Manifold decomposition
# Manifold decomposition

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.

| 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 |
