Suppose $M$ and $N$ are both compact, connected, oriented $m$-manifolds without boundary and $f: M \to N$ is smooth. What additional condition(s) must $f$ satisfy so that there exists at least one point $q \in N$ such that the preimage of $q$ is finite and nonempty? I know it holds if $f$ is an immersion, but I was hoping for a less stringent condition.

According to Sard's Theorem, the set of critical values in $N$ has Lebesgue measure zero, so if we impose the condition that $Im(f)$ has nonzero measure in $N$, does that suffice?

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    $\begingroup$ There's always such a point. Sard's theorem shows that there is a regular value in $N$, and the preimage of a regular value is a closed, embedded $0$-dimensional submanifold of $M$, which in this case is a finite set of points. Or did you mean to require that the preimage of $q$ is finite and nonempty? $\endgroup$ – Jack Lee Jul 13 '14 at 20:56
  • $\begingroup$ Yes, I meant finite and nonempty. Thanks for pointing that out! I've edited the question. $\endgroup$ – Andrew Jul 13 '14 at 22:19
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    $\begingroup$ Yes, $Im(f)$ having nonzero measure works fine since then it must contain a regular point by Sard's theorem, the preimage of which will be a nonempty, finite set of points. $\endgroup$ – Tom Oldfield Jul 13 '14 at 22:21
  • $\begingroup$ @TomOldfield, no, it's going to be finite only when $\dim M=\dim N$. $\endgroup$ – Ted Shifrin Jul 13 '14 at 22:22
  • $\begingroup$ @TedShifrin Yes, which is assumed in the question. $\endgroup$ – Tom Oldfield Jul 13 '14 at 22:23

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