René Thom proved that for any topological space $X$, any integral homology class of $X$ (of any degree) has an odd multiple that is the pushforward of the fundamental class of a smooth compact oriented manifold mapping contin to $X$. So, if we pass to $rational$ homology, then any rational homology class of $X$ is the pushforward of an at most rational multiple of the fundamental class of a smooth compact oriented manifold mapping to $X$. Have there appeared other, different proofs (i.e. different from Thom's) of this last rational result since?
