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Let's consider the related problem of whether or not the diagram $$ \begin{CD} G_{n+1} @> \pi >> S^n \\ @V\cap VV @VV E V \\ F_{n+1} @>>\lambda > \Omega^{n+2} S^{2n+2} \end{CD} $$ commutes. If it does then Greg's question (1) will commute if we replace $H$ by $\lambda$, since the map $O(n+1) \to S^n$ factors through $G_{n+1}$.

Reformulate the claim as follows: let $B = \Sigma A$ be a suspension and let $B\to G_{n+1}$ be any map. Then the claim is equivalent to the assertion that the diagram $$ \begin{CD} B @> \pi >> S^n \\ @V\cap VV @VV E V \\ F_{n+1} @>>\lambda > \Omega^{n+2} S^{2n+2} \end{CD} $$ commutes (by taking $B = \Sigma \Omega G_{n+1}$).

Let's consider the related problem of whether or not the diagram $$ \begin{CD} G_{n+1} @> \pi >> S^n \\ @VVV @VV E V \\ F_{n+1} @>>\lambda > \Omega^{n+2} S^{2n+2} \end{CD} $$ commutes. If it does then Greg's question (1) will commute if we replace $H$ by $\lambda$, since the map $O(n+1) \to S^n$ factors through $G_{n+1}$.

Reformulate the claim as follows: let $B = \Sigma A$ be a suspension and let $B\to G_{n+1}$ be any map. Then the claim is equivalent to the assertion that the diagram $$ \begin{CD} B @> \pi >> S^n \\ @VVV @VV E V \\ F_{n+1} @>>\lambda > \Omega^{n+2} S^{2n+2} \end{CD} $$ commutes (by taking $B = \Sigma \Omega G_{n+1}$).

Since $\Sigma F$ is a suspension, its Hopf invariant $\lambda(\Sigma F)$ is trivial. So, if we take the Hopf invariant of the composite $$ \Sigma F = (f,g,h_F) \circ (p_1 + p_2 + p_{12}) $$ and use the composition formula and the Cartan formula (in Boardman and Steer), we obtain, after some rewriting, the formula $$ \lambda(h_F) = (f \wedge g) \circ \Sigma p_{12} $$ Implicit in this calculation, which I've omitted, is the fact that the reduced diagonal maps of both $B$ and $X$ are trivial since these spaces are suspensions. I refer the reader to Boardman and Steer's paper for the details.

Returning to our original notation, note that the map $f$ is the suspension of composition $$ \begin{CD} B @>>> G_{n+1} @>\pi >> S^n \end{CD} $$ and the map $\lambda(h_F)$ is adjoint to the composite $$ \begin{CD} B\to G_{n+1} \to F_{n+1} @>\lambda >> \Omega^{n+2} S^{2n+2}\, , \end{CD} $$ whereas the map $(f \wedge g) \circ \Sigma p_{12}$ is the composite $$ \begin{CD} \Sigma^2 (B \times S^n) @>>> \Sigma^2 (G_{n+1} \times S^n) @>q >> \Sigma^2 G_{n+1} \wedge S^n @>\Sigma \pi \wedge 1 >> \Sigma S^n \wedge \Sigma S^n \end{CD} $$ where $q$ is the quotient map. By definition, the last composition is adjoint to the composition $$ \begin{CD} B @>>> G_{n+1} @> \pi >> S^n @> E >> \Omega^{n+2} S^{2n+2} \end{CD} $$ and the claim is proved!

Comments: (1). The above argument shows that the diagram of Greg's question (1) commutes after looping. The equation we derived $$ \lambda(h_F) = (f \wedge g) \circ \Sigma p_{12} $$ isn't generally valid when $B$ (or $X$) isn't a suspension. For general $B$, there is a correction term in the formula involving the reduced diagonal $B\to B \wedge B$. This gives evidence that Greg's diagram will fail to commute before taking loops.

Since $\Sigma F$ is a suspension, its Hopf invariant $\lambda(\Sigma F)$ is trivial. So, if we take the Hopf invariant of the composite $$ \Sigma F = (f,g,h_F) \circ (p_1 + p_2 + p_{12}) $$ and use the composition formula and the Cartan formula (in Boardman and Steer), we obtain, after some rewriting, the formula $$ \lambda(h_F) = (f \wedge g) \circ \Sigma p_{12} \qquad (*) $$ Implicit in this calculation, which I've omitted, is the fact that the reduced diagonal maps of both $B$ and $X$ are trivial since these spaces are suspensions. I refer the reader to Boardman and Steer's paper for the details.

Returning to our original notation, note that the map $f$ is the suspension of composition $$ \begin{CD} B @>>> G_{n+1} @>\pi >> S^n \, , \end{CD} $$ and the map $g$ is the identity map of $S^{n+1}$. The map $\lambda(h_F)$ is adjoint to the composite $$ \begin{CD} B\to G_{n+1} \to F_{n+1} @>\lambda >> \Omega^{n+2} S^{2n+2}\, \end{CD} $$ which is one of the composites of the reformulated claim.

The map $(f \wedge g) \circ \Sigma p_{12}$ is the composite $$ \begin{CD} \Sigma^2 (B \times S^n) @>>> \Sigma^2 (G_{n+1} \times S^n) @>q >> \Sigma^2 G_{n+1} \wedge S^n @>\Sigma \pi \wedge 1 >> \Sigma S^n \wedge \Sigma S^n \end{CD} $$ where $q$ is the quotient map. By definition, the last composition is adjoint to the composition $$ \begin{CD} B @>>> G_{n+1} @> \pi >> S^n @> E >> \Omega^{n+2} S^{2n+2} \end{CD} $$ which is the other composite of the reformulated claim. Thus the claim follows from equation $(\ast)$.

Comments: (1). The above argument shows that the diagram of Greg's question (1) commutes after looping. The equation $(\ast)$ isn't generally valid when $B$ (or $X$) isn't a suspension. 

For general $B$, there is a correction term in the formula involving the reduced diagonal $B\to B \wedge B$. This gives evidence that Greg's diagram will fail to commute before taking loops.

(2). The reader might ask why we've worked with the Boardman-Steer Hopf invariant rather than the one of James. The answer is that James' invariant doesn't satisfy a Cartan formula––––there are "partial" Cartan formulae, which were known to Barratt, but this is lost knowledge...

Boardman, J. M.; Steer, B. Axioms for Hopf invariants. Bull. Amer. Math. Soc. 72 1966 992–994.

In fact, Bill Richter (unpublished) shows that if $n > 2$ then Greg's diagram fails to homotopy commute (Richter also explicitly identifies the deviation from the diagram commuting).

(2). The reader might ask why we've worked with the Boardman-Steer Hopf invariant rather than the one of James. The answer is that James' invariant doesn't satisfy a Cartan formula––––there are "partial" Cartan formulae, which were known to Barratt, but this is lost knowledge...

Boardman, J. M.; Steer, B. Axioms for Hopf invariants. Bull. Amer. Math. Soc. 72 1966 992–994.

(4) The result of James about the diagram looping after one suspension isn't stated by James. In effect, James proves the (reformulated) claim in the special case when $B = S^p$ is a sphere (so Greg's diagram will commute on homotopy groups). The result is given by Corollary 15.9 of the paper:

James, I. M. On the suspension triad. Ann. of Math. (2) 63 (1956), 191–247.