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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.