart.
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