Alexander's Trick

Get Alexander's Trick essential facts below. View Videos or join the Alexander's Trick discussion. Add Alexander's Trick to your PopFlock.com topic list for future reference or share this resource on social media.
## Statement

## Proof

## Radial extension

### Exotic spheres

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Alexander's Trick

**Alexander's trick**, also known as the **Alexander trick**, is a basic result in geometric topology, named after J. W. Alexander.

Two homeomorphisms of the *n*-dimensional ball which agree on the boundary sphere are isotopic.

More generally, two homeomorphisms of *D*^{n} that are isotopic on the boundary are isotopic.

**Base case**: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.

If satisfies , then an isotopy connecting *f* to the identity is given by

Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' down to the origin. William Thurston calls this "combing all the tangles to one point". In the original 2-page paper, J. W. Alexander explains that for each the transformation replicates at a different scale, on the disk of radius , thus as it is reasonable to expect that merges to the identity.

The subtlety is that at , "disappears": the germ at the origin "jumps" from an infinitely stretched version of to the identity. Each of the steps in the homotopy could be smoothed (smooth the transition), but the homotopy (the overall map) has a singularity at . This underlines that the Alexander trick is a PL construction, but not smooth.

**General case**: isotopic on boundary implies isotopic

If are two homeomorphisms that agree on , then is the identity on , so we have an isotopy from the identity to . The map is then an isotopy from to .

Some authors use the term *Alexander trick* for the statement that every homeomorphism of can be extended to a homeomorphism of the entire ball .

However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true piecewise-linearly, but not smoothly.

Concretely, let be a homeomorphism, then

- defines a homeomorphism of the ball.

The failure of smooth radial extension and the success of PL radial extension yield exotic spheres via twisted spheres.

- Hansen, Vagn Lundsgaard (1989).
*Braids and coverings: selected topics*. London Mathematical Society Student Texts.**18**. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511613098. ISBN 0-521-38757-4. MR 1247697. - Alexander, J. W. (1923). "On the deformation of an
*n*-cell".*Proceedings of the National Academy of Sciences of the United States of America*.**9**(12): 406-407. Bibcode:1923PNAS....9..406A. doi:10.1073/pnas.9.12.406.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists