0708-1300/Class notes for Thursday, October 18

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Class Notes

The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

Outline

Today we stated the Whitney Embedding Theorem and began to discuss its proof. Along the way, we also encountered some related notions that will serve us well in the future. We begin by stating the theorem:

Theorem (Whitney Embedding)

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} be a smooth Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\!} -manifold. Then there exists an embedding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi : M \to \mathbb{R}^{2m+1}} .

Proof

Outline

We will break the proof of the theorem into three parts:

1. Find an embedding of a compact Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^N\!} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\!} .
2. Use Sard's Theorem to reduce Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\!} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2m+1\!} .
3. Use the "Zebra Trick" to prove the theorem for non-compact Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} .

Parts two and three shall be left to the next lecture.

Part 1

Suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} is compact. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\phi_\alpha : U_\alpha \to \mathbb{R}^m \}_{\alpha \in A}} be an atlas for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} , and note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_\alpha\}_{\alpha \in A}\!} is an open cover of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} . Hence it possesses a finite subcover Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_j\}_{1 \le j \le J}\!} , and the corresponding collection $\{\phi _{j}\}_{1\leq j\leq J}$ of charts is an atlas.

Choose smooth functions $\{\lambda _{j}:M\to \mathbb {R} _{\geq 0}\}_{1\leq j\leq J}$ with the following properties:

1. $\lambda _{j}|_{M\setminus U_{j}}=0$
2. $\bigcup _{j=1}^{J}\lambda _{j}^{-1}\left((0,\infty )\right)=M$
3. $\mathrm {supp} \left(\lambda _{j}\right)\subset U_{j}$ for $1\leq j\leq J$

where $\mathrm {supp} (f)\!$ for $f:M\to \mathbb {R}$ is the support of $f\!$, ie. the closure of $f^{-1}(\mathbb {R} \setminus \{0\}).$ The existence of such functions follows from the existence of smooth partitions of unity for manifolds---a concept that will be discussed later on.

Now define $\Phi :M\to \mathbb {R} ^{J+mJ}$ by $\Phi (p)=\left(\lambda _{1}(p),\ldots ,\lambda _{J}(p),\lambda _{1}{\tilde {\phi }}_{1}(p),\ldots ,\lambda _{J}{\tilde {\phi }}_{J}(p)\right)$, where ${\tilde {\phi }}_{j}:M\to \mathbb {R} ^{m}$ is defined by ${\tilde {\phi }}_{j}|_{U_{j}}=\phi _{j}$ and ${\tilde {\phi }}_{j}|_{M\setminus U_{j}}=0$.

We claim that $\Phi \!$ is an embedding. That $\Phi \!$ is smooth follows immediately from its construction (the $\lambda _{j}\!$s have been used to smear out the $\phi _{j}\!$ to smooth functions on all of $M\!$). That $\Phi \!$ is injective is also clear. It takes a bit of work to show that $\Phi \!$ is an immersion, but this is left as an exercise. It remains to see that $\Phi \!$ is a homeomorphism, but this fact follows from the following topological lemma. $\Box \!$

Lemma

Let $(X,\tau )\!$ and $(Y,\sigma )\!$ be topological spaces. Suppose that $X\!$ is compact, $Y\!$ is Hausdorff, and that $f:X\to Y$ is continuous and injective. Then $f:X\to f(X)$ is a homeomorphism.

Proof

Since $f\!$ is an injection onto its image, it is a bijection. Since $f\!$ is continuous, it remains to show that $f^{-1}\!$ is continuous. Thus, it suffices to see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\!} takes closed sets to closed sets. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subset X} be closed. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\!} is compact, so is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\!} . Hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(A)\!} is compact since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\!} is continuous. But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y\!} is Hausdorff, and every compact subset of a Hausdorff space is closed. Hence $f(A)\!$ is closed. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Box}

Remarks

The smearing functions we used in Part 1 of the proof of the Whitney Embedding Theorem are very similar to partitions of unity---collections of functions that break the constant function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \mapsto 1} into a bunch of bump functions. We will now formalize this notion and show that such collections of functions exist for smooth manifolds.

Definition

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \{U_\alpha\}_{\alpha \in A}} be an open cover of a topological space (manifold) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} . A partition of unity subordinated to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\!} is a collection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\lambda_\beta : M \to \mathbb{R}_{\ge 0} \}_{\beta \in B}} of continuous (smooth) functions such that

1. For every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \in B} there is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in A} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{supp}(\lambda_\beta) \subset U_\alpha}
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathrm{supp}(\lambda_\beta)\}_{\beta \in B}} is locally finite, ie. for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in M} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in \mathrm{supp}(\lambda_\beta)} for only finitely many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta\!} .
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\beta \in B} \lambda_\beta = 1}

A local refinement of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\!} is an open cover Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{V_\gamma\}_{\gamma \in C}} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} such that for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \in C} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\gamma \subset U_\alpha} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in A} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} is called paracompact if every open cover of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\!} has a locally finite refinement.

The following result will be useful for constructing partitions of unity on manifolds:

Theorem

Manifolds are paracompact. In particular, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_\alpha\}_{\alpha \in A}} is locally finite then there is an open cover Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{V_\alpha\}_{\alpha \in A}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{V}_\alpha \subset U_\alpha} for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in A} .