= Universal cover of a closed three-manifold with finite fundamental group
{title2=$H_*(\widetilde M;\mathbb Z)\cong H_*(S^3;\mathbb Z)$}
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.
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