Cokernel of the stable J-homomorphism at odd primes

Question

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.

Answer 1

Mahowald's $\eta_j$ family (of dimension $2^j$, for $j\neq 2$) forms a collection of elements in the 2-power torsion stable stems that are not in the image of the J-homomorphism. In the Adams spectral sequence, $\eta_j$ is represented by $h_1 h_j$. Mahowald's construction in the paper "A new infinite family in ${_2}\pi_* S$" is beautiful, short, and somewhat mysterious. It arises from a map related to the J-homomorphism, Snaith's splitting of $\Omega^n \Sigma^n X$ for $X$ connected (in this case, $n=2$, and $X$ is a sphere), and the $\Lambda$-algebra description of the homotopy of the pieces in this splitting.

This family is phenomenological, and its discovery perhaps more attributable to Mahowald's genius than structural reasons about why it should exist. In contrast is the chromatic approach to stable homotopy theory, which is very structural: this gives a filtration of the stable stems coming from truncations of the Adams-Novikov spectral sequence associated the height of a formal group law (via Quillen's theorem on what the MU-based Adams-Novikov spectral sequence is actually computing).

An output of this machinery are the infinite families of "Greek letter" elements. The $\alpha$-family is precisely the image of $J$, and is associated with $K$-theory (and the multiplicative group, the formal group of height 1). The $\beta$-family is associated with a cohomology theory whose associated formal group is like that of a supersingular elliptic curve (i.e., height 2). Ravenel's book is a great reference on this, as is the original reference by Miller, Ravenel, and Wilson: "Periodic Phenomena in the Adams-Novikov Spectral Sequence," as well as the many homotopy theorists who are undoubtedly going to chime in on the subject.

Answer 2

I wish to record some other families of elements in $\mathrm{coker}(J)$ arising from computations of Toda and of Oka of stable homotopy groups at odd primes. These computations aren't merely for small stems (as I incorrectly believed from reading Toda's book and looking at tables in Hatcher or Ravenel texts).

Since the image of the stable J-homomorphism is known explicitly at any prime $p$, we can determine $\mathrm{coker}(J)$ in Toda-Oka range.

As is explained in Appendix B of Milnor-Stasheff, the image of the J-homomorphism in the $k$th stem is a cyclic group $\mathrm{Im}(J_k)$ whose $p$-component $\mathrm{Im}(J_k: p)$ is as follows for an odd prime $p$ (the case $p=2$ is slighly different but is just as easy to describe):

1. If $\frac{k+1}{2(p-1)}\notin\mathbb N$, then $\mathrm{Im}(J_k: p)$ is zero.

2. If $\frac{k+1}{2(p-1)}\in\mathbb N$, then $\mathrm{Im}(J_k: p)$ is isomorphic to $\mathbb Z_{p^{r+1}}$ where $p^r$ is the largest power of $p$ that divides $\frac{k+1}{2(p-1)}$.

Toda computed the $p$-component of the $k$th stem for $k<2p^2(p-1)-3$. I won't analyse $\mathrm{coker}(J_k)$ for his range except for one obvious example: If $k=2p(p-1)^2-1$, Toda shows that the $k$th stem has $p$ component $\mathbb Z_p\times\mathbb Z_{p^2}$, which is non-cyclic, and hence $\mathrm{coker}(J_k)$ is nontrivial; in fact $\mathrm{coker}(J_k: p)$ is $\mathbb Z_p$.

Oka in a series of papers, see here and references therein, extended Toda's range and constructed for each $p>3$, some elements (in his notations $\phi$, $\mu$, $\beta$) such that $\mathrm{coker}(J_k: p)$ is $\mathbb Z_{p^2}$. In particular, for the $\beta$-elements the degree $k$ is even, and for $\phi$-elements $\frac{k+3}{2(p-1)}\in\mathbb N$, so in these cases the image of $J$ is zero at $p$.

I do not know any examples where $\mathrm{coker}(J_k: p)$ has elements of order $>p^2$, and wonder if this is due to natural limitations of Toda-Oka range, or is there some other explanation of this phenomenon?