Exercise 9, § 3

Let $G$ be a connected Lie group, let $(\widetilde{G}, q)$ be a universal covering of G and let $\widetilde{e}$ be the identity element of $\widetilde{G}$. Let H be a closed connected subgroup of G; denote by $i: H\rightarrow G$ the canonical injection and by $p: G\rightarrow G/H$ the canonical surjection.

a) Let $H_1$ be the identity component of $\overset{-1}{q}(H)$ and let $q_1:\widetilde{G}/H_1\rightarrow G/H$ be the mapping deduced from $q$ by passing to quotients. Prove that $q_1$ makes $\widetilde{G}/H_1$ a covering of $G/H$ whose fibre at $p(e)$ is isomorphic to $\pi_1(G/H, p(e))$. b) Deduce that there exists an exact sequence of groups