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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?

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  • $\begingroup$ Try chapter 2 of the book by Bott-Tu. $\endgroup$ – Liviu Nicolaescu Jan 10 '14 at 10:17
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Yes, there is a perfect pairing $$ H^p(M;L)\otimes H^q(M;L^*)\to\mathbb{R}, $$ $p+q=n$.

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    $\begingroup$ In fact, even if $M$ is not oriented you get something similar only instead of $L^*$ you get $L$ (dualized and) twisted by the orientation sheaf. $\endgroup$ – Mariano Suárez-Álvarez Jan 10 '14 at 9:37

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