Universal covering group

Remarks

Existence: let $p \colon \tilde G \to G$ be the universal cover of the manifold $G$, and choose $\tilde e$ over the identity $e$. There is a unique Lie group structure on $\tilde G$ with identity $\tilde e$ for which $p$ is a homomorphism. The universal covering group is unique up to isomorphism over $G$. The kernel of $p$ is a discrete central subgroup of $\tilde G$, isomorphic to $\pi_1(G)$.

Example: $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$, is a homomorphism from $(\R, +)$, a covering map, and $\R$ is simply connected; the kernel is $\Z = \pi_1(S^1)$.

Example: identify $SU(2)$ with the unit quaternions and $\R^3$ with the purely imaginary quaternions. The map sending $q$ to the rotation $v \mapsto q v q^{-1}$ is a universal covering group $SU(2) \to SO(3)$ with kernel $\set{\pm 1}$.

Non-example: the identity map $SO(3) \to SO(3)$ is a homomorphism and a covering map, but $SO(3)$ is not simply connected, since $\pi_1(SO(3)) \cong \Z/2$.