Consider $X$ $n-$dimensional smooth compact oriented manifold. Denote $H^i(X,\Omega)$ as the cohomology of smooth differential form, $H_i(X)$ as the $i-$th cycles and $C_i(X)$ as the singular chains and $Z^i(X)$ as the $i-$the cocycles of differential forms.
There are 2 different pairings.
$C_i(X)\times Z^i(X)\to R$ by $\int_c\omega$ where $c\in C_i(X),\omega\in Z^i(X)$. This obviously descends to $H_i(X)\times H^i(X,\Omega)\to R$ by Stokes Thm. Clearly I have embedding $H_i(X)\to (H^i(X,\Omega))^\star$ by shrinking the target cycle I need.
$\textbf{Q:}$ Is this isomorphism?
The other one is $C^i(X)\times C^{n-i}(X)\to R$ by $\int_X \omega_i\wedge\omega_{n-i}$ with $\omega_k\in C^k(X)$. Clearly this also descends to $H^i(X)\times H^{n-i}(X,\Omega)\to R$. This pairing is non degenerate.
$\textbf{Q:}$ What is the relationship between the 2 pairings? By poincare duality, I have $H_i(X)\cong H^{n-i}(X)$. So second pairing can be replaced by $H_i(X)\times H^i(X,\Omega)\to R$. Should I say if $\omega_2\in H^{n-i}(X,\Omega)$ is identified with a cycle $c\in H_i(X)$, given $\omega_1\in H^i(X)$, I consider $\int_c\omega_1=\int_X\omega_1\wedge\omega_2$? Is this even correct?