TITLE: Weight lattice and the fundamental group
QUESTION [9 upvotes]: Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively.  Then, $\frak{t}$ can be considered as a Cartan subalgebra of $\frak{g}$.  It is known that  the kernel of exponential map $exp : \frak{t} \to$ $T$ is the lattice of all integral weights of $\frak{g}$, i.e. weihts $\lambda \in (it)^*$ such that $\lambda(H)\in 2\pi i\mathbb{Z},$  whenever $exp H= I$ for $H\in\frak{t}$.
I have the following questions: 
1) What is the relation between  the fundamental group $\pi_{1}(G)$ of $G$ with the integral lattice described above? I am trying to find any good references about this fact, but it seems difficult. 
2) How we can use the fibration $T\to G$$\to G/T$    to compute  $\pi_{1}(G/T)$? (answered)
3) What we can say about  the second homotopy group $\pi_{2}(G)$? (answered)
4) Is it true, that if $G$   is semisimple, then $\pi_{1}(G)$ is finite? (answered)
Thank you!

REPLY [2 votes]: Regarding Q3, one more explanation of why $\pi_2(G)=0$ (which is similar to what Claudio Gorodski points out above) can be found here. A nice observation made there is that one can use the same arguments to show that $\pi_3(G)$ is torsion-free! (which I guess was not in your list of facts about homotopy of compact Lie groups).