Let $F \xrightarrow{f} E \rightarrow B$ be a fiber bundle and $L$ is the locally constant sheaf on $E$. $B$ is NOT simply connected.
We can apply Leray-Serre spectral sequence to compute the local cohomology of $E$, and assume that $E_2$ page of the spectral sequence converged.
My question is that how I can read the information of cocycles in $H^{p+q}(E;L)$ from cocycles in $H^p(B;R^{q}f_*L)$?
More explicitly, what is the explicit map from $H^p(B;R^{q}f_*L)$ in the language of deRham cohomology.