# Alexander trick

Want to extend a homeomorphism  of the circle $S^{1}$ to the whole disk $D^{2}$?

Let $f\colon S^{1}\to S^{1}$ be a homeomorphism. Then the formula

 $F(x)=||x||f(x/||x||)$

allows you to define a map $F\colon D^{2}\to D^{2}$ which extends $f$, for if $x\in S^{1}\subset D^{2}$ then $||x||=1$ and $F(x)=1\cdot f(x/1)=f(x)$. Clearly this map is continuous  , save (maybe) the origin, since this formula is undefined there. Nevertheless this is removable.

To check continuity at the origin use: “A map $f$ is continuous at a point $p$ if and only if for each sequence $x_{n}\to p$, $f(x_{n})\to f(p)$.

So take a sequence $u_{n}\in D^{2}$ such that $u_{n}\to 0$ (i.e. which tends to the origin). Then $F(u_{n})=||u_{n}||f(u_{n}/||u_{n}||)$ and since $f(u_{n}/||u_{n}||)\neq 0$, hence $||u_{n}||\to 0$ implies $F(u_{n})\to 0$, that is $F$ is also continuous at the origin.

The same method works for $f^{-1}$.

In the same vein one can extend homeomorphisms $S^{n}\to S^{n}$ to $D^{n+1}\to D^{n+1}$.

Title Alexander trick AlexanderTrick 2013-03-22 15:53:38 2013-03-22 15:53:38 juanman (12619) juanman (12619) 7 juanman (12619) Definition msc 37E30 msc 57S05 Homeomorphism