I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of compact manifolds.
From the classification of compact surfaces I can see that the closed ones cannot be embedded in the plane, and that the ones that can be embedded are disks and annuli, which have non empty boundary.
I'd like to know if this intuition is still sound in dimension $n\geq 3$, so my question is: if a compact manifold of dimension $n$ embeds in $\mathbb{R}^n$, is it forced to have a non empty boundary?
I've read the wiki article about Whitney embedding and it's section about "sharper results", but there they give general estimates for the whole class of compact $n$-manifolds, whereas I would be interested in just a single example of a compact boundaryless $n$-manifold embedded in $\mathbb{R}^n$ (possibly in low dimensions), or a proof/reference if this can never happen. Notice I'm not interested in distinction between orientable/non-orientable manifolds, but in the presence of a boundary.
Thanks in advance!
Best Answer
Here's an alternate proof which doesn't use invariance of domain. It also gives a slightly stronger result.
Theorem:
Let $M^n$ be compact without boundary. Then there is no immersion $f:M\rightarrow \mathbb{R}^n$.
Proof: (sketch). Assume for a contradiction there is such an $f$. Since $M$ is compact, $f$ is a closed map, that is, it maps closed sets to closed sets. To see this, let $F\subseteq M$ be closed. Then it's compact because $M$ is, so $f(F)$ is compact, hence closed. (Here, we just use the fact that $f$ is continuous).
Further, $f$ is an open map. That it, it maps open sets to open sets. To see this, note that it is enough to map really small open sets to open sets because $f(\bigcup U_i) = \bigcup f(U_i)$.
It's a fact (a consequence of the implicit function theorem, if I recall) that every immersion locally looks like the natural inclusion $\mathbb{R}^k\rightarrow \mathbb{R}^n$ into the first $k$ coordinates. (This uses the boundarylessness of $M$ - if $p$ is on the boundary, this part doesn't work.)
Said more precisely, given our immersion $f$ and a point $p\in M$, there is a chart around $p$ and around $f(p)$ so that in these chart coordinates, $f$ looks like the inclusion.
Now, in our case $k = n$ and then the natural inclusion is a homeomorphism. This implies open sets in these special charts are mapped to open sets, so $f$ is open.
Putting this together, we've now seen that $f$ is an open and closed mapping. Now, what is $f(M)$? It must be compact because $M$ is, but it must also be both open and closed in $R^n$ because $M$ is open and closed in itself. This implies $f(M) = \mathbb{R}^n$ since that's the only nonempty clopen subset of $\mathbb{R}^n$, but $\mathbb{R}^n$ isn't compact, so we've reached a contradiction.