0708-1300/Class notes for Tuesday, October 30

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Today's Agenda

Debts

A bit more about proper functions on locally compact spaces.

Smooth Retracts and Smooth Brouwer

Theorem. There does not exist a smooth retract 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 r:D^{n+1}\to S^n} .

Corollary. The sphere 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 S^n} is not smoothly contractible.

Challenge. Remove the word "smooth" everywhere above.

Smooth Approximation

Theorem. 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} be a closed subset of a smooth 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} , 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 f:M\to{\mathbb R}} be a continuous function whose restriction 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} 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 A} is smooth, and 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 \epsilon} be your favourite small number. Then there exists a smooth $g:M\to {\mathbb {R} }$ 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 f|_A=g|_A} 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 ||f-g||<\epsilon} . Furthermore, 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} 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 g} are homotopic via 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 \epsilon} -small homotopy.

Theorem. The same, with the target space replaced by an arbitrary compact metrized 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 N} .

Tubular Neighborhoods

Theorem. Every compact smooth submanifold 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} 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 {\mathbb R}^n} has a "tubular neighborhood".

Entertainment

A student told me about this clip on YouTube (lyrics). Enjoy!

There is this one too but it is in Spanish. Romance of the Derivative and the Arctangent

Further Notes

Corollary. (The Brouwer Fixed Point Theorem) Every 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 f:D^n\to D^n} has a fixed point.

Suggestion for a good deed. Tell Dror if he likes the Brouwer fixed point theorem, for he is honestly unsure. But first hear some drorpaganda on what he likes and what he doesn't quite.

With Brouwer's fixed point theorem you can prove amazing things

1) There are to antipodal points in the equator with the same temperature.

2) There are two antipodal points with the same temperature and the same pressure.

3) You can through three potatoes in the air and with just one swing cut all of them in half.

4) Every non-bold person has a swirl of hair or some oder problem ordering their hair...

5) If you have a car with a loose antenna and you always go in your car in a trip exactly the same way every day then there is an initial position of the antenna such that it wont fall during your trip.