Is it possible to construct explicitely a homologue between two smooth simplicial cycles $A,B$ in a manifold that satisfy $\int_A w = \int_B w$?

Question

Suppose that $M$ is a compact manifold, and $[A]$ and $[B]$ are two cycles in the (real coefficients) homology of $M$ that are represented by smooth simplices $A$ and $B$. If we assume that $\int_A w = \int_B w$ for each closed differential form, then I think it follows from De Rham's theorem that $[A] = [B]$. But is it possible to build out of this data the simplex $ s : \Delta \to M$ that has as boundary $A - B$?