In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. Theoreme II.27).
I have the following variation of this question:
Let $M$ be a finite-dimensional manifold. Given $\lambda \in H^k(M, \mathbb R)^*$, does there exists a compact oriented submanifold $S$ of $M$ and a closed form $\alpha$ on $S$ such that $$\lambda([\beta]) = \int_S \beta \wedge \alpha,$$ where $\beta$ is a closed $k$-form on $M$, and we used the de Rham isomorphism to identify de Rham cohomology with singular cohomology. Of course, the dimension of $S$ and the degree of $\alpha$ need to satify $\dim S = k + \# \alpha$ for this integral to make sense.
I'm interested in conditions on $M$, $k$ and/or $\lambda$ that ensure such a represenation. Moreover, I suspect the following:
There exists a lattice in $H^k(M, \mathbb R)^*$ whose elements can be realized as above such that $\alpha$ has integral periods (i.e. the integrals of $\alpha$ over cycles are integers).