# Differentials in the Adams Spectral Sequence for spheres at the prime p=2

How does one compute the differentials in the Adams Spectral Sequence for spheres at the prime 2 in the range $13\le t-s\le 20$? There seem to be 6 nonzero differentials, and at this point I only understand $d_2(h_4)=h^2_3h_0$.

There seem to be two methods that are used or referenced in various texts, but I haven't figured out exactly how to apply either in this context. The first is the Massey Product/Toda Product (apparently they are the same, but Massey is algebraic and works in $E_2$, and Toda is topological and works in $\pi_*^s$). The second is by building a cofiber sequences $S^0\to S^0\cup_f e^i\to S^i$ which gives a long exact sequence in both the $\pi^s_*$ and the spectral sequence itself.

If possible, could somebody point me to a resource where they use these methods in this range, or give me a hint on how I can try to do this?

Thanks a bunch -Joseph

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Another option is Kochman's book "Bordism, Stable Homotopy and Adams Spectral Sequences". If I recall correctly the differentials are fully calculated in this range, using Massey products. –  Drew Heard Jul 17 '12 at 15:21

As far as I know, there are no systematic methods known to guarantee complete information. One uses the algebra structure to deduce differentials from known calculations wherever possible. One uses relations between Massey products and Toda brackets to deduce some differentials, and one uses relations between Steenrod operations and their homotopical analogues to deduce others. Some systematic discussion is in Bruner's contribution to "$H_{\infty}$ ring spectra and their applications'' and there are many more recent sources.
@Joseph: the section Peter is referring to has a nice explanation for the differential you list. It is not specific to $d_2$, $h_i$ or the range you give. –  Sean Tilson Jul 20 '12 at 1:19