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

Statement

Two homeomorphisms of the ''n''-dimensional ball $D^n$ which agree on the boundary sphere $S^$ 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 $f\backslash colon\; D^n\; \backslash to\; D^n$ satisfies $f(x)\; =\; x\; \backslash text\; x\; \backslash in\; S^$, then an isotopy connecting ''f'' to the identity is given by : $J(x,t)\; =\; \backslash begin\; tf(x/t),\; \&\; \backslash text\; 0\; \backslash leq\; \backslash |x\backslash |\; <\; t,\; \backslash \backslash \; x,\; \&\; \backslash text\; t\; \backslash leq\; \backslash |x\backslash |\; \backslash leq\; 1.\; \backslash end$ 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 $t\backslash rightarrow\; 0$ 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 $f,g\backslash colon\; D^n\; \backslash to\; D^n$ are two homeomorphisms that agree on $S^$, then $g^f$ is the identity on $S^$, so we have an isotopy $J$ from the identity to $g^f$. 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 $S^$ 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 $f\backslash colon\; S^\; \backslash to\; S^$ be a homeomorphism, then :$F\backslash colon\; D^n\; \backslash to\; D^n\; \backslash text\; F(rx)\; =\; rf(x)\; \backslash text\; r\; \backslash in$ defines a homeomorphism of the ball.

[[Exotic spheres

The failure of smooth radial extension and the success of PL radial extension yield [[exotic spheres via [[exotic sphere#Twisted spheres|twisted spheres.

See also

* [[Clutching construction

References

* * {{cite journal|first=J. W.|last= Alexander|authorlink=James Waddell Alexander II| title=On the deformation of an ''n''-cell|journal= Proceedings of the National Academy of Sciences of the United States of America |volume=9|issue=12 |year=1923|pages= 406-407|doi=10.1073/pnas.9.12.406|bibcode=1923PNAS....9..406A|doi-access=free Category:Geometric topology Category:Homeomorphisms

Statement

Two homeomorphisms of the ''n''-dimensional ball $D^n$ which agree on the boundary sphere $S^$ are isotopic. More generally, two homeomorphisms of ''D''

Proof

Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary. If $f\backslash colon\; D^n\; \backslash to\; D^n$ satisfies $f(x)\; =\; x\; \backslash text\; x\; \backslash in\; S^$, then an isotopy connecting ''f'' to the identity is given by : $J(x,t)\; =\; \backslash begin\; tf(x/t),\; \&\; \backslash text\; 0\; \backslash leq\; \backslash |x\backslash |\; <\; t,\; \backslash \backslash \; x,\; \&\; \backslash text\; t\; \backslash leq\; \backslash |x\backslash |\; \backslash leq\; 1.\; \backslash end$ 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 $t\backslash rightarrow\; 0$ 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 $f,g\backslash colon\; D^n\; \backslash to\; D^n$ are two homeomorphisms that agree on $S^$, then $g^f$ is the identity on $S^$, so we have an isotopy $J$ from the identity to $g^f$. 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 $S^$ 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 $f\backslash colon\; S^\; \backslash to\; S^$ be a homeomorphism, then :$F\backslash colon\; D^n\; \backslash to\; D^n\; \backslash text\; F(rx)\; =\; rf(x)\; \backslash text\; r\; \backslash in$ defines a homeomorphism of the ball.

[[Exotic spheres

The failure of smooth radial extension and the success of PL radial extension yield [[exotic spheres via [[exotic sphere#Twisted spheres|twisted spheres.

See also

* [[Clutching construction

References

* * {{cite journal|first=J. W.|last= Alexander|authorlink=James Waddell Alexander II| title=On the deformation of an ''n''-cell|journal= Proceedings of the National Academy of Sciences of the United States of America |volume=9|issue=12 |year=1923|pages= 406-407|doi=10.1073/pnas.9.12.406|bibcode=1923PNAS....9..406A|doi-access=free Category:Geometric topology Category:Homeomorphisms