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The universal cover of a connected Lie group carries a unique lifted Lie group structure with a chosen identity, making its covering map a Lie group homomorphism. The kernel is a discrete central subgroup: conjugation of any kernel element is a continuous map from a connected group into a discrete set, hence constant. Every connected Lie group is therefore a quotient of its simply connected cover by a discrete central subgroup.

The universal cover of a connected closed manifold of dimension three with finite fundamental group is a closed simply connected three-manifold. It is orientable, since its orientation character has trivial domain. Its first integral homology vanishes, and the universal coefficient theorem for cohomology then gives $H^1=0$. Integral Poincare duality gives $H_2=H^1=0$, while $H_0$ and $H_3$ are $\mathbb{Z}$. Thus it is an integral homology sphere. This proof does not require classifying simply connected three-manifolds.