0708-1300/Class notes for Thursday, November 1

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Today's Agenda

• HW4 and TE1.
• Continue with Tuesday's agenda:
  • Debt on proper functions.
  • Prove that "the sphere is not contractible".
  • Complete the proof of the "tubular neighborhood theorem".

Proper Implies Closed

Theorem. A proper 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 f:X\to Y} from a topological space 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} to a locally compact (Hausdorff) topological space 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 closed.

Proof. Let $B$ be closed 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 X} , we need to show 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(B)} is closed 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 Y} . Since closedness is a local property, it is enough to show that every point 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\in Y} has a neighbourhood 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} such that $f(B)\cap U$ is closed in $U$. Fix $y\in Y$, and by local compactness, choose a neighbourhood $U$ of $y$ whose close ${\bar {U}}$ is compact. Then

$f(B)\cap U=f(B\cap f^{-1}(U))\cap U\subset f(B\cap f^{-1}({\bar {U}}))\cap U\subset f(B)\cap U$,

so that $f(B)\cap U=f(B\cap f^{-1}({\bar {U}}))\cap U$. But ${\bar {U}}$ is compact by choice, so $f^{-1}({\bar {U}})$ is compact as $f$ is proper, so $B\cap f^{-1}({\bar {U}})$ is compact as $B$ is closed, so $f(B\cap f^{-1}({\bar {U}}))$ is compact (and hence closed) as a continuous image of a compact set, so $f(B)\cap U$ is the intersection $f(B\cap f^{-1}({\bar {U}}))\cap U$ of a closed set with $U$, hence it is closed in $U$.

Note

The example of a non-contractible "comb" seen today is, in fact, "Cantor's comb". See, for example, page 25 of www.karlin.mff.cuni.cz/~pyrih/e/e2000v0/c/ect.ps

Typed Notes

Definition 1

X is contractible to $x_{0}$ if there exists a continuous function $H:X\times I\rightarrow X$ where

1) $H(x,0)=x$ $\forall x$

2) $H(x,1)=x_{0}$ $\forall x$

3) $H(x_{0},t)=x_{0}$ $\forall t$

Example 1

The singleton $\{x_{0}\}$ is contractible

Example 2

$B^{n}$, $D^{n}$ and $\mathbb {R} ^{n}$ are all contractible. For instance, $\mathbb {R} ^{n}$ is contractible via the function $H:\mathbb {R} ^{n}\times I\rightarrow \mathbb {R} ^{n}$ given by H(x,t) = (1-t)x.

Example 3

$S^{\infty }$ the unit sphere in a Hilbert space is contractible

Example 4

$S^{n}$ is NOT contractible for all n (as we shall prove)

Example 5

Consider the Cantor Comb consisting of the subset of $\mathbb {R} ^{2}$ consisting of the unit interval along the x axis with a spike of height 1 going up perpendicularly to the x axis at the location 1/n for n = 0,1,2,...

This set is contractible to (0,0) via the mapping that shrinks all the spikes on the comb to the real axis in time t=1/2 and then shrinks the interval to the point (0,0) by time t=1.

However, this set is NOT contrabible to (0,1). This is because should we try to flatten to the real axis points on the teeth of the comb within an epsilon neighborhood of (1,0) they would pull (1,0) down to the real axis as well otherwise continuity would be broken. Hence the third requirement of a contraction is broken.

Proposition

$S^{n}$ is not contractible

Assume not, thus it is contractible with contraction H(x,t).

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 D^{n+1}} and consider the spherical shell of diameter less than or equal to 1 inside 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 D^{n+1}} where each shell comes to rest tangentially to a point 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_0} on 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} . This is like the idea of making a new spherical shell for each such diameter inside 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 D^{n+1}} and letting all the shells "fall" to the bottom.

We now define a retract r on 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 D^{n+1}} by associating H(-,t) with the spherical shell of diameter t. I.e. the outside shell 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 D^{n+1}, S^{n}} is associated with H(x,0).

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 r|_{\partial D^{n+1}} = Id|_{S^n}} . R is clearly continuous and is a retract. But we proved last class that such retracts are impossible and this establishes the contradiction.

Q.E.D