Proposition 3

Let G be a connected compact Lie group, and H a connected closed subgroup of maximal rank.

a) The compact manifold $G/H$ is simply-connected.

b) The homomorphism $\pi_1(H)\rightarrow \pi_1(G)$, induced by the canonical injection of H into G, is surjective.

Since H is connected, we have an exact sequence (General Topology, Chap. XI, in preparation)

where $\overline{e}$ is the image in $G/H$ of the identity element of G. Since $G/H$ is connected, this immediately implies the equivalence of assertions a) and b). Moreover, if $f: G'\rightarrow G$ is a surjective morphism of connected compact Lie groups whose kernel is central, proving the proposition (in the form $a$)$)$ for G is the same as proving it for $G'$ (Prop. 2). Thus, we can first of all replace G by Ad(G), then assume that G is semi-simple, and then by replacing G by a universal covering (§1, no. 4, Cor. 2), assume that G is simply-connected. But then assertion b) is trivial.