In ["Quelques propriétés globales des variétés différentiables"](https://eudml.org/doc/139072), 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).
> 
> 
> 



Realizing cohomology classes by submanifolds
Your question is just a reformulation of what Thom did, so the answer is always yes.


Since the Stokes map from de~Rham cohomology to singular cohomology (with real coefficients) is an isomorphism, your problem is equivalent to that of understanding the degree to which the map
$$
\Omega\_\bullet(M)\otimes \Bbb R \to H^{\bullet}\_{DR}(M;\Bbb R)^\ast
$$
is surjective. Here, the source is oriented bordism of manifolds mapping to $M$ and the target is the linear dual of de Rham cohomology. The identification $H^\bullet\_{DR}(M;\Bbb R)^\ast \cong H\_\bullet(M;\Bbb R)$ (singular homology on the right) enables one to reformulate the problem as to that of studying the degree to which
$$
\Omega\_\bullet(M) \to H\_\bullet(M)
$$
is surjective modulo torsion. The latter homomorphism assigns to a representative of a bordism class $\Sigma\to M$ the image of the fundamental class $[\Sigma]$ in $H\_\bullet(M)$.


It is known that that real (or rational) homology classes are representable by oriented smooth manifolds. Here is a dumb reason: there is a Hurewicz map 
$$\pi\_\bullet^{\text{st}}(M)\to \Omega\_\bullet(M)$$
from stable homotopy to oriented bordism. The composite
$$\pi\_\bullet^{\text{st}}(M)\to \Omega\_\bullet(M)\to H\_\bullet(M)
$$ is just the usual Hurewicz map. 


It follows from Serre's thesis that the composite is an isomorphism modulo torsion (more specifically, Serre showed that the map $S\to h\Bbb Z$ from the sphere to the Eilenberg Mac Lane spectrum is a rational homotopy equivalence. If we smash this map with $M\_+$ and take homotopy groups we get the statement about the Hurewicz map for $M$). Then it follows that the map $\Omega\_\bullet(M)\to H\_\bullet(M)$ is a surjection modulo torsion.


