Universal cover of a closed three-manifold with finite fundamental group ($H_{\ast }(\tilde{M};\mathbb{Z})\cong H_{\ast }(S^3;\mathbb{Z})$)

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.