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
More generally, two homeomorphisms of
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.
, 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
at a different scale, on the disk of radius
it is reasonable to expect that
merges to the identity.
"disappears": the germ at the origin "jumps" from an infinitely stretched version of
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
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.
be a homeomorphism, then The failure of smooth radial extension and the success of PL radial extension yield exotic spheres via twisted spheres.