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!

`$H_{\infty}$`

-ring spectra volume, it is on Peter May's website in the Books section. $\endgroup$ – Sean Tilson May 5 '15 at 7:52