Alexander's trick

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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[edit]

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[edit]

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[edit]

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[edit]

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

References[edit]

  • Hansen, V.L. (1989). Braids and Coverings. Cambridge University Press. ISBN 0-521-38757-4. 
  • J. W. Alexander, On the deformation of an n-cell, Proc. Nat. Acad. Sci. USA 9 (1923), 406-407