 Poincare duality with the orientation local system

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 $H^n(M;\mathbb{Z})=\mathbb{Z}/2$.