Let $F \hookrightarrow E \to B$ be a fibration, and let everything be over a field $k$. The leray hirsch theorem in its usual form says that the (homology) cohomology spectral sequence degenerates at ($E^2$) $E_2$ if $H^*(E) \to H^*(F)$ has a splitting $(H_*(F) \to H_*(E)$ has a splitting). The condition imposed, however, that $H^*(E) \to H^*(F)$ be surjective does not allow for the possibility of a nontrivial local coefficient system.
Suppose we instead impose the condition on that $H_*(E) \to H_*(B)$ is surjective. Using the edge maps $H_*(E) \to E^\infty_{p,*} \subset E^2_{p,*}=H_*(B)$, we get that $d_r^{*,0}$ is 0 for all $r$ and using the coalgebra structure on $E^r$, we get that this spectral sequence degenerates at $E_2$.
This allows for a nontrivial local system since for any fiber bundle with nontrivial local coefficient system, one can take the product with a trivial fibration over a point, to get a bundle with a nontrivial local coefficient system with a topological splitting.
Do you know any examples where this is fruitful?
