smooth surjective map from lower dimensional manifold onto higher dimensional manifold?

Question

I'm thinking about whether there exists a smooth surjective map $f: M^m \twoheadrightarrow N^n$ where both $M$ and $N$ are smooth manifolds, and $\dim M = m < n = \dim N$. (We might assume that $M$ and $N$ are boundaryless, although I don't think this is necessary.)

Intuition tells me the answer is no, since smooth manifolds and smooth maps between them ought to behave nicely. Things like space-filling curves are obviously excluded from this class of nice objects.

However, I find it hard to justify such a claim. In particular, I tried to approach from the measure of $M$ and $N$ but failed. Any ideas? Thanks.

Answer 1

You're looking for Sard's theorem. If $m<n$, the image of $f$ consists solely of critical values, and thus must have measure zero in $N$.

Note that the proof of Sard's theorem is extremely difficult, so you shouldn't feel too bad about not discovering it!

Answer 2

Sard's theorem tells us that the image has measure zero.