Let $X$ be a topological space (say a manifold). A result of R. Thom states that the pushforwards of fundamental classes of closed, smooth manifolds generate the rational homology of $X$. This work of Thom predates the development of bordism. Is there now a more elementary proof of this result that does not rely on spectral sequence techniques?



Representing rational homology by manifolds
A nice, direct combinatorial construction was given by Gaifullin, see his [papers](https://arxiv.org/abs/0712.1709) [on the](https://arxiv.org/abs/0806.3580) [arXiv](https://arxiv.org/abs/0912.3933) (equivalently: *[Explicit construction of manifolds realizing the prescribed homology classes](https://arxiv.org/abs/0712.1709)*, *[Realisation of cycles by aspherical manifolds](https://doi.org/10.4213/rm9198)* and *[Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class](https://doi.org/10.1134/S0081543810010074)*). A drawback of this approach is that if you think of it as realizing some multiple $m\alpha$ of the given integral homology class $\alpha$ by an oriented smooth manifold, then $m$ is not bounded in terms of the dimension of $\alpha$.


There has also been another geometric approach.
Thom also proved that $\bmod2$ homology classes are representable by maps of smooth (possibly unorientable) manifolds. This was reproved geometrically in



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> S. Buoncristiano and D. Hacon, *An elementary geometric proof of two theorems of Thom,* Topology 20 (1981), no. 1, 97–99 ([Core pdf](https://core.ac.uk/download/pdf/82811869.pdf))
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The other theorem of their title is that unoriented bordism is determined by Stiefel-Whitney numbers, and it is used in their proof that mod 2 homology classes are representable by smooth manifolds.


I believe the same geometric argument should also work to show that rational homology classes are representable by oriented smooth manifolds - *modulo* the fact that Pontryagin numbers determine oriented bordism tensored by $\Bbb Q$. This fact I'm afraid I don't know how to prove geometrically (for some proof, see e.g. the Milnor-Stasheff book). But note that in a subsequent paper Buouncristiano and Hacon also gave a geometric proof that Chern numbers determine complex bordism (Ann. of Math., 118 (1983), 1-7 <https://doi.org/10.2307/2006950>). Their other papers may also be of interest if you care about geometric proofs of classical results on bordism.


