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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

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up vote 9 down vote accepted

In that range of dimensions one can cheat, as I did in my 1964 thesis. That is available on MathSciNet, and the differentials are penciled in on page A.2 (near the end). It was an easy exercise then to deduce the differentials algebraically as the only ones consistent with Toda's calculations of the homotopy groups slightly beyond that range, from 1962. There is a nice fun contrast made between calculating Ext and calculating differentials in Adams' 1961 Berkeley lecture notes. As evidence that your question is not trivial, Adams had a mistake in the differentials, in the range of your question if my memory is correct, which I pointed out to him in the first of hundreds of letters between us. His answer was that he didn't have Toda's calculations in front of him when he wrote that.

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.

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Another good way to "cheat" is to compare the ASS and the ANSS in that range. This is done in Ravenel's green book, chapter 4, pg. 143-144 (2nd edition). The book is available online: math.rochester.edu/u/faculty/doug/mu.html –  Christian Nassau Jul 16 '12 at 6:50
Thank you. I have your thesis "The Cohomology of Restricted Lie Algebras and of Hopf Algebras", and I'm going to try to work through it today and tomorrow. I'm afraid I'm not as up on the contents of this paper as I probably should be, but I'm sure I can figure it out. Thanks again. –  Joseph Victor Jul 16 '12 at 21:03
@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

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