It is known that $\pi_*^{st}$ defines a generalized homology theory, where $\pi_n^{st}(X) = \text{colim}_{k \geq 0} \pi_{n+k}(\Sigma^k X)$ is the $n$th stable homotopy group of the based space $X$. That is, the functors $\pi_n^{st}$ satisfy the Eilenberg-Steenrod axioms for generalized homology theories. As such, among other properties, the following two conditions hold :
- $\pi_n^{st}(X)$ is an abelian group.
- The homotopy axiom is satisfied. That is, continuous, homotopic maps $f,g : X \rightarrow Y$ induce the same maps $\pi_n^{st}(f), \pi_n^{st}(g) : \pi_n^{st}(X) \rightarrow \pi_n^{st}(Y)$ for all $n$.
I'd like a better understanding for why each of these conditions holds for stable homotopy groups. In the colimit that defines $\pi_n^{st}(X)$, each homotopy group that appears isn't necessarily abelian, since the fundamental group of a space may certainly be non-abelian. Thus, why does it follow that $\pi_n^{st}(X)$ is abelian ? Further, how can we rigorously show the homotopy axiom holds ?
Thanks !