Alexander's trick explained

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

Statement

D ^{n} which agree on the boundary sphere S ^{n-1} are isotopic.

More generally, two homeomorphisms of

D ^{n} that are isotopic on the boundary are isotopic.

Proof

Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.

If

f\colon D ^{n} \to D ^{n} satisfies f(x) = x forall x \in S ^{n-1} , then an isotopy connecting f to the identity is given byJ(x,t) = \begin{cases} tf(x/t), & if 0 \leq \|x\| < t, \ x, & if t \leq \|x\| \leq 1 . \end{cases}

Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing'

f 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 t> 0 the transformation J _{t} replicates f at a different scale, on the disk of radius t , thus as t → 0 it is reasonable to expect that J _{t} merges to the identity.

The subtlety is that at

t =0 , f "disappears": the germ at the origin "jumps" from an infinitely stretched version of f 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 (x,t) = ( 0 , 0 ) . This underlines that the Alexander trick is a PL construction, but not smooth.

General case: isotopic on boundary implies isotopic

If

f,g\colon D ^{n} \to D ^{n} are two homeomorphisms that agree on S ^{n-1} , then g ^{-1} f is the identity on S ^{n-1} , so we have an isotopy J from the identity to g ^{-1} f . The map gJ is then an isotopy from g to f .

Radial extension

Some authors use the term Alexander trick for the statement that every homeomorphism of

S ^{n-1} can be extended to a homeomorphism of the entire ball D ^{n} .

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

f\colon S ^{n-1} \to S ^{n-1} be a homeomorphism, thenF\colon D ^{n} \to D ^{n} with F(rx) = rf(x) forall r \in [ 0 , 1 ] and x \in S ^{n-1} defines a homeomorphism of the ball.

Exotic spheres

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

See also

• Clutching construction

References

• Book: Hansen, Vagn Lundsgaard . Braids and coverings: selected topics. 1989 . . Cambridge. London Mathematical Society Student Texts. 18. 10.1017/CBO9780511613098. 0-521-38757-4. 1247697.
• J. W.. Alexander. James Waddell Alexander II. On the deformation of an n-cell. . 9. 12 . 1923. 406–407. 10.1073/pnas.9.12.406. 16586918. 1085470. 1923PNAS....9..406A. free.