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.



Homotopy groups of an infinite wedge of 2-spheres
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).


