Weight lattice and the fundamental group

Question

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!

Answer 1

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)

Answer 2

The fundamental group $\pi_1(G)$ is isomorphic to the quotient of the integral lattice by the inverse roots.

A nice exposition (also including answers to the other questions) is in Chapter V (7) of "Broecker , tom Dieck - Representations of compact Lie groups (Springer 1985)."

Answer 3

Jesper Grodal's reference to Bourbaki is a reasonable one for these questions, including 1). There are also two volumes in the Springer GTM series which treat many aspects of compact Lie groups, including the book by Bump indicated and the earlier text GTM 98 (1985) by Brocker & Tom Dieck on Representations of Compact Lie Groups where V.7 treats the fundamental group and related matters thoroughly. Naturally the notation differs somewhat in such sources, but the answers to the questions raised here are all standard and arise from early work of Cartan, Weyl, and others. In general, the topology of a semisimple Lie group depends just on the topology of a maximal compact subgroup.

In the setting of abstract root systems, motivated by the theory of semisimple Lie groups over the complex field and their Lie algebras or by compact semisimple Lie groups, the notion of "fundamental group" focuses on the quotient of the abstract weight lattice by the abstract root lattice. In semisimple groups or Lie algebras, the actual weight lattice (or character group) of a maximal torus can vary from the root lattice to the full abstract weight lattice, but the quotients in any case are finite and easily computable for each simple type. Moreover, the abstract fundamental group for a given root system is realized internally as the center of a simply connected group.

Answer 4

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).

Answer 5

Question 3 has also been answered in here.