Strong Whitney embedding theorem for non-compact manifolds

Question

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.

The strong Whitney embedding theorem is usually stated as follows.

Theorem: If $M$ is a smooth $n$-dimensional manifold, then $M$ admits a smooth embedding into $\RR^{2n}$.

In fact, the theorem is stated in essentially this form in Whitney's original article "The self-intersections of a smooth $n$-manifold in $2n$-space". For definiteness, I will assume that all manifolds are Hausdorff, second countable, and smooth.

Question 1: Can we always take the embedding in the above theorem to be closed? If so, is there a reference for such a statement of the theorem?

It seems that Whitney's original proof produces an embedding whose image is not closed when $M$ is open. In fact, immediately after the construction, Whitney explicitly poses the following problem: "Does there exist an imbedding, for $M$ open, with no limit set?"

Having thought about the matter for a short while, I am inclined to believe that Whitney's trick (introduced in the aforementioned article by Whitney) allows the cancellation of infinitely many double points in a manner that preserves closed immersions. Is this correct? Or is my argument getting trapped in some pitfall?

My second question concerns possible dimensional restrictions in the above embedding theorem, stemming from the failure of Whitney's trick for $n=2$.

Question 2: Does every $2$-dimensional manifold embed in $\RR^4$? If so, can we also take the embedding to be closed in this case?

Here is the suggested proof in Whitney's article: "For $n=2$, we imbed the sphere, projective plane, or Klein bottle in $E^4$, and add the necessary number of handles to obtain the given manifold." I can see that this procedure should work for compact surfaces, but I am unable to carry it out in the non-compact case.

Finally, I would also be interested to hear about more recent, good references concerning Whitney's strong embedding theorem.

Answer 1

Regarding question 1, yes you can always ensure the image is closed. You prove the strong Whitney by perturbing a generic map $M \to \mathbb R^{2m}$ to an immersion, and then doing a local double-point creation/destruction technique called the Whitney trick. So instead of using any smooth map $M \to \mathbb R^{2m}$, start with a proper map -- one where the pre-image of compact sets is compact. You can then inductively perturb the map on an exhausting collection of compact submanifolds of $M$, making the map into an immersion that is also proper.

Regarding question 2, generally speaking if a manifold is not compact the embedding problem is easier, not harder. Think of how your manifold is built via handle attachments. You can construct the embedding in $\mathbb R^4$ quite directly. Think of $\mathbb R^4$ with its standard height function $x \longmapsto |x|^2$, and assume the Morse function on $M$ is proper and takes values in $\{ x \in \mathbb R : x > 0 \}$. Then I claim you can embed $M$ in $\mathbb R^4$ so that the Morse function is the restriction of the standard Morse function. The idea is every $0$-handle corresponds to creating an split unknot component in the level-sets, etc.

edit: The level sets of the standard morse function on $\mathbb R^4$ consists of spheres of various radius. So when you pass through a critical point (as the radius increases) either you are creating an split unknot component, doing a connect-sum operation between components (or the reverse, or a self-connect-sum), or you are deleting a split unknot component. By a split unknot component, I'm referring to the situation where you have a link in the $3$-sphere. A component is split if there is an embedded 2-sphere that contains only that component, and no other components of the link. So a split unknot component means that component bounds an embedded disc that's disjoint from the other components.

Regarding your last question, the Whitney embedding theorem isn't written up in many places since all the key ideas appear in the proof of the h-cobordism theorem. So Milnor's notes are an archetypal source. But Adachi's Embeddings and Immersions in the Translations of the AMS series is one of the few places where it occurs in its original context. You can find the book on Ranicki's webpage.