# Alexander trick

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

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

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

allows you to define a map $F:{D}^{2}\to {D}^{2}$
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 ${x}_{n}\mathrm{\to}p$, $f\mathit{}\mathrm{(}{x}_{n}\mathrm{)}\mathrm{\to}f\mathit{}\mathrm{(}p\mathrm{)}$”.

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 $f({u}_{n}/||{u}_{n}||)\ne 0$, 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 |
---|---|

Canonical name | AlexanderTrick |

Date of creation | 2013-03-22 15:53:38 |

Last modified on | 2013-03-22 15:53:38 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 7 |

Author | juanman (12619) |

Entry type | Definition |

Classification | msc 37E30 |

Classification | msc 57S05 |

Related topic | Homeomorphism |