For an H-space \((X, x_0)\) with multiplication \(\mu: X \times X \to X\), show that the group operation in \(\pi_n(X, x_0)\) can also be defined by the rule \((f + g)(x) = \mu\bigl(f(x), g(x)\bigr)\).
Solution
Write the usual group operation on \(\pi_n(X,x_0)\) additively. Since \[ \pi_n(X\times X,(x_0,x_0))\cong \pi_n(X,x_0)\times\pi_n(X,x_0), \] the multiplication \(\mu:X\times X\to X\) induces a homomorphism \[ \mu_*:\pi_n(X)\times\pi_n(X)\longrightarrow\pi_n(X). \] Because \(x\mapsto\mu(x,x_0)\) and \(x\mapsto\mu(x_0,x)\) are homotopic to the identity, \[ \mu_*(a,0)=a, \qquad \mu_*(0,b)=b. \] Hence, using that \(\mu_*\) is a homomorphism, \[ \mu_*(a,b) =\mu_*((a,0)+(0,b)) =\mu_*(a,0)+\mu_*(0,b) =a+b. \] If \(a=[f]\) and \(b=[g]\), the class \(\mu_*(a,b)\) is represented by \[ x\longmapsto\mu(f(x),g(x)). \] Therefore this pointwise H-space multiplication gives exactly the usual operation on \(\pi_n(X,x_0)\).