Dehn's lemma

Dehn's lemma is in the topology of a fundamental theorem from the theory of 3-dimensional manifolds. It was originally developed from Max Dehn, but was not proved until 1957 by Christos Papakyriakopoulos together with a more general statement, the so-called loop set ( engl. loop theorem).

Just as the spheres set it establishes a connection between the ( be formulated in algebraic terms ) homotopy theory and geometric topology of 3-manifolds now, both sets provide the basis for a large part of the theory of 3-manifolds.

Dehn's lemma

Let be a 3-manifold and a continuous map of the circular disk that is on a neighborhood of the boundary is an embedding with.

Then there is an embedding with.

Loopnest

Let be a 3-manifold, a connected component of the boundary.

If it is not injective, then there is a real embedding with

More generally, when under the above conditions and is a normal subgroup, then there is a real embedding with

Application: Incompressible surfaces

One in a 3-manifold actually embedded ( or embedded in the edge ) surface of genus is called incompressible if there is no embedded in a circular disk with and.

Direct application of the loop set supplies the following homotopy - theoretical characterization of two-sided incompressible surfaces of genus.

A connected in a 3-manifold actually embedded ( or embedded in the edge ) two-sided surface of genus is incompressible if and only if

Is injective.

Application: knot theory

In knot theory follows from Dehn's lemma that the trivial knot by means of the node group, that is the fundamental group of Knotenkomplements can be characterized.

A node is then exactly trivial when

Applies.