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For a connected closed oriented four-manifold, Poincare duality identifies $H_3$ with $H^1$. The universal coefficient theorem for cohomology identifies $H^1$ with $\mathrm{Hom}(H_1,\mathbb{Z})$ because $H_0=\mathbb{Z}$ has zero Ext. Consequently $H_3$ is torsion-free and its rank equals that of $H_1$.

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.

A rational Witt space is a stratified pseudomanifold such that the link $L$ of every stratum of odd codimension $2r+1$ satisfies $I{H}_r^{\bar{m}}(L;\mathbb{Q})=0$. This removes the only difference between lower and upper middle extensions and gives rational middle-perversity Poincare duality on closed oriented Witt spaces.