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!
Best Answer
A good reference for 1) is Bourbaki: Lie groups and Lie algebras Chapter 9. See in particular Section 4.6.
In particular it follows that 2) $\pi_1(G/T) = 0$ and that 4) $\pi_1(G)$ is finite if and only if $G$ is semisimple.
Concerning 3) $\pi_2(G) = 0$ always, which is a theorem of Cartan. I don't recall Cartan's proof, but it follows from Bott's analysis of the cell structure of G/T, and can also be proved using that $H^*(G)$ is a Hopf algebra (See Browder: Torsion in H-spaces. Ann. of Math. (2) 74 1961 24–51.).
EDIT (3 years later..): Just to elaborate, 1) is completely answered the above Bourbaki reference. The formula is that $\pi_1(G) = L/L_0$, where L is the integral lattice from above and $L_0$ is the coroot lattice. For the connoisseurs out there I mention that there is also a homotopical version, in that the formula also holds for p-compact groups (see Section 8 of my paper with Kasper Andersen on the classification of 2-compact groups linked here: http://www.math.ku.dk/~jg/papers/2classification.pdf)