Third homology of a closed oriented four-manifold
= Third homology of a closed oriented four-manifold
{title2=$H_3(M;\mathbb Z)\cong\operatorname{Hom}(H_1(M;\mathbb Z),\mathbb Z)$}
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 $\operatorname{Hom}(H_1,\mathbb Z)$ because $H_0=\mathbb Z$ has zero Ext. Consequently $H_3$ is torsion-free and its rank equals that of $H_1$.