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I have heard that the following is a "well-known"

Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite fibers, i.e., $f^{-1}(n)$ is a finite set for all $n\in N$.

(Here "generic" is with respect to the $C^\infty$ topology, so the claim is that there is an open-dense or perhaps only residual subset $S\subset C^\infty(M,N)$ such that $f^{-1}(n)$ is a finite set for all $f\in S$ and $n\in N$.)

However, I have been unable to find a reference or proof of the claim, and the topologists who told me that the claim is "well-known" also told me that they do not know a proof or reference.

Question. Is the above claim true? If so, what is a reference or sketch of proof?

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    $\begingroup$ My comment was incorrect! Hirsch's exercise assumes $\dim N > \dim M$ (in which case you get the desired result with a bound on the size of $f^{-1}(n)$ depending on $\dim N / \dim M$.) Of course, you should expect no such bound here depending only on $M, N$, as $z^k: S^1 \to S^1$ has arbitrarily large degree. $\endgroup$
    – mme
    Commented 2 days ago
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    $\begingroup$ If $\Sigma_g$ is a closed orientable surface of genus $g>0$, how can one construct a smooth finite and surjective map $f \colon S^2 \to \Sigma_g$? $\endgroup$
    – Francesco Polizzi
    Commented 2 days ago
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    $\begingroup$ @FrancescoPolizzi I think one can work as follows. Take $\pi: S^2 \to D^2$ projection onto the first two coordinates, and then construct a surjective map $g: D^2 \to \Sigma_g$ by projecting a large disc in $\mathbb H^2$ to $\Sigma_g$ --- large enough that it contains a fundamental domain for the action of $\pi_1(\Sigma_g)$. $\endgroup$
    – mme
    Commented 2 days ago
  • $\begingroup$ @mme I think that your construction of a surjective map $S^2\to D^2 \to \Sigma_g$ seems correct, so I'm not sure if I'm missing something, or if your other comment is referring to something else as being "incorrect"? Also, what is the Hirsch exercise? (And if I'm not making a mistake, I think that multijet transversality is one way to get your mentioned a priori bound on the size of $f^{-1}(n)$ when $\dim N > \dim M$. Are you reasoning differently about that?) $\endgroup$
    – Matthew Kvalheim
    Commented yesterday
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    $\begingroup$ I deleted a comment referring to Hirsch, exercise 3.2.7(a). I have an idea about how to prove the result of this exercise --- similar to the proof that the space of embeddings is residual in $C^\infty(M,N)$ when $\dim N > 2\dim N$ --- but the idea does not help me with your question. $\endgroup$
    – mme
    Commented yesterday

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Yes, the claim is true, and here's a reference. For $M$ compact, your condition is satisfied by a "finite mapping." Such finite mappings form a residual set when $\dim M \leq \dim N$. See pp. 167-169 of Golubitsky & Guillemin's Stable Mappings and their Singularities for the details.

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    $\begingroup$ The answer was in my bookcase all along. Thank you. It looks like the relevant result, Theorem 2.6, is proved by Tougeron in a paper written in French. Do you know if the proof appears in English anywhere? $\endgroup$
    – Matthew Kvalheim
    Commented yesterday
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    $\begingroup$ I don't know an English reference, unfortunately. It is a little surprising how hard it is to track this down! $\endgroup$
    – Martin M. W.
    Commented yesterday
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    $\begingroup$ @MatthewKvalheim A proof in English appears in the following paper: Chaperon and Meyer, On a theorem of René Thom in géométrie finie. The relevant theorem is on p.6. $\endgroup$
    – Derived Cats
    Commented yesterday
  • $\begingroup$ @DerivedCats thank you so much. $\endgroup$
    – Matthew Kvalheim
    Commented yesterday

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