# Question about John Lee's Proof of Whitney's Embedding Theorem on the noncompact case

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.

• The key word you are missing is "regular domain" (the sublevel set of a regular value). Thus, a submanifold with boundary. Commented Oct 23, 2021 at 14:30
• @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. Commented Oct 23, 2021 at 14:41
• 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. Commented Oct 23, 2021 at 14:44
• @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? Commented Oct 23, 2021 at 14:46

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

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.

• 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? Commented Oct 24, 2021 at 23:23
• Oh, you're right. I guess I have to work a little harder. I've updated the correction. Commented Oct 25, 2021 at 21:05
• 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. Commented Dec 26, 2023 at 20:12
• @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. Commented Dec 26, 2023 at 21:23
• Oh, that makes much more sense, thank you! Commented Dec 27, 2023 at 23:29