Otdfbt

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).

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 $Sigmato M$ 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
$$pi_bullet^textst(M)to Omega_bullet(M)$$
from stable homotopy to oriented bordism. The composite
$$pi_bullet^textst(M)to Omega_bullet(M)to H_bullet(M)
$$ 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 $Sto hBbb Z$ 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?