Введение
Обобщение леммы Дена в топологии 3-многообразий. В математике, в топологии 3-многообразий, теорема о петле является обобщением леммы Дена. Теорема о петле была впервые доказана Христосом Папакириакопулосом в 1956 году, одновременно с леммой Дена и теоремой о сфере. Простая и полезная формулировка теоремы о петле утверждает, что если для некоторого 3-многообразия M с границей ∂M существует отображение f: M → S², такое что f не гомотопно нулю в S², то существует вложение с тем же свойством. Следующая версия теоремы о петле, предложенная Джоном Сталлингсом, приведена в стандартных трудах по 3-многообразиям (таких как Хемпель или Джако):
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 for some 3 dimensional manifold M with boundary ∂M there is a map
with 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 Let be a continuous map such that and Then there exists an embedding such that and
Furthermore 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.
Пусть M — 3-многообразие и пусть S — связная поверхность в M. Пусть N — нормальная подгруппа π₁(M), такая что π₁(M)/N = π₁(S). Пусть f: S → M — непрерывное отображение, такое что f|π₁(S) = id и f(S) ⊂ M. Тогда существует вложение g: S → M, такое что g|π₁(S) = id и g(S) ⊂ M.
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 for some 3 dimensional manifold M with boundary ∂M there is a map
with 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 Let be a continuous map such that and Then there exists an embedding such that and
Furthermore 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.
Более того, если начать с отображения f в общем положении, то для любой окрестности U сингулярного множества f, можно найти такое отображение g, образ которого лежит внутри объединения образа f и U. Доказательство Сталлингса использует адаптацию, предложенную Уайтхедом и Шапиро, "построения башни" Папакириакопулоса. "Башня" относится к специальной последовательности накрытий, предназначенной для упрощения поднятий данного отображения. Та же конструкция башни была использована Папакириакопулосом для доказательства теоремы о сфере (3-многообразия), которая утверждает, что нетривиальное отображение сферы в 3-многообразие влечет за собой существование нетривиального вложения сферы. Существует также версия леммы Дена для минимальных дисков, разработанная Миксом и С.Т. Яу, которая также существенно опирается на конструкцию башни. Существует доказательство первой версии теоремы о петле, не использующее конструкцию башни. Это было по существу сделано 30 лет назад Фридхельмом Вальдхаузеном как часть его решения проблемы слова для многообразий Хакена; хотя он признал, что это даёт доказательство теоремы о петле, он не представил подробного доказательства. Существенным элементом этого доказательства является концепция иерархии Хакена. Позже доказательства были представлены Клаусом Йоханнсоном, Марком Лакенби и Иэном Эйчисоном совместно с Хайамом Рубинштейном.
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 for some 3 dimensional manifold M with boundary ∂M there is a map
with 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 Let be a continuous map such that and Then there exists an embedding such that and
Furthermore 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.
Следующее
Один из простых следствий теоремы о петле заключается в следующем: пусть M — компактное ориентируемое неразложимое трёхмерное многообразие. Тогда M несжимаема тогда и только тогда, когда отображение включения инъективно для каждой компоненты связности M.