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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.

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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.

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  • $\begingroup$ Hi John - Thanks. I'm not aware of the generalized Whitehead product map you are referring to. I understand Whitehead products only through the most elementary definitions - the standard one using the attaching map for a torus (which is just in homotopy), and the space level commutator map $G \times G \to G$ which defines the Samelson product and coincides with the Whitehead product (up to sign) when $G = \Omega X$. If you could elaborate on $P$ and perhaps give a reference I'd appreciate it. $\endgroup$
    – Dev Sinha
    Commented Feb 11, 2011 at 5:15
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    $\begingroup$ The map $P: \Sigma Y \wedge Y\to \Sigma Y$ is a composition of the map $f:\Sigma Y \wedge Y \to \Sigma Y \vee \Sigma Y$ with the fold map $\Sigma Y \vee \Sigma Y \to \Sigma Y$. Here $f$ has homotopy cofiber identified with $\Sigma Y \times \Sigma Y$. Alternatively, $P$ can be defined explicitly as in my next comment. $\endgroup$
    – John Klein
    Commented Feb 11, 2011 at 11:45
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    $\begingroup$ Method 2: Ganea defines the homotopy fiber of the inclusion $i: \Sigma Y \vee \Sigma Y \to \Sigma Y \times \Sigma Y$ and shows that it can be identified with $\Sigma (\Omega \Sigma Y) \wedge (\Omega \Sigma Y)$. Now use the evident map $ u: \Sigma Y \wedge Y \to \Sigma (\Omega \Sigma Y) \wedge (\Omega \Sigma Y)$ to obtain $P$ as the composite of $u$ followed by the usual map from a homotopy fiberof $i$ into the domain of $i$. $\endgroup$
    – John Klein
    Commented Feb 11, 2011 at 12:00
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    $\begingroup$ Method 3 elaborated: take the Samelson product of $G = \Omega \Sigma Y$, to get a map $G \wedge G \to G$. Precompose it with the evident map $Y \wedge Y \to G \wedge G$ to get a map $Y \wedge Y \to \Omega \Sigma Y$. Now take the adjoint. $\endgroup$
    – John Klein
    Commented Feb 11, 2011 at 12:12
  • $\begingroup$ I found a reference for $P$ (probably there are more definitive ones): projecteuclid.org/DPubS/Repository/1.0/… $\endgroup$
    – John Klein
    Commented Feb 11, 2011 at 20:05

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