Realizing cohomology classes by submanifolds

Question

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).

Answer 1

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.

Answer 2

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?