In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 Dehn's lemma
asserts that a piecewise-linear map of a disk

Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

 into a 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

, with the map's singularity set in the disc's interior, implies the existence of another piecewise-linear map of the disc which is an embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

 and is identical to the original on the boundary of the disc.

This theorem was thought to be proven by , but found an error. The status of Dehn's lemma remained in doubt until proved it using his "tower construction". He also generalized the theorem to the loop theorem

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....

 and sphere theorem

Sphere theorem (3-manifolds)

In mathematics, in the topology of 3-manifolds, the sphere theorem of gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.One example is the following:...

.

Tower construction

Papakyriakopoulos proved Dehn's lemma using a tower of 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 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 disc 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.