  Suppose $X^{N}$ is a closed oriented manifold, $\alpha \in H^*(X;\mathbb{R})$
is a cohomology class, and $Z \in H_{N-k}(X)$ is an integral homology class. We
ask the following question: is there an oriented embedded submanifold $Y^{N-k}
\subset X$ with homology class $Z$ such that $\alpha|_Y = 0 \in
H^*(Y;\mathbb{R})$? In this article, we provide a family of computable
obstructions to the existence of such 'exact' submanifolds in a given homology
class which arise from studying formal deformations of the de Rham complex. In
the final section, we apply these obstructions to prove that the following
symplectic manifolds admit no non-separating exact (a fortiori contact-type)
hypersurfaces: K\"ahler manifolds, symplectically uniruled manifolds, and the
Kodaira--Thurston manifold.
