Exercise 8, § 3

Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$; we denote by $H_0$ the neutral component of $H$ and by $\pi_0(H)$ the group $H/H_0$. We also denote by $p$ the canonical surjection of $G$ onto $G/H$.

a) The mapping $p$ has the path lifting property.

b) If $H$ is connected, the homomorphism $p_*:\pi_1(G, e)\rightarrow \pi_1(G/H, p(e))$ is surjective.

c) There exists a unique mapping $\varphi :\pi_1(G/H, p(e))\rightarrow \pi_0(H)$ which, for every path $\widetilde{c}$ with origin $e$ in $G$ whose endpoint belongs to $H$, associates to the class of the loop $p\circ \widetilde{c}$ in $G/H$ the class of $\widetilde{c}(1)$ modulo $H_0$; it is a group morphism.

d) If $G$ is simply connected, then $\varphi$ is an isomorphism. In particular, $H$ is connected if and only if $G/H$ is simply connected.