Dehn's lemma

In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, 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 of the disk.

This theorem was thought to be proven by Max Dehn (1910), but Hellmuth Kneser (1929, page 260) found a gap in the proof. The status of Dehn's lemma remained in doubt until Christos Papakyriakopoulos (1957, 1957b) using work by Johansson (1938) proved it using his "tower construction". He also generalized the theorem to the loop theorem and sphere theorem.