In
mathematics, Dehn's lemma asserts that a
piecewise-linear map of a
disk into a
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
, with the map's
singularity set in the disk's
interior, implies the existence of another piecewise-linear map of the disk which is an
embedding and is identical to the original on the
boundary
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* Boundary (cricket), the edge of the pl ...
of the disk.
This theorem was thought to be proven by , but found a gap in the proof. The status of Dehn's lemma remained in doubt until using work by
Johansson
Johansson is a patronymic family name of Swedish origin meaning ''"son of Johan"'', or ''"Johan's son"''. It is the most common Swedish family name, followed by Andersson. (First 18 surnames ends -sson.)
The Danish, Norwegian, German and Dutc ...
(1938) proved it using his "tower construction". He also generalized the theorem to the
loop theorem
In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem.
A simple and useful ver ...
and
sphere theorem
In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. ...
.
Tower construction
Papakyriakopoulos proved Dehn's lemma using a
tower of
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
s. Soon afterwards gave a substantially simpler proof, proving a more powerful result. Their proof used Papakyriakopoulos' tower construction, but with double covers, as follows:
*Step 1: Repeatedly take a connected double cover of a
regular neighborhood of the image of the disk to produce a tower of spaces, each a connected double cover of the one below it. The map from the disk can be lifted to all stages of this tower. Each double cover simplifies the singularities of the embedding of the disk, so it is only possible to take a finite number of such double covers, and the top level of this tower has no connected double covers.
*Step 2. If the 3-manifold has no connected double covers then all its boundary components are 2-spheres. In particular the top level of the tower has this property, and in this case it is easy to modify the map from the disk so that it is an embedding.
*Step 3. The embedding of the disk can now be pushed down the tower of double covers one step at a time, by cutting and pasting the 2-disk.
References
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*{{citation , last1= Shapiro , first1= Arnold, author-link1=Arnold S. Shapiro, last2=Whitehead , first2=J.H.C. , author-link2= J.H.C. Whitehead , title=A proof and extension of Dehn's lemma , volume= 64 , issue= 4, year=1958 , publisher=AMS , pages=174–178 , journal =
Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...
, doi= 10.1090/S0002-9904-1958-10198-6, doi-access=free
External link
proof by Papakyriakopoulos from 1957
3-manifolds
Lemmas