I'm looking at this paper, where the authors state the following (highlighted)
Hopefully this gives enough context. Basically, the idea is that even though the homology class of a hypersurface $\Sigma$ is zero, it doesn't necessarily mean that the class $[\omega_\Sigma \otimes \tau]$ in cohomology with local coefficients obtained by tensoring a Poincare dual form $\omega_\Sigma$ of $[\Sigma]$ with a parallel section $\tau$ of some flat vector bundle is trivial.
I'm a beginner when it comes to algebraic topology (I know categories, sheaves, etc. but I guess I don't know a lot of basic algebraic topology) so this seems very weird to me from my point of view. Isn't this construction just a cup product? Shouldn't cupping with something that's zero (say if $[\Sigma]=0$ and hence $[\omega_\Sigma]=0$) give us zero?