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
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al
ball
A ball is a round object (usually spherical
of a sphere
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Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
which agree on the
boundary
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sphere
A sphere (from Greek#REDIRECT Greek
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Greece
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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
satisfies
, then an isotopy connecting ''f'' to the identity is given by
:
Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing'
down to the origin.
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician
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Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

calls this "combing all the tangles to one point". In the original 2-page paper, J. W. Alexander explains that for each
the transformation
replicates
at a different scale, on the disk of radius
, thus as
it is reasonable to expect that
merges to the identity.
The subtlety is that at
,
"disappears": the
germ at the origin "jumps" from an infinitely stretched version of
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
. This underlines that the Alexander trick is a
PL construction, but not smooth.
General case: isotopic on boundary implies isotopic
If
are two homeomorphisms that agree on
, then
is the identity on
, so we have an isotopy
from the identity to
. The map
is then an isotopy from
to
.
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
can be extended to a homeomorphism of the entire ball
.
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
be a homeomorphism, then
:
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=
, 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