Dehn's lemma

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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.

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This theorem was thought to be proven by MaxDehn  ( 1910 ), but HellmuthKneser  ( 1929 , page 260) found a gap in the proof. The status of Dehn's lemma remained in doubt until ChristosPapakyriakopoulos  ( 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.

Tower construction

Papakyriakopoulos proved Dehn's lemma using a tower of covering spaces. Soon afterwards ArnoldShapiro and J.H.C. Whitehead  ( 1958 ) gave a substantially simpler proof, proving a more powerful result. Their proof used Papakyriakopoulos' tower construction, but with double covers, as follows:

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