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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 the fiber of $$E: \Sigma Y \to \Omega \Sigma (\Sigma Y)$$ 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.

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.

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 the fiber of $$E: \Sigma Y \to \Omega \Sigma (\Sigma Y)$$ 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 is seems to live in the world of 2-localized spheres, and not the spheres themselves.

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 the fiber of $$E: \Sigma Y \to \Omega \Sigma (\Sigma Y)$$ 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.

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 the fiber of $$E: \Sigma Y \to \Omega \Sigma (\Sigma Y)$$ 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.

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

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 the fiber of $$E: \Sigma Y \to \Omega \Sigma (\Sigma Y)$$ 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 is seems to live in the world of 2-localized spheres, and not the spheres themselves.