# Realizing cohomology classes by submanifolds

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).

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.
It seems to me, that the original question asked about the cokernel of the map $$\Omega_*(M) \to H_*(M).$$ You are right, Thom's theorem claims it is finite. But can one say more about it?