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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 D^n which agree on the boundary sphere S^ are isotopic. More generally, two homeomorphisms of ''D''''n'' 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 f\colon D^n \to D^n satisfies f(x) = x \text x \in S^, then an isotopy connecting ''f'' to the identity is given by : J(x,t) = \begin tf(x/t), & \text 0 \leq \|x\| < t, \\ x, & \text t \leq \|x\| \leq 1. \end Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' f 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 t>0 the transformation J_t replicates f at a different scale, on the disk of radius t, thus as t\rightarrow 0 it is reasonable to expect that J_t merges to the identity. The subtlety is that at t=0, f "disappears": the germ at the origin "jumps" from an infinitely stretched version of f 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 (x,t)=(0,0). This underlines that the Alexander trick is a PL construction, but not smooth. General case: isotopic on boundary implies isotopic If f,g\colon D^n \to D^n are two homeomorphisms that agree on S^, then g^f is the identity on S^, so we have an isotopy J from the identity to g^f. The map gJ is then an isotopy from g to f.

Radial extension

Some authors use the term ''Alexander trick'' for the statement that every homeomorphism of S^ can be extended to a homeomorphism of the entire ball D^n. 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 f\colon S^ \to S^ be a homeomorphism, then : F\colon D^n \to D^n \text F(rx) = rf(x) \text r \in ,1) \text x \in S^ 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 [[exotic sphere#Twisted spheres|twisted spheres.

See also

* [[Clutching construction

References

* * {{cite journal|first=J. W.|last= Alexander|authorlink=James Waddell Alexander II| title=On the deformation of an ''n''-cell|journal= Proceedings of the National Academy of Sciences of the United States of America |volume=9|issue=12 |year=1923|pages= 406-407|doi=10.1073/pnas.9.12.406|bibcode=1923PNAS....9..406A|doi-access=free Category:Geometric topology Category:Homeomorphisms