“Alexander's trick”的意思、由来-开放百科全书

词条

Alexander's trick

释义

Statement
Proof
Radial extension
Exotic spheres
References

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

Statement

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

More generally, two homeomorphisms of Dⁿ that are isotopic on the boundary are isotopic.

Proof

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 .

Radial extension

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.

Exotic spheres

The failure of smooth radial extension and the success of PL radial extension

yield exotic spheres via twisted spheres.

References

{{cite book |last=Hansen |first=V.L. |title=Braids and Coverings |year=1989 |publisher=Cambridge University Press |isbn=0-521-38757-4}}
J. W. Alexander, On the deformation of an n-cell, Proc. Natl. Acad. Sci. USA 9 (1923), 406-407

2 : Geometric topology|Homeomorphisms

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。