By R. Göbel, L. Lady, A. Mader
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Additional resources for Abelian Group Theory. Proc. conf. Honolulu, 1983
3. We are including this survey here because no comparable exposition exists in the literature and we believe these results should be understood by more than a handful of experts. In particular, this section could serve as an introduction to Mahowald . For computations at the prime 3, see Toda , which extends the known range for unstable 3-primary homotopy groups from 55 to 80. The EHP sequences are the long exact sequences of homotopy groups associated with certain fibration constructed by James  and Toda .
Recall that H ∗ (X) is a module over the mod (p) Steenrod algebra A, to be described explicitly in the next chapter. 1. Theorem (Adams ). Let X be a spectrum as above. There is a spectral sequence E∗∗∗ (X) with dr : Ers,t → Ers+r,t+r−1 41 42 2. SETTING UP THE ADAMS SPECTRAL SEQUENCE such that ∗ (a) E2s,t = Exts,t A (H (X), Z/(p)).
18(b). The first thing we must do is show that the elements produced are actually nontrivial in the E2 -term. This has been done only for α’s, β’s, and γ’s. For p = 2, β1 and γ1 are zero but for t > 1 βt and γt are nontrivial; these results are part of the recent computation of E22,∗ at p = 2 by Shimomura , which also tells us which generalized β’s are defined and are nontrivial. The corresponding calculation at odd primes was done in Miller, Ravenel, and Wilson , as was that of E21,∗ for all primes.