Alexander's trick
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]\displaystyle{ D^n }[/math] which agree on the boundary sphere [math]\displaystyle{ S^{n-1} }[/math] are isotopic.
More generally, two homeomorphisms of [math]\displaystyle{ 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]\displaystyle{ f\colon D^n \to D^n }[/math] satisfies [math]\displaystyle{ f(x) = x \text{ for all } x \in S^{n-1} }[/math], then an isotopy connecting f to the identity is given by
- [math]\displaystyle{ J(x,t) = \begin{cases} tf(x/t), & \text{if } 0 \leq \|x\| \lt t, \\ x, & \text{if } t \leq \|x\| \leq 1. \end{cases} }[/math]
Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' [math]\displaystyle{ 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]\displaystyle{ t\gt 0 }[/math] the transformation [math]\displaystyle{ J_t }[/math] replicates [math]\displaystyle{ f }[/math] at a different scale, on the disk of radius [math]\displaystyle{ t }[/math], thus as [math]\displaystyle{ t\rightarrow 0 }[/math] it is reasonable to expect that [math]\displaystyle{ J_t }[/math] merges to the identity.
The subtlety is that at [math]\displaystyle{ t=0 }[/math], [math]\displaystyle{ f }[/math] "disappears": the germ at the origin "jumps" from an infinitely stretched version of [math]\displaystyle{ 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]\displaystyle{ (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]\displaystyle{ f,g\colon D^n \to D^n }[/math] are two homeomorphisms that agree on [math]\displaystyle{ S^{n-1} }[/math], then [math]\displaystyle{ g^{-1}f }[/math] is the identity on [math]\displaystyle{ S^{n-1} }[/math], so we have an isotopy [math]\displaystyle{ J }[/math] from the identity to [math]\displaystyle{ g^{-1}f }[/math]. The map [math]\displaystyle{ gJ }[/math] is then an isotopy from [math]\displaystyle{ g }[/math] to [math]\displaystyle{ f }[/math].
Radial extension
Some authors use the term Alexander trick for the statement that every homeomorphism of [math]\displaystyle{ S^{n-1} }[/math] can be extended to a homeomorphism of the entire ball [math]\displaystyle{ 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]\displaystyle{ f\colon S^{n-1} \to S^{n-1} }[/math] be a homeomorphism, then
- [math]\displaystyle{ 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
- 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.
- 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. doi:10.1073/pnas.9.12.406. PMID 16586918. Bibcode: 1923PNAS....9..406A.
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