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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
    
I did read Hatcher's explanations of the table but it does not answer all the questions. Here are some for the top table. What does "essentially the same as the image of $J$" mean? Does "essentially" refer to some abbreviation reflecting the size of $2$-components? If a graph in the 2nd band sits over several stems, is its group present in all of them. Does the isolated $\sigma^2 dot on the left of the 3rd band live both in 15th and 11th (it is kind of between them but closer to 15). Same for the isolated dot in the 2nd band. –  Igor Belegradek Mar 6 at 14:17
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I can also recommend Mark Behrens' talk recently at MSRI, which can be found msri.org/workshops/685. He explicitly shows the periodic families on Hatcher's charts –  Drew Heard Mar 6 at 22:21

2 Answers 2

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.

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Thank you! Is the order of $\eta_j$ known? –  Igor Belegradek Mar 6 at 4:32
    
I think that it's order 2; this is at least suggested by the Adams spectral sequence (since $h_0 h_1 h_j = 0$), but I don't actually know. –  Craig Westerland Mar 6 at 4:43
    
Ack, sorry; I never even saw the odd-primary part of your request. –  Craig Westerland Mar 6 at 4:48
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There is an odd-primary analogue of the $\eta_j$ family discovered by Ralph Cohen in "Odd primary infinite families in stable homotopy theory." That text is hard to find on the internet, but a more recent approach to the subject is Hunter-Kuhn –  Craig Westerland Mar 6 at 5:22
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A much simpler argument: at the prime $p$, the only contributions to the image of $j$ are in degrees of the form $* = 2(p-1)k-1$, which cannot be the dimension of Cohen's family. –  Craig Westerland Mar 6 at 14:52

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?

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