TITLE: Homotopy groups of an infinite wedge of 2-spheres
QUESTION [8 upvotes]: I know Hilton's result about a finite wedge of spheres, and I know that certain homotopy groups (such as the third homotopy group) can be directly calculated for an infinite wedge too.
My question is -- Is there some general result that gives the homotopy groups of an (uncountable) infinite wedge of 2-spheres in terms of the homotopy groups of 2-spheres (or other simpler spaces)?
Thanks.

REPLY [14 votes]: For any infinite wedge $X=\bigvee_{i\in I} X_i$, the homotopy groups are just the colimit $$\pi_n(X)=\operatorname{colim}_F \pi_n\left(\bigvee_{i\in F} X_i\right),$$
where $F$ ranges over all finite subsets of $I$.  The reason for this is simple: $S^n$ is compact, so the image of any map $S^n\to X$ must be contained in $\bigvee_F X_i$ for some finite $F$; this means the natural map from the colimit to $\pi_n(X)$ is surjective.  Similarly, since $S^n\times I$ is compact, any homotopy also lands in a finite wedge, so the natural map is injective.  The same reasoning would apply if you replaced $S^n$ with any compact space, and the infinite wedge with any colimit for which any compact subset factors through some stage (eg, any filtered system of sub-CW complexes).