Show that in the Postnikov tower of an H-space, all the spaces are H-spaces and the maps are H-maps, commuting with the multiplication, up to homotopy.
Solution
Let \(P_nX\) denote the \(n\)-th Postnikov stage. The Postnikov construction is functorial up to homotopy, and \[ P_n(X\times X)\simeq P_nX\times P_nX \] because both sides have the same homotopy groups through degree \(n\) and none above.
If \[ \mu:X\times X\to X \] is an H-space multiplication, apply \(P_n\) to obtain \[ P_n\mu:P_n(X\times X)\to P_nX. \] Using the displayed product identification gives a multiplication \[ \mu_n:P_nX\times P_nX\to P_nX. \] The unit homotopies, associativity homotopy, and any inverse homotopy for \(\mu\) pass functorially to \(P_nX\), so each \(P_nX\) is an H-space.
Naturality of Postnikov truncation makes every tower map \[ P_nX\to P_{n-1}X \] commute with these multiplications up to homotopy. Hence \[ \boxed{\text{all stages are H-spaces and all Postnikov tower maps are H-maps}.} \]