0708-1300/Class notes for Tuesday, October 23

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Announcements go here

Dror's Notes

• You're all invited to my talk today at 12, "Non-Commutative Gaussian Elimination and Rubik's Cube".
• Today's office hours will go 1-2.
• Our handout today is a printout of a Mathematica notebook that demonstrates a "space-filling" Peano curve. Here's the notebook, and here's a PDF version. Also, here's the main picture on that notebook:

Typed Notes

First Hour

Diversion

It is noted that there are no smooth curves that cover the plane. However, there ARE continuous curves that cover the plane.

As an example we consider the continuous function from the unit interval to the unit square that is defined iteratively that looks like the function above.

The construction is done as follows:

$f_{0}$ draws a diagonal line from the bottom left corner to the top right corner of the unit square. For $f_{1}$ one breaks the unit interval into 7 sections. The map 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_1} takes the 1st, 3rd, 5th, 7th sections respectively to a diagonal line in the bottom left, bottom right, top left and top right subboxes respectively where the diagonal line goes from the bottom left to the top right corner of each subbox. The 2nd, 4th and 6th "filler" sections of the unit interval simply draw the lines that map the end of one diagonal to the start of the other. The 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_i} is defined iteratively. See the diagram above to see what this looks like.

We note that this is obviously continuous and we get uniform convergence to a continuous function into the unit square. As every point in the square get approached arbitrary close to a point in the image of one of the iterates of the function, compactness tells us the entire square is covered.

Definition

A general definition of "locally something" is typically that every point has a neighborhood in which this something property holds. Or perhaps a neighborhood basis where this property holds.

Hence, precisely:

1) A 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 \{U_{\alpha}\}} is locally finite if every point has a neighborhood V such that V intersects 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 U_{\alpha}} 's.

2) A space is paracompact if:

a) Every open cover has a locally finite refinement

b) If a 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 \{U_{\alpha}\}} is locally finite then 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 \exists\ \{V_{\alpha}\}} so 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 \{V_{\alpha}\}} still covers the space but 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}}

Note: Manifolds are paracompact

Whitney Embedding Theorem

We recall the 3 steps to prove this theorem mentioned last class. It is noted that step three will actually be broken up into two steps:

3a: For an arbitrary 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^m} we can embed $M^{m}$ in 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}^{4m+3}}

3b: We can then embed it in 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}^{2m+1}}

Proof of Theorem

We had previously proved most of part 1, but what we still had to show was that such partitions of unity actually exist.

Reminder of Definition:

A partition of unity subordinate 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_{\alpha}\}} is a collection of functions 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_{\alpha}:M\rightarrow\mathbb{R}} such that:

1) Supp 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_{\alpha}\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 \sum_{\alpha} \lambda_{\alpha} = 1}

Claim: 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 \phi:U\rightarrow\mathbb{R}^n} is a chart and $K\subset U$ is compact then we can find a 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 \lambda:M\rightarrow\mathbb{R}} that is compact and 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 K\subset} supp 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 \subset U}

Proof of Claim

Because we are inside a chart, it is enough to just do this in 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 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 K} we can find a radius r 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 B_{r(p)}(p)\subset U} . By compactness we can take only finitely many such p's. 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 \{B_{r(p_i)}(p_i)\}} cover K.

We want to put a bump function of each ball and sum them up to give us our 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} .

Let $f_{r}(x)=e^{\frac {1}{x^{2}-r^{2}}}$ for x<r and 0 otherwise.

Now 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 \lambda(p):=\sum_i f_{r(p_i)}(d^2(p,p_i))}

Q.E.D

Theorem

On a manifold, given an open cover, you can find a partition of unity subordinate to a locally finite refinement of it.

Proof

WLOG, the cover is by charts and each one is bounded and the cover is locally finite 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}\}}

By paracompactness, find 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}\subset U_{\alpha}} 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}} and 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 \cup V_{\alpha} = M} . By the previous claim can find 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} supp 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^t_{\alpha}\subset U_{\alpha}}

Now consider 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 \lambda_{alpha}^t =: \lambda^t} . This is a finite sum.

By local finiteness, it is smooth and so we define $\lambda _{\alpha }:={\frac {\lambda _{\alpha }^{t}}{\lambda }}$

Q.E.D. for part 1 of Whitney

Second Hour

coming soon to a wikipedia near you.