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.

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