Why is the group $[\Sigma\Sigma X, Y]_{\ast}$ commutative?

Question

Can anyone give a reference (or explain here), why the group $[\Sigma\Sigma X,Y]_*$ is commutative? How is it related to the fact that $\Sigma X$ is a co-H-space?

Answer 1

This is not my area of expertise, so this is a rough idea of why $[\Sigma\Sigma X, Y]_{\ast}$ is a commutative group.

First of all, $\Sigma$ is left adjoint to $\Omega$ so $[\Sigma\Sigma X, Y]_{\ast} \cong [\Sigma X, \Omega Y]_{\ast}$. Now $\Sigma X$ is a cogroup object in $\mathsf{hTop}_{\ast}$ (which is even stronger than being a co-H-space) and $\Omega Y$ is a group object in $\mathsf{hTop}_{\ast}$, so $[\Sigma X, \Omega Y]_{\ast}$ has two natural group structures. It should then follow from the Eckmann-Hilton argument that the two group structures coincide and that the group is abelian.

May gives a sketch of a more direct proof which doesn't use any such language in a lemma at the end of Chapter $8$, section $2$ of A Concise Course in Algebraic Topology.

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This fact can be used to show that higher homotopy groups are abelian: for $n \geq 2$,

$$\pi_n(Y) = [S^n, Y]_{\ast} = [\Sigma\Sigma S^{n-2}, Y]_*$$

so $\pi_n(Y)$ is abelian.