##  [Alexander Trick](/alexander-trick-0) 

 Definition

A topological technique that extends a homeomorphism or an isotopy of the (n−1)-sphere which fixes a hemisphere (or is suitably controlled on the boundary) to a homeomorphism or isotopy of the n‑ball by radial contraction: points in the interior are moved radially in proportion to their distance from the center while the boundary motion is prescribed by the sphere map.

 

 

 

 

 

 





## Principle

Principle

Radial extension: use the radial coordinate to interpolate between the identity at the center and the prescribed boundary homeomorphism, yielding a continuous (often isotopic) extension of sphere data to the whole ball.

 

 

 

 

 





## Demonstration

Demonstration

In dimension two, a homeomorphism of the circle that is the identity on an arc (a hemisphere analogue) can be extended to the disk by mapping each radius to itself and applying the circle homeomorphism at the outer endpoint scaled by radius; similarly, an isotopy of the circle fixing an arc extends to an isotopy of the disk by the same radial scaling.

 

 

 

 

## Misapplication

Misapplication

Expecting the trick to preserve differentiability class (a continuous radial extension need not be smooth even if the boundary map is smooth), or attempting to extend maps that do not fix (or are not controlled on) a hemisphere without checking the required boundary condition. Attempting to apply the trick in contexts where the radial parametrization is ill-defined (nonstar-shaped domains, wild embeddings) fails.

 

 

 

 

 





## Consequence

Consequence

Provides elementary proofs of extension and isotopy results for balls, shows contractibility properties of certain homeomorphism groups relative to a hemisphere, and supplies explicit ambient isotopies used in low‑dimensional topology and in simplifying boundary behaviour of maps.

 

 

 

 

## Reversal

Reversal

If the boundary map is changed arbitrarily (no fixed hemisphere or no compatible radial behaviour), a naive radial extension can break continuity or injectivity; conversely, fixing the interior instead of the boundary leads to different extension problems not solved by this trick.

 

 

 

 

 





## Boundary

Boundary

Applies to maps of the standard sphere S^{n−1} and the standard ball B^n using the radial coordinate; typically requires the boundary map to be continuous (homeomorphism) and to fix a hemisphere (or to be isotopic to the identity via maps fixing a hemisphere) when the isotopy extension version is invoked. Does not directly apply to arbitrary manifolds without a radial structure or to differentiability classes without extra smoothing arguments.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often juxtaposed with the general isotopy extension theorem or the collar theorem; the Alexander Trick is an elementary, explicit radial construction valid for balls and spheres, while those theorems are broader and more abstract and may require stronger hypotheses.

 

 

 

 

 





## Synthesis

Synthesis

The Alexander Trick is the radial contraction method for promoting controlled sphere homeomorphisms or isotopies to the ball: interpolate linearly along radii from the identity at the center to the given boundary motion, obtaining an explicit extension used to build isotopies and to control boundary behaviour in manifold constructions.