Alexander's trick

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.

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.

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.

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