I'm reading Hatcher's survey on 3-manifolds (https://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf) and noting that the fundamental group determines all homology groups of a closed orientable 3-manifold $M$.
$\pi_1(M)$ gives us $H_1(M)$ by Hurewicz. Duality says $H_2(M) \cong H^1(M)$. I don't quite understand why $H^1(M)$ is $H_1(M)$ mod torsion. Hatcher states that this is due to the universal coefficient theorem but I can't connect the theorem to the claim (I'm probably missing something obvious).