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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?

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This seems to be asked and answered here mathoverflow.net/questions/79084/… –  Mark Grant Oct 19 '12 at 13:32
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Yes. But nobody pointed out there that if all that you want is the rational surjectivity (as opposed to rational isomorphism) of the stable Hurewicz map from stable homotopy to homology then all you need is to know is that $\pi_n(S^n)\to H_n(S^n)$ is surjective and $\pi_k(S^n)=0$ when $k<n$. –  Tom Goodwillie Oct 19 '12 at 16:30

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