Английская Википедия:Alexander's trick
Шаблон:Short description 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 <math>D^n</math> which agree on the boundary sphere <math>S^{n-1}</math> are isotopic.
More generally, two homeomorphisms of <math>D^n</math> 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 <math>f\colon D^n \to D^n</math> satisfies <math>f(x) = x \text{ for all } x \in S^{n-1}</math>, then an isotopy connecting f to the identity is given by
- <math> J(x,t) = \begin{cases} tf(x/t), & \text{if } 0 \leq \|x\| < t, \\ x, & \text{if } t \leq \|x\| \leq 1. \end{cases} </math>
Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' <math>f</math> 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 <math>t>0 </math> the transformation <math>J_t </math> replicates <math>f</math> at a different scale, on the disk of radius <math>t</math>, thus as <math>t\rightarrow 0</math> it is reasonable to expect that <math>J_t </math> merges to the identity.
The subtlety is that at <math>t=0</math>, <math>f</math> "disappears": the germ at the origin "jumps" from an infinitely stretched version of <math>f</math> 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 <math>(x,t)=(0,0)</math>. This underlines that the Alexander trick is a PL construction, but not smooth.
General case: isotopic on boundary implies isotopic
If <math>f,g\colon D^n \to D^n</math> are two homeomorphisms that agree on <math>S^{n-1}</math>, then <math>g^{-1}f</math> is the identity on <math>S^{n-1}</math>, so we have an isotopy <math>J</math> from the identity to <math>g^{-1}f</math>. The map <math>gJ</math> is then an isotopy from <math>g</math> to <math>f</math>.
Radial extension
Some authors use the term Alexander trick for the statement that every homeomorphism of <math>S^{n-1}</math> can be extended to a homeomorphism of the entire ball <math>D^n</math>.
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 <math>f\colon S^{n-1} \to S^{n-1}</math> be a homeomorphism, then
- <math> F\colon D^n \to D^n \text{ with } F(rx) = rf(x) \text{ for all } r \in [0,1) \text{ and } x \in S^{n-1}</math> 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.
See also
References
