Manifold decomposition

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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 decompositionMHow it is decomposedThe piecesHow 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 decompositionCertain 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 handlebodiesUnions along subsurfaces on boundaries of handlebodies

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