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I am reading the proof of the Whitney Embedding Theorem from John Lee's Introduction to Smooth Manifolds. However, I cannot figure out some statements from the proof. The proof relies on Lemma 6.14 that states if $M$ is a smooth $n$-manifold with or without boundary and $M$ admits a smooth embedding into $\mathbb{R}^N$ for some $N$, then it admits a proper smooth embedding into $\mathbb{R}^{2n+1}$.

The first part of the proof proves the theorem in the case $M$ is compact. Then according to the errata of the book, this argument applies to the case when $M$ is an arbitrary compact subset of a larger manifold $\tilde{M}$ with or without boundary, by covering $M$ with finitely many coordinate balls or half-balls for $\tilde{M}$. The result is a smooth injective map $F:M\to \mathbb{R}^{nm+m}$ whose differential is injective at each point. [This is needed in the ensuing argument for the noncompact case, because the sets $E_i$ might not be regular domains when $\partial M \neq 0$.]

However, in the second part of the proof, I cannot see how lemma 6.14 applies to the compact sets $E_i$. So the first part shows that for each $i$ there is a smooth injective map of $E_i$ into some Euclidean space whose differential is injective eat each point. But $E_i$ here, in the case $\partial M \neq 0$, is only a compact subset of $M$, so without a reference as to it being a codimension-0 submanifold of $M$, the hypothesis of a smooth $n$-manifold with or without boundary is not satisfied for $E_i$. Indeed, the definition of smooth embedding requires the domain to be a smooth manifold but here we only have that $E_i$ is a compact subset of a smooth manifold. So how do we ensure an embedding $\varphi: E_i \to \mathbb{R}^{2n+1}$?

Finally, in the $F$ constructed, how do we know that $F$ is proper because $f$ is? I would greatly appreciate some help here.

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  • $\begingroup$ The key word you are missing is "regular domain" (the sublevel set of a regular value). Thus, a submanifold with boundary. $\endgroup$ Commented Oct 23, 2021 at 14:30
  • $\begingroup$ @MoisheKohan The regular domain only applies when $M$ is a smooth manifold without boundary. This was noted in the errata of the text, hence the extra comments about the modification of the first part of the proof. $\endgroup$ Commented Oct 23, 2021 at 14:41
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    $\begingroup$ Oh, I see, I was assuming that $M$ has no boundary. Then, indeed, the treatment is insufficient and one has to introduce manifolds with corners to fix the problem. $\endgroup$ Commented Oct 23, 2021 at 14:44
  • $\begingroup$ @MoisheKohan If we reduce the domain of $\varphi_i$ to Int $E_i$, then doesn't it reduce to a smooth embedding, and Int $E_i$ is an open submanifold of $M$ with codimension $0$, hence Lemma 6.14 applies? But how does the argument for $F$ being proper work here? $\endgroup$ Commented Oct 23, 2021 at 14:46

2 Answers 2

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Let me answer your last question.

Lemma. Suppose that $X, Y, Z$ are Hausdorff topological spaces, $f:X\to Y, g: X\to Z$ are continuous and $f$ is proper. Then $F=f\times g: X\to Y\times Z$ is also proper.

Proof. Consider a compact $K\subset Y\times Z$; I'll prove that its preimage is compact. Let $A, B$ denote the projections of $K$ to $Y, Z$ respectively; both are compact. Then $F^{-1}(K)\subset f^{-1}(A)\cap g^{-1}(B)$, a closed subset of the compact $f^{-1}(A)$. Hence $F^{-1}(K)$ is compact. qed

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Good point. I overlooked that when I posted the correction for that proof.

Fortunately, it's easily fixed -- the argument in the first part of the proof actually produces an embedding of an open neighborhood of $E_i$, which is itself a manifold with or without boundary. I've just posted a correction.

This whole proof really should be reworked for the case of manifolds with boundary, but that would be too much to try to cram into a correction. One way to avoid the problem would be to hold off on claiming the version for manifolds with boundary until Chapter 8, when I show that every smooth manifold with boundary can be embedded in a smooth manifold without boundary; but I haven't taken the time to check whether there are arguments before Chapter 8 that use the boundary case in an essential way.

I've made myself a note to rethink this when and if I ever publish a third edition of the book. Meanwhile, I guess we'll all have to make do with these awkward corrections.

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  • $\begingroup$ One issue I have with this solution is that the first part of the proof only shows that $F$ is an injective smooth immersion. $F$ being an embedding derives from the condition that the domain of $M$ is compact. So the new argument gives that $F$ is an injective smooth immersion on the neighborhood, say $U$ of a compact subset $M$. How do we derive that $F: U \to F(U)$ is a homeomorphism? $\endgroup$ Commented Oct 24, 2021 at 23:23
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    $\begingroup$ Oh, you're right. I guess I have to work a little harder. I've updated the correction. $\endgroup$
    – Jack Lee
    Commented Oct 25, 2021 at 21:05
  • $\begingroup$ In the posted errata, I'm not sure why $\bigcup_{i}\overline{B_{i}}$is a compact manifold with boundary. I guess I'll wait for the proof in Chapter 8. $\endgroup$ Commented Dec 26, 2023 at 20:12
  • $\begingroup$ @KarthikKannan: It's not necessarily a compact manifold with boundary. But the argument in the proof produces an injective immersion from the open set $\bigcup_i B'_i$ (which is a manifold) into $\mathbb R^{nm+m}$, and the restriction of this to the compact set $\bigcup_i \overline{B}_i$ is an embedding. $\endgroup$
    – Jack Lee
    Commented Dec 26, 2023 at 21:23
  • $\begingroup$ Oh, that makes much more sense, thank you! $\endgroup$ Commented Dec 27, 2023 at 23:29

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