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-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.
{| class="wikitable"
|-
! 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
|}
See also
- Surgery theory