Poincare duality with the orientation local system ($H^q(M;\mathcal{A})\cong H_{n-q}(M;{\mathcal{O}}_M\otimes \mathcal{A})$)

For a closed connected manifold, cap product with $[M]\in H_n(M;{\mathcal{O}}_M)$ gives the displayed isomorphism. The ordinary integral theorem results when the orientation system is trivial. Modulo two the sign representation is always trivial, so duality holds without an orientability assumption. In particular a nonorientable connected closed $n$-manifold has $H_n(M;\mathbb{Z})=0$ and $H^n(M;\mathbb{Z})=\mathbb{Z}/2$.

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 $\mathrm{Ext}(H_2(M;\mathbb{Z}),\mathbb{Z})$, so the only torsion in $H_2$ is one copy of $\mathbb{Z}/2$. In particular $b_1\ge 1$.