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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|