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:

Step 1: Repeatedly take a connected double cover of a regular neighborhood of the image of the disk to produce a tower of spaces, each a connected double cover of the one below it. The map from the disk can be lifted to all stages of this tower. Each double cover simplifies the singularities of the embedding of the disk, so it is only possible to take a finite number of such double covers, and the top level of this tower has no connected double covers.
Step 2. If the 3-manifold has no connected double covers then all its boundary components are 2-spheres. In particular the top level of the tower has this property, and in this case it is easy to modify the map from the disc so that it is an embedding.
Step 3. The embedding of the disk can now be pushed down the tower of double covers one step at a time, by cutting and pasting the 2-disk.

References

Bing, R.H. (1983), The Geometric Topology of 3-manifolds, American Mathematical Society, p. 183, ISBN 0-8218-1040-5
Dehn, Max (1910), "Über die Topologie des dreidimensionalen Raumes", Math. Ann., 69: 137–168, doi:10.1007/BF01455155
Jaco, William; Rubinstein, Hyam (1989), "PL Equivariant Surgery and Invariant Decompositions of 3-Manifolds", Advances in Mathematics, 73 (2): 149–191, doi:10.1016/0001-8708(89)90067-4
Kneser (1929), "Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten", Jber. Deutsch. Math. Verein., 38: 248–260
Papakyriakopoulos, C. D. (1957), "On Dehn's Lemma and the Asphericity of Knots", Proc. Natl. Acad. Sci. USA, 43 (1): 169–172, Bibcode:1957PNAS...43..169P, doi:10.1073/pnas.43.1.169, MR 0082671, PMC 528404, PMID 16589993
Papakyriakopoulos, C. D. (1957b), "On Dehn's Lemma and the Asphericity of Knots", Ann. Math., 66 (1): 1–26, doi:10.2307/1970113, JSTOR 1970113, MR 0090053
Rubinstein, J.H. (2003), Dehn's lemma and the loop theorem, Low-dimensional topology, new studies in advanced mathematics, Vol 3 International Press, pp. 61–68
Stallings, J.R. (1971), Group theory and three-dimensional manifolds, Yale University Press, ISBN 0-300-01397-3
Shapiro, Arnold; Whitehead, J.H.C. (1958), "A proof and extension of Dehn's lemma", Bull. Am. Math. Soc., AMS, 64: 174–178

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