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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}.
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
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 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 homo
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 subalgebr
A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.