Local gauge invariance of field's homotopy class? Every map $S^2\rightarrow \mathrm{group } G$ is homotopic to a constant map?

Question

In a discussion of a gauge field theory with gauge group $G$, someone says we can use a celebrated result of E. Cartan to show the gauge invariance of matter field's homotopy class. And Cartan's result is that every map $S^2\rightarrow G$ is homotopic to a constant map, and since we are assuming G to be connected this constant may be taken to be the identity element of G. I reckon that this argument makes sense.

But could you tell me more about this Cartan's theorem (proof, explanation or so) or at least, where I can find some introduction of it?

Answer 1

The statement is that if $G$ is any Lie group, then its second homotopy group $\pi_2(G)$ vanishes. You can find a number of proofs and references to proofs at this MO question.