Let and be compact smooth manifolds of dimensions and , respectively. Given a map , we give homological conditions under which has nontrivial cohomology (with local coefficients) for any map homotopic to . We also show that a certain cohomology class in is Poincaré dual (with local coefficients) under to the image of a corresponding class in when is transverse to . This generalizes a similar formula of D Gottlieb in the case of simple coefficients.
"Cohomology of preimages with local coefficients." Algebr. Geom. Topol. 6 (3) 1471 - 1489, 2006. https://doi.org/10.2140/agt.2006.6.1471