# Cokernel of the stable J-homomorphism at odd primes

Where can one learn about odd-primary components of the cokernel of the stable J-homomorphism?

According to wonderful Wikipedia article on Homotopy groups of spheres, the "hard" part of the stable stem is the cokernel of $J$. I am not an expert and have trouble finding what is known. After an extensive search all I found was

1. some low-dimensional computations (in $k$th stem for $k\le 17$) which do not work for my current purpose.

2. Theorem 1.1.14 in Ravenel's "green book" book which gives some infinite families. Embarassingly, I do not even understand the statement of the theorem and cannot locate its proof (which is probably implicit in Section 4 of Chapter 4).

Theorem 1.1.14 says in particular "For $p\ge 3$ the $p$-component of $\mathrm{coker}\, J$ has the following generators in dimensions $\le 3pq − 6$ (where $q = 2p − 2$), each with order $p$", and then it goes on to list two generators in $(pq-2)$ stem and $(pq+q-3)$ stem, and six of their products.

Does this mean that the $p$-component of $\mathrm{coker}\, J$ in the $k$th stem (with $k\le 3pq-6$) is zero unless $\pi_k^S$ contains one of the either elements mentioned above? Or does this merely mean that these eight elements are nonzero?

Is there a more comprehensive account of what is known (preferably with proofs or references)? Again, I am after infinite families of nonzero elements.

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There are some nice pictures on Alan Hatcher's webpage: math.cornell.edu/~hatcher/stemfigs/stems.html. In particular the bottom forms the image of the $J$ homomorphism, and you can see how large the coker J is! –  Drew Heard Mar 6 at 2:49
@DrewHeard: I still need to learn how to read the diagrams, how does one see the order of $\mathrm{coker} J$? –  Igor Belegradek Mar 6 at 4:45
Each dot represents a copy of $Z/p$, whilst connected vertical dots are meant to represent a non-trivial extension (e.g. at $p=2,\eta^3$ corresponds to a $\mathbb{Z}/8$) –  Drew Heard Mar 6 at 6:02

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