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Let $m>n$ and suppose $M$ is a smooth $m$-manifold, $N$ is a smooth $n$-manifold.

Can there be an injective smooth map $f:M\to N$?

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2 Answers 2

Hint: Considering the problem locally, this requires an injective smooth map $\mathbb R^m\to \mathbb R^n$.

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True, but why does no such smooth map exist? That's not obvious. –  Zhen Lin Jan 20 '13 at 13:44
    
I've been thinking about it since you posted it. I see Zhen Lin's comment concurs with my thoughts... –  Bruno Stonek Jan 20 '13 at 13:57
    
@BrunoStonek Yes, it's not as obvious as it seems and the full answer is best given using homology. At least the case $n=1$ is simple: Removing a single point from $\mathbb R^m$ leaves the space connected, hence also the image is connected, but that can't be if it misses an interior point. –  Hagen von Eitzen Jan 20 '13 at 15:11

The answer is no, and in fact there cannot even be an injective continuous map $M\to N$.

As noted by Hagen von Eitzen, we look at the map locally, so that it suffices to show that there is no continuous injection $\mathbb{R}^m\to\mathbb{R}^n$. This can be done using the Borsuk-Ulam theorem, as done by Jason DeVito in his answer to Why isn't there a continuously differentiable injection into a lower dimensional space?.

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