Second homology of a closed nonorientable three-manifold
= Second homology of a closed nonorientable three-manifold
{title2=$H_2(M;\mathbb Z)\cong\mathbb Z^{b_1-1}\oplus\mathbb Z/2$}
Zero <Euler characteristic> and $H_3=0$ give free rank $b_1-1$. The <universal coefficient theorem for cohomology> identifies $H^3(M;\mathbb Z)=\mathbb Z/2$ with $\operatorname{Ext}(H_2(M;\mathbb Z),\mathbb Z)$, so the only torsion in $H_2$ is one copy of $\mathbb Z/2$. In particular $b_1\geq1$.