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Alexander's trick, also known as the Alexander trick, is a basic result in
geometric topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, named after J. W. Alexander.


Statement

Two
homeomorphism and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
s of the ''n''-
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
al
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
D^n which agree on the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
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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\colon D^n \to D^n satisfies f(x) = x \text x \in S^, then an isotopy connecting ''f'' to the identity is given by : J(x,t) = \begin tf(x/t), & \text 0 \leq \, x\, < t, \\ x, & \text t \leq \, x\, \leq 1. \end Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' f down to the origin.
William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

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\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\colon D^n \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 and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
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\colon S^ \to S^ be a homeomorphism, then : F\colon D^n \to D^n \text F(rx) = rf(x) \text r \in ,1) \text x \in S^ 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 ''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-review Peer review is the evaluation of work by one or more people with similar competencies as the ...
, volume=9, issue=12 , year=1923, pages= 406-407, doi=10.1073/pnas.9.12.406, bibcode=1923PNAS....9..406A, doi-access=free Geometric topology Homeomorphisms