TITLE: twisted poincare duality
QUESTION [5 upvotes]: Let $M$ be a closed smooth oriented manifold of dimension $n$. Suppose that $\pi:L\longrightarrow M$ is a line bundle with a flat connection $\nabla$. Consider the space of $L$-valued differential forms: $$\Omega^{*}(L):=\Omega^{*}(M)\otimes L$$
The flat connection induce a differential operator on $\Omega^{*}(L)$, therefore we can define the $L$-valued de Rham cohomology (or twisted cohomology) $H^{*}(M;L)$. So is there a Poincar$\acute{e}$ duality for such twisted cohomology by a flat line bundle?

REPLY [4 votes]: Yes, there is a perfect pairing
$$
H^p(M;L)\otimes H^q(M;L^*)\to\mathbb{R},
$$
$p+q=n$.