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 (1911), but Hellmuth Kneser (1929, page 260) found an error. 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 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 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 0821810405 
• Dehn, Max (1911), "Über die Topologie des dreidimensionalen Raumes", Math. Ann. 69: 137–168, doi:10.1007/BF01456932 
• Kneser (1929), Jber. Deutsch. Math. Verein. 38: 248–260 
• Papakyriakopoulos, C. D. (1957), "On Dehn's Lemma and the Asphericity of Knots", Proc. Nat. Acad. Sci. USA 43 (1): 169–172, doi:10.1073/pnas.43.1.169, MR0082671, PMC 528404, PMID 16589993 
• Papakyriakopoulos, C. D. (1957b), "On Dehn's Lemma and the Asphericity of Knots", Ann. Math. (The Annals of Mathematics, Vol. 66, No. 1) 66 (1): 1–26, doi:10.2307/1970113, JSTOR 1970113, MR0090053 
• Stallings, J.R. (1971), Group theory and three-dimensional manifolds, Yale University Press, ISBN 0300013973 

This topology-related article is a stub. You can help Wikipedia by expanding it.v · Categories: 3-manifolds Lemmas Topology stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Live on Stage (Chuck Berry album) E.Town Concrete Look at other dictionaries: Lemma (mathematics) — In mathematics, a lemma (plural lemmata or lemmascite book |last= Higham |first= Nicholas J. |title= Handbook of Writing for the Mathematical Sciences |publisher= Society for Industrial and Applied Mathematics |year= 1998 |isbn= 0898714206 |pages …   Wikipedia Dehn, Max — ▪ German mathematician in full  Max Wilhelm Dehn  born Nov. 13, 1878, Hamburg, Ger. died June 27, 1952, Black Mountain, N.C., U.S.       German mathematician and educator whose study of topology in 1910 led to his theorem on topological manifolds …   Universalium Max Dehn — Born November 13, 1878(1878 11 13) …   Wikipedia List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia Christos Papakyriakopoulos — Christos Dimitriou Papakyriakopoulos, commonly known as Papa (Χρήστος Δημητρίου Παπακυριακόπουλος in Greek; born in Chalandri, Athens, Greece on June 29, 1914, died on June 29, 1976, Princeton, New Jersey), was a Greek mathematician specializing… …   Wikipedia Christos Papakyriakopoulos — (griechisch Χρίστος Παπακυριακόπουλος, * 1914 in Chalandri, Athen; † 29. Juni 1976) war ein griechischer Mathematiker, der sich mit geometrischer Topologie beschäftigte. Inhaltsverzeichnis 1 Leben 2 Wirken …   Deutsch Wikipedia Loop theorem — In the topology of 3 manifolds, the loop theorem is a generalization of Dehn s lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn s lemma and the sphere theorem.A simple and useful version of the loop… …   Wikipedia List of lemmas — This following is a list of lemmas (or, lemmata , i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. 0 to 9 *0/1 Sorting Lemma ( comparison… …   Wikipedia List of geometric topology topics — This is a list of geometric topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of algebraic topology topics Publications in topology Contents 1 Low dimensional topology 1.1 …   Wikipedia Sphere theorem (3-manifolds) — In the topology of 3 manifolds, the sphere theorem denotes a family of statements which show us how the image of a 2 sphere, under a continuous map into a 3 manifold, may behave.One example is the following:Let M be an orientable 3 manifold such… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”