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?