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- Loop_theorem abstract "In mathematics, 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 theorem states that if there is a mapwith not nullhomotopic in , then there is an embedding with the same property.The following version of the loop theorem, due to John Stallings, is given in the standard 3-manifold treatises (such as Hempel or Jaco):Let be a 3-manifold and let be a connected surface in . Let be a normal subgroup such that .Letbe a continuous map such thatandThen there exists an embeddingsuch thatandFurthermore if one starts with a map f in general position, then for any neighborhood U of the singularity set of f, we can find such a g with image lying inside the union of image of f and U.Stalling's proof utilizes an adaptation, due to Whitehead and Shapiro, of Papakyriakopoulos' "tower construction". The "tower" refers to a special sequence of coverings designed to simplify lifts of the given map. The same tower construction was used by Papakyriakopoulos to prove the sphere theorem (3-manifolds), which states that a nontrivial map of a sphere into a 3-manifold implies the existence of a nontrivial embedding of a sphere. There is also a version of Dehn's lemma for minimal discs due to Meeks and S.-T. Yau, which also crucially relies on the tower construction.A proof not utilizing the tower construction exists of the first version of the loop theorem. This was essentially done 30 years ago by Friedhelm Waldhausen as part of his solution to the word problem for Haken manifolds; although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of Haken hierarchy. Proofs were later written up, by Klaus Johannson, Marc Lackenby, and Iain Aitchison with Hyam Rubinstein.".
- Loop_theorem wikiPageExternalLink 3Mdownloads.html.
- Loop_theorem wikiPageID "4350138".
- Loop_theorem wikiPageRevisionID "606751280".
- Loop_theorem hasPhotoCollection Loop_theorem.
- Loop_theorem subject Category:3-manifolds.
- Loop_theorem subject Category:Continuous_mappings.
- Loop_theorem subject Category:Geometric_topology.
- Loop_theorem subject Category:Theorems_in_topology.
- Loop_theorem type Abstraction100002137.
- Loop_theorem type Communication100033020.
- Loop_theorem type ContinuousMappings.
- Loop_theorem type Function113783816.
- Loop_theorem type MathematicalRelation113783581.
- Loop_theorem type Message106598915.
- Loop_theorem type Proposition106750804.
- Loop_theorem type Relation100031921.
- Loop_theorem type Statement106722453.
- Loop_theorem type Theorem106752293.
- Loop_theorem type TheoremsInTopology.
- Loop_theorem comment "In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma.".
- Loop_theorem label "Loop theorem".
- Loop_theorem label "Teorema del lazo".
- Loop_theorem sameAs Teorema_del_lazo.
- Loop_theorem sameAs m.0byjlw.
- Loop_theorem sameAs Q3822213.
- Loop_theorem sameAs Q3822213.
- Loop_theorem sameAs Loop_theorem.
- Loop_theorem wasDerivedFrom Loop_theorem?oldid=606751280.
- Loop_theorem isPrimaryTopicOf Loop_theorem.