The E and H maps in the EHP sequence have models that aren't too hard to define. To review, the E map is induced on homotopy by $E: \Omega^n S^n \to \Omega^{n+1} S^{n+1}$ sending a map to its suspension. The H map has a nice model once one knows about James's theorem that $\Sigma \Omega \Sigma X$ splits as $\bigvee\_i \Sigma X^{\wedge i},$ where $X^{\wedge i}$ is $X$ smashed with itself $i$ times. When $X$ is $S^n$ one has $\Sigma \Omega S^{n+1}$ then projects
onto the "quadratic" wedge factor $\Sigma (S^n)^{\wedge 2} \cong S^{2n + 1}$. The adjoint of this map is $H : \Omega S^{n+1} \to \Omega S^{2n + 1}$. 


My question is "what about $P$"? That is, what is a model for this map? As I understand it, the usual approach is to show that $E$ and $H$ fit into a fiber sequence (when localized at two, or integrally through a range(?)) and then $P$ is the connecting map. I was disappointed to learn that despite its name $P$ only coincides with a Whitehead product in some cases. Of course, there are general nonsense constructions of connecting maps for fibrations in terms of the maps one already has in hand, but I want to know:


Is there an "intrinsic" model for $P$, one which doesn't rely on the $E$ and $H$ maps in its definition?


I would also be interested in an intrinsic model for the map on homotopy, say in terms of framed bordism.



Models for P map in EHP sequence
Dear Dev, 


You probably know this, but the map $P$ is the Whitehead product map in the metastable range in the following sense: Suppose $X = \Sigma Y$ is a suspension. Then the generalized Whitehead product map $$P:\Sigma Y \wedge Y\to \Sigma Y$$ coincides with map
from the homotopy fiber of $$E: \Sigma Y \to \Omega \Sigma (\Sigma Y)$$ into the domain of $E$ in the metastable range (roughly thrice the connectivity of $Y$). 


By "coincide," I mean that since $E\circ P$ has a preferred null-homotopy, one has a preferred map 
$$
\Sigma Y \wedge Y \to \text{hofiber}(E)
$$
which is a metstable equivalence. It is in this sense that we write "$P$ " for the
connecting map in the EHP sequence (which is a metastable homotopy fiber sequence).


However, I do not believe this map will *integrally* factor up to homotopy through $\Omega^2\Sigma (Y\wedge Y)$ even when $Y$ is a sphere (except in the cases when $Y = S^{n-1}$ is an $H$-space: $n=2,4,8$).


So your question seems to live in the world of 2-localized spheres, and not the spheres themselves. 


