Show that all Whitehead products in a path-connected H-space are trivial.
Solution
A Whitehead product \[ [f,g]\in\pi_{p+q-1}(X) \] measures the obstruction to extending \[ f\vee g:S^p\vee S^q\to X \] over \(S^p\times S^q\), whose top cell is attached by \([\iota_p,\iota_q]\).
If \(X\) is an H-space with multiplication \(\mu:X\times X\to X\), define \[ F:S^p\times S^q\to X, \qquad F(x,y)=\mu(f(x),g(y)). \] On the two coordinate axes this restricts, up to the unit homotopies of the H-space, to \(f\) and \(g\). Using the homotopy extension property, adjust \(F\) near the wedge so that its restriction is exactly \(f\vee g\). Thus \(f\vee g\) extends over the product, so the obstruction vanishes: \[ \boxed{[f,g]=0.} \] Since \(f\) and \(g\) were arbitrary, every Whitehead product in a path-connected H-space is trivial.