jiggaloon Posted:

Let G be a connected, simply connected compact Lie group, for example the special unitary group SU(n), and let Γ be a finite subgroup of G. Then the homogeneous space X = G/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space G. Among the many variants of this construction, one of the most important is given by locally symmetric spaces X = Γ\G/K, where

G is a non-compact simply connected, connected Lie group (often semisimple),

K is a maximal compact subgroup of G

Γ is a discrete countable torsion-free subgroup of G.

In this case the fundamental group is Γ and the universal covering space G/K is actually contractible (by the Cartan decomposition for Lie groups).

As an example take G = SL(2, R), K = SO(2) and Γ any torsion-free congruence subgroup of the modular group SL(2, Z).

From the explicit realization, it also follows that the universal covering space of a path connected topological group H is again a path connected topological group G. Moreover, the covering map is a continuous open homomorphism of G onto H with kernel Γ, a closed discrete normal subgroup of G:

{\displaystyle 1\to \Gamma \to G\to H\to 1.} 1\to \Gamma \to G\to H\to 1.

Since G is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the center of G. In particular π1Log in to see images! = Γ is an abelian group; this can also easily be seen directly without using covering spaces. The group G is called the universal covering group of H.

As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at Lattice of covering groups.

Just checked and yeah not sure my calculator has a bumon for all that.