Oops, one more thing

So first of all, I realized that in my sleepiness yesterday, I left off the last part of the story of why homotopy groups are groups. More precisely, we need to show that if ${X, Y}$ are any pointed spaces, then

$\displaystyle \hom_{\mathbf{PT}}(\Sigma^2 X, Y)$

is an abelian group under the cogroup law of ${\Sigma^2 X}$ (the double suspension). But this group is just

$\displaystyle \hom_{\mathbf{PT}}(\Sigma X, \Omega Y)$

and one can check that the adjointness between ${\Sigma, \Omega}$ respects the group structure. And we showed by the Eckmann-Hilton argument yesterday that this is abelian under either the group law of ${\Omega Y}$ or the cogroup law of ${\Sigma X}$ ; they’re also both the same. So in particular, the homotopy classes out of a higher suspension form an abelian group. Since the homotopy groups ${\pi_n, n \geq 2}$ are defined in this way, they are abelian.