0708-1300/Class notes for Tuesday, October 30

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In Preparation

The information below is preliminary and cannot be trusted! (v)

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 $r:D^{n+1}\to S^{n}$.

Corollary. (The Brouwer Fixed Point Theorem) Every smooth $f:D^{n}\to D^{n}$ 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.

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

Challenge. Remove the word "smooth" everywhere above.

Smooth Approximation

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

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

Tubular Neighborhoods

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