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2025-06-19T01:26:11Z

See also: ==References== {{reflist}}

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{{Unreferenced|date=December 2009}}
In [[topology]], a branch of [[mathematics]], a [[manifold]] ''M'' may be decomposed or split by writing ''M'' as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form ''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-manifold]]s 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.

{{Expand list|date=August 2008}}

{| class="wikitable"
|-
! Type of decomposition
! ''M''
! How it is decomposed
! The pieces
! How they are combined
|-
! [[Triangulation (topology)|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.
|
| [[Simplex|Simplices]]
| Glue together pairs of codimension-one faces
|-
! [[JSJ decomposition|Jaco-Shalen/Johannson torus decomposition]]
| [[Irreducible (mathematics)|Irreducible]], [[orientable]], [[compact space|compact]] [[3-manifold]]s
| Cut along embedded [[Torus|tori]]
| [[Atoroidal]] or [[Seifert-fibered space|Seifert-fibered]] 3-manifolds
| [[Adjunction space|Union along their boundary, using the trivial homeomorphism]]
|-
! [[Prime decomposition (3-manifold)|Prime decomposition]]
| Essentially [[Surface (topology)|surface]]s and [[3-manifold]]s. The decomposition is unique when the manifold is orientable.
| Cut along embedded [[Sphere#Topology|spheres]]; then [[Adjunction space|union by the trivial homeomorphism along the resultant boundaries]] with disjoint [[Ball#Topology|balls]].
| [[Prime manifold]]s
| [[Connected sum]]
|-
! [[Heegaard splitting]]
| [[Closed manifold|Closed]], [[orientable]] [[3-manifold]]s
|
| Two [[Handlebody|handlebodies]] of equal genus
| [[Adjunction space|Union along the boundary by some homeomorphism]]
|-
! [[Handle decomposition]]
| Any compact ([[smooth manifold|smooth]]) [[n-manifold]] (and the decomposition is never unique)
| Through [[Morse function]]s a handle is associated to each [[critical point (mathematics)|critical point]].
| [[Ball (mathematics)#Topology|Balls]] (called [[Handle (mathematics)|handles]])
| [[Adjunction space|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
| [[Ball (mathematics)#Topology|3-balls]]
|
|-
! Disk decomposition
| Certain [[Compact space|compact]], [[orientable]] [[3-manifold]]s
| [[Sutured manifold|Suture]] the manifold, then cut along special surfaces (condition on boundary curves and sutures...)
| [[Ball (mathematics)#Topology|3-balls]]
|
|-
! [[Open book decomposition]]
| Any [[Closed manifold|closed]] [[orientable]] [[3-manifold]]
|
| A [[Link (knot theory)|link]] and a family of [[Surface (topology)|2-manifold]]s that share a [[Manifold|boundary]] with that link
|
|-
! [[Trigenus]]
| [[Compact space|Compact]], [[closed manifold|closed]] 3-manifolds
| [[Surgery theory|Surgeries]]
| Three orientable handlebodies
| Unions along subsurfaces on boundaries of handlebodies
|}

==See also==
* [[Surgery theory]]

==References==
{{reflist}}

[[Category:Geometric topology]]

Manifold decomposition - Revision history

imported>Bearian: /* See also */==References== {{reflist}}