Existence of universal covering groups

Theorem (existence and uniqueness)

Let $G$ be a connected Lie group . Then there exists a Lie group $\widetilde G$ and a smooth map $p:\widetilde G\to G$ such that:

1. $p$ is a covering map of manifolds and a Lie group homomorphism .
2. $\widetilde G$ is simply connected .
3. The pair $(\widetilde G,p)$ is unique up to unique Lie group isomorphism over $G$ (i.e. it is the universal covering group of $G$).

Moreover, the induced map on Lie algebras

is an isomorphism (by functoriality of the differential ).

Construction idea (context)

One standard construction starts with the universal cover $\widetilde{G}_{\mathrm{top}}$ of the underlying manifold of $G$. The Lie group operations on $G$ lift uniquely to $\widetilde{G}_{\mathrm{top}}$ once a basepoint over $e\in G$ is fixed, producing a Lie group structure on the universal cover so that the covering projection becomes a homomorphism. This gives a canonical way to pass from a connected Lie group to a simply connected one with the same Lie algebra , which underlies many “Lie algebra determines the simply connected group” results (compare simply connected groups are determined by their Lie algebras ).