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 disc's interior, implies the existence of another piecewise-linear map of the disc which is an embedding and is identical to the original on the boundary of the disc.

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) 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 Arnold Shapiro 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:

References

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