Dehn's lemma

Dehn's lemma states that a simple closed curve on the boundary of a 3-manifold that bounds a singular disk in the manifold also bounds an embedded disk.

Singular disk: A map of a disk into a space that need not be injective or preserve local geometry. The lemma starts with this possibly self-intersecting disk and replaces it with an embedded disk.
Max Dehn: A German mathematician whose work helped establish combinatorial and geometric methods in topology. The lemma bears Dehn's name and arose from his work on 3-manifolds.
Knot complement: The 3-manifold obtained by removing a neighborhood of a knot from the 3-sphere. Dehn's lemma helps analyze which curves on its boundary torus bound disks.
Sphere theorem: A theorem guaranteeing an embedded sphere under suitable nontriviality conditions on the second homotopy group. It concerns spherical homotopy classes rather than boundary curves bounding disks.
Embedded disk: A disk mapped injectively into a space, with its topology matching the topology induced from that space. This is the disk whose existence Dehn's lemma guarantees.
Christos Papakyriakopoulos: A Greek mathematician known for foundational results in 3-manifold topology. He published the first accepted proof of Dehn's lemma in 1957.
Knot group: The fundamental group of the complement of a knot in the 3-sphere. The lemma supports arguments connecting boundary curves to nontriviality in knot groups.
Jordan curve theorem: A theorem stating that a simple closed curve in the plane separates the plane into two regions. It concerns separation in the plane, not filling a boundary curve by a disk in a 3-manifold.
Loop theorem: A theorem relating nontrivial boundary loops in a 3-manifold to embedded disks. Its proof uses Dehn's lemma as a key step.
Dehn's lemma controversy: The episode in which an influential purported proof of Dehn's lemma was found to be incomplete. The flawed proof delayed acceptance of the theorem until Papakyriakopoulos's work.

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Geometric group theoryRelated: Dehn's geometric methods helped shape the study of groups through topology.