Введение
В топологии, отрасли математики, многообразие M может быть разложено или разделено, представляя M как комбинацию более мелких частей. При этом необходимо указать, что представляют собой эти части и каким образом они соединяются вместе, образуя M.
Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold. In particular, it has been useful in attempts to classify 3 manifolds and also in proving the higher dimensional Poincaré conjecture. The table below is a summary of the various manifold decomposition techniques. The column labeled "M" indicates what kind of manifold can be decomposed; the column labeled "How it is decomposed" indicates how, starting with a manifold, one can decompose it into smaller pieces; the column labeled "The pieces" indicates what the pieces can be; and the column labeled "How they are combined" indicates how the smaller pieces are combined to make the large manifold. Type of decomposition M How it is decomposed The pieces How they are combined Triangulation Depends on dimension. In dimension 3, a theorem by Edwin E. Moise gives that every 3 manifold has a unique triangulation, unique up to common subdivision. In dimension 4, not all manifolds are triangulable. For higher dimensions, general existence of triangulations is unknown. Simplices Glue together pairs of codimension one faces Jaco Shalen/Johannson torus decomposition Irreducible, orientable, compact 3 manifolds Cut along embedded tori Atoroidal or Seifert fibered 3 manifolds Union along their boundary, using the trivial homeomorphism Prime decomposition Essentially surfaces and 3 manifolds. The decomposition is unique when the manifold is orientable. Cut along embedded spheres; then union by the trivial homeomorphism along the resultant boundaries with disjoint balls. Prime manifolds Connected sum Heegaard splitting Closed, orientable 3 manifolds Two handlebodies of equal genus Union along the boundary by some homeomorphism Handle decomposition Any compact (smooth) n manifold (and the decomposition is never unique) Through Morse functions a handle is associated to each critical point. Balls (called handles) Union along a subset of the boundaries. Note that the handles must generally be added in a specific order. Haken hierarchy Any Haken manifold Cut along a sequence of incompressible surfaces 3 balls Disk decomposition Certain compact, orientable 3 manifolds Suture the manifold, then cut along special surfaces (condition on boundary curves and sutures ) 3 balls Open book decomposition Any closed orientable 3 manifold A link and a family of 2 manifolds that share a boundary with that link Trigenus Compact, closed 3 manifolds Surgeries Three orientable handlebodies Unions along subsurfaces on boundaries of handlebodies
Разложение многообразия работает в двух направлениях: можно начать с меньших частей и построить многообразие, либо начать с большого многообразия и разложить его. Последний подход оказался очень полезным для изучения многообразий: без таких инструментов, как разложение, понимание многообразия иногда бывает крайне затруднительным. В частности, он был полезен в попытках классификации 3-мерных многообразий, а также в доказательстве обобщенной гипотезы Пуанкаре. В таблице ниже представлен обзор различных техник разложения многообразий. Колонка с надписью "M" указывает, какой тип многообразия может быть разложен; колонка с надписью "Как оно разлагается" описывает, как, начиная с многообразия, его можно разложить на более мелкие части; колонка с надписью "Части" указывает, какими могут быть эти части; а колонка с надписью "Как они объединяются" описывает, как меньшие части соединяются вместе, образуя большое многообразие.
Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold. In particular, it has been useful in attempts to classify 3 manifolds and also in proving the higher dimensional Poincaré conjecture. The table below is a summary of the various manifold decomposition techniques. The column labeled "M" indicates what kind of manifold can be decomposed; the column labeled "How it is decomposed" indicates how, starting with a manifold, one can decompose it into smaller pieces; the column labeled "The pieces" indicates what the pieces can be; and the column labeled "How they are combined" indicates how the smaller pieces are combined to make the large manifold. Type of decomposition M How it is decomposed The pieces How they are combined Triangulation Depends on dimension. In dimension 3, a theorem by Edwin E. Moise gives that every 3 manifold has a unique triangulation, unique up to common subdivision. In dimension 4, not all manifolds are triangulable. For higher dimensions, general existence of triangulations is unknown. Simplices Glue together pairs of codimension one faces Jaco Shalen/Johannson torus decomposition Irreducible, orientable, compact 3 manifolds Cut along embedded tori Atoroidal or Seifert fibered 3 manifolds Union along their boundary, using the trivial homeomorphism Prime decomposition Essentially surfaces and 3 manifolds. The decomposition is unique when the manifold is orientable. Cut along embedded spheres; then union by the trivial homeomorphism along the resultant boundaries with disjoint balls. Prime manifolds Connected sum Heegaard splitting Closed, orientable 3 manifolds Two handlebodies of equal genus Union along the boundary by some homeomorphism Handle decomposition Any compact (smooth) n manifold (and the decomposition is never unique) Through Morse functions a handle is associated to each critical point. Balls (called handles) Union along a subset of the boundaries. Note that the handles must generally be added in a specific order. Haken hierarchy Any Haken manifold Cut along a sequence of incompressible surfaces 3 balls Disk decomposition Certain compact, orientable 3 manifolds Suture the manifold, then cut along special surfaces (condition on boundary curves and sutures ) 3 balls Open book decomposition Any closed orientable 3 manifold A link and a family of 2 manifolds that share a boundary with that link Trigenus Compact, closed 3 manifolds Surgeries Three orientable handlebodies Unions along subsurfaces on boundaries of handlebodies
Тип разложения | M | Как оно разлагается | Части | Как они объединяются
---|---|---|---|---
Триангуляция | Зависит от размерности. В размерности 3 теорема Эдвина Э. Мойса утверждает, что каждое 3-многообразие имеет единственную триангуляцию, единственную с точностью до общего подразделения. В размерности 4 не все многообразия триангулируемы. Для более высоких размерностей общее существование триангуляций неизвестно. | Симплексы | Склеивание пар граней коразмерности один
Декомпозиция Яко-Шалена/Йоханссона по торам | Нередуцируемые, ориентируемые, компактные 3-многообразия | Разрез вдоль вложенных торов | Атороидальные или Сейфертовы волокнистые 3-многообразия | Объединение вдоль их границы с использованием тривиального гомеоморфизма
Первичное разложение | По сути, поверхности и 3-многообразия. Разложение единственно, когда многообразие ориентируемо. | Разрез вдоль вложенных сфер; затем объединение тривиальным гомеоморфизмом вдоль полученных границ с непересекающимися шарами. | Первичные многообразия | Связанная сумма
Разделение Хигаарда | Закрытые, ориентируемые 3-многообразия | Два тела-ручки одинакового рода | Объединение вдоль границы посредством некоторого гомеоморфизма
Разложение на тела-ручки | Любое компактное (гладкое) n-многообразие (и разложение никогда не является единственным) | Через функции Морса каждому критическому пункту сопоставляется тело-ручка. | Шары (так называемые ручки) | Объединение вдоль подмножества границ. Следует отметить, что ручки, как правило, должны добавляться в определенном порядке.
Иерархия Хакена | Любое многообразие Хакена | Разрез вдоль последовательности несжимаемых поверхностей | 3 шара |
Разложение на диски | Некоторые компактные, ориентируемые 3-многообразия | Сшивание многообразия, затем разрез вдоль специальных поверхностей (условие на граничные кривые и швы) | 3 шара |
Разложение по открытой книге | Любое закрытое ориентируемое 3-многообразие | Связь и семейство из 2-многообразий, имеющих общую границу с этой связью |
Триген | Компактные, закрытые 3-многообразия | Хирургии | Три ориентируемых тела-ручки | Объединения вдоль подповерхностей на границах тел-ручек
Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold. In particular, it has been useful in attempts to classify 3 manifolds and also in proving the higher dimensional Poincaré conjecture. The table below is a summary of the various manifold decomposition techniques. The column labeled "M" indicates what kind of manifold can be decomposed; the column labeled "How it is decomposed" indicates how, starting with a manifold, one can decompose it into smaller pieces; the column labeled "The pieces" indicates what the pieces can be; and the column labeled "How they are combined" indicates how the smaller pieces are combined to make the large manifold. Type of decomposition M How it is decomposed The pieces How they are combined Triangulation Depends on dimension. In dimension 3, a theorem by Edwin E. Moise gives that every 3 manifold has a unique triangulation, unique up to common subdivision. In dimension 4, not all manifolds are triangulable. For higher dimensions, general existence of triangulations is unknown. Simplices Glue together pairs of codimension one faces Jaco Shalen/Johannson torus decomposition Irreducible, orientable, compact 3 manifolds Cut along embedded tori Atoroidal or Seifert fibered 3 manifolds Union along their boundary, using the trivial homeomorphism Prime decomposition Essentially surfaces and 3 manifolds. The decomposition is unique when the manifold is orientable. Cut along embedded spheres; then union by the trivial homeomorphism along the resultant boundaries with disjoint balls. Prime manifolds Connected sum Heegaard splitting Closed, orientable 3 manifolds Two handlebodies of equal genus Union along the boundary by some homeomorphism Handle decomposition Any compact (smooth) n manifold (and the decomposition is never unique) Through Morse functions a handle is associated to each critical point. Balls (called handles) Union along a subset of the boundaries. Note that the handles must generally be added in a specific order. Haken hierarchy Any Haken manifold Cut along a sequence of incompressible surfaces 3 balls Disk decomposition Certain compact, orientable 3 manifolds Suture the manifold, then cut along special surfaces (condition on boundary curves and sutures ) 3 balls Open book decomposition Any closed orientable 3 manifold A link and a family of 2 manifolds that share a boundary with that link Trigenus Compact, closed 3 manifolds Surgeries Three orientable handlebodies Unions along subsurfaces on boundaries of handlebodies