Can stable stems be generated by homotopy operations?

The motivation for this question comes from J. Cohen's result; at the prime $p=2$ his result says that any element in ${_2\pi_*^s}$ can be written as a (higher) Toda bracket of $2,\eta,\nu,\sigma$, indeed modulo indeterminacy in choosing various extension and coextension maps; it has to be said that this is mainly based on the way that he defines the higher brackets as I understand it.

Now, for a moment, let's think of Toda bracketing as a specific kind of homotopy operation. Then, how much of ${_2\pi_*}$ can be obtained by applying various homotopy operations to $2,\eta,\nu,\sigma$.

As an another motivation, consider the Kahn-Priddy theorem, or rather delooped version of it (look at "The Homology of the James-Hopf maps" by Kuhn); it asserts that there is a map

$$t:S^1\times QD_2S^1\to QS^1$$ inducing an epimorphism on ${_2\pi_*}$. Consequently, in dimensions $>1$ we may `capture' ${_2\pi_*}QS^1\simeq{_2\pi_{*+1}^s}$ just by $_2\pi_*^sD_2S^1$, that is any element in $\pi_{*+1}^s$ maybe written as $\alpha^*(1)$ defined by the stable composition

$$S^{n+1} \stackrel{\alpha}{\rightarrow} D_2S^1\stackrel{D_21}{\rightarrow} D_2S^1\stackrel{t}{\to} S^1$$

where $\alpha$ varies in ${_2\pi_*^s}D_2S^1={_2\pi_*^s}\Sigma\mathbb{R}P$. Now, I wonder, if such classic theorems can be interpreted in any such a way to show that we can generate ${_2\pi_*^s}$ by homotopy operations possibly coming from $D_2^iS^n$ with $D_2^i=D_2D_2^{i-1}$ and $n>-1$.

I wonder if anything on this is known, maybe in the land of $E_\infty$ operations on homotopy groups, something that we can do sums with it!

• I think that you should be a bit more specific about the class of operations that you admit, since any (stable) homotopy group of spheres determines a (stable) homotopy operation and is thus trivially generated by it. – Anton Fetisov May 4 '15 at 19:25
• @AntonFetisov. Take the operations coming from the $D_2$ construction and its iterations as above. – user51223 May 4 '15 at 19:29
• You should look at Bruner's chapters in the $H_{\infty}$-ring spectra volume, it is on Peter May's website in the Books section. – Sean Tilson May 5 '15 at 7:52
• @SeanTilson Thanks. Although, I am not sure if any answer to the above question is in there; as far I known something like this also has not appeared in his later works. – user51223 May 5 '15 at 10:31
• @user51223 I was not claiming he answered your question explicitly but that if you want to know something about homotopy operations then you should look there. He has tables of operations that might help give intuition or even show such a thing is not possible as some element is not in the image of the operations in the range he computed. – Sean Tilson May 5 '15 at 11:11