TITLE: First and second homotopy groups of a connected Lie group
QUESTION [6 upvotes]: I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$?
Thanks for anyone who give me references for a 'simple proof' of these results

REPLY [8 votes]: Some references:

This MO post describes the Eckmann-Hilton argument. 
Here's a blog post about abelian $\pi_1$. 
This other MO post has answers concerning the nullity of the second fundamental group.