Definition
Alexander's trick is a result in geometric topology stating that any homeomorphism of the $(n-1)$-dimensional sphere $S^{,n-1}$ that fixes the sphere pointwise (or, more generally, is isotopic to the identity) can be extended to a homeomorphism of the $n$-dimensional closed ball $B^{,n}$. The extension is obtained by radially interpolating the boundary homeomorphism toward the centre of the ball.
Overview
The theorem is typically presented for the 2‑dimensional case, where a homeomorphism $h : S^{1}\to S^{1}$ of the unit circle is extended to a homeomorphism $\tilde h : D^{2}\to D^{2}$ of the closed unit disk by the formula
$$ \tilde h(r e^{i\theta}) = r,h(e^{i\theta}),\qquad 0\le r\le 1 . $$
For higher dimensions the same radial construction applies: for a point $x\in B^{,n}$ written as $x = r\cdot u$ with $0\le r\le 1$ and $u\in S^{,n-1}$, define
$$ \tilde h(x)= r;h(u). $$
The map $\tilde h$ is continuous, bijective, and its inverse is obtained by the same formula applied to $h^{-1}$; thus $\tilde h$ is a homeomorphism. The result shows that the inclusion $S^{,n-1}\hookrightarrow B^{,n}$ induces a surjection on the group of self‑homeomorphisms, a fact that underlies several classical theorems, such as the planar Schoenflies theorem and various isotopy extension results.
Etymology / Origin
The trick is named after the American topologist James Waddell Alexander II (1888–1971), who introduced the argument in his work on the topology of manifolds and the behavior of embeddings in Euclidean space. Alexander employed the construction in the early 20th century while studying properties of spheres and balls, and the method subsequently became a standard tool in the field, retaining his name.
Characteristics
| Aspect | Details |
|---|---|
| Dimensionality | Valid for all $n\ge 1$; the case $n=1$ is trivial, while the case $n=2$ is most frequently illustrated. |
| Construction | Radial extension: points are written in polar (or spherical) coordinates and the angular component is acted on by the boundary homeomorphism, while the radial coordinate is left unchanged. |
| Continuity | The extension is continuous because both the boundary homeomorphism and the radial coordinate vary continuously; bijectivity follows from the bijectivity of the boundary map. |
| Isotopy Extension | If the boundary homeomorphism is isotopic to the identity, the radial extension provides an explicit isotopy of the ball to the identity map. |
| Applications | Used in proofs of the Schoenflies theorem, in establishing that the group of homeomorphisms of a ball is contractible, and in arguments concerning the tameness of embeddings (e.g., Alexander’s horned sphere). |
| Limitations | The construction requires the boundary map to be defined on the entire sphere and to be a homeomorphism; it does not directly apply to more general maps such as embeddings with self‑intersections. |
Related Topics
- Schoenflies theorem – a planar result concerning the extension of circle homeomorphisms to disk homeomorphisms.
- Isotopy extension theorem – a general principle that isotopies of a subspace extend to isotopies of the ambient manifold.
- Homeomorphism groups of manifolds – the study of the topology of spaces of self‑homeomorphisms; Alexander's trick shows contractibility for balls.
- Alexander's horned sphere – another concept named after James Alexander, dealing with pathological embeddings of the sphere in $\mathbb{R}^3$.
- Alexander duality – a relation in algebraic topology linking the homology of a subspace of a sphere to the cohomology of its complement.
This entry reflects the established usage of the term “Alexander's trick” within the mathematical literature.