In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Dehn's lemma asserts that a
piecewise-linear map of a
disk into a
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
, 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
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
and is identical to the original on the
boundary 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 (1938) proved it using his "tower construction". He also generalized the theorem to the
loop theorem and
sphere theorem.
Tower construction
Papakyriakopoulos proved Dehn's lemma using a
tower of
covering space
In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...
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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External links
proof by Papakyriakopoulos from 1957
{{Authority control
3-manifolds
Lemmas