= Poincare duality with the orientation local system
{c}
{title2=$H^q(M;\mathcal A)\cong H_{n-q}(M;\mathcal O_M\otimes\mathcal A)$}
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 $H^n(M;\mathbb Z)=\mathbb Z/2$.
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