Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.
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.
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".
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 .
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.
The failure of smooth radial extension and the success of PL radial extension
yield exotic spheres via twisted spheres.