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 decompositionMHow it is decomposedThe piecesHow they are combined
TriangulationDepends 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.simplicesGlue together pairs of codimension-one faces
Jaco-Shalen/Johannson torus decompositionIrreducible, orientable, compact 3-manifoldsCut along embedded toriAtoroidal or Seifert-fibered 3-manifoldsUnion along their boundary, using the trivial homeomorphism
Prime decompositionEssentially 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 manifoldsConnected sum
Heegaard splittingclosed, orientable 3-manifoldsTwo handlebodies of equal genusUnion along the boundary by some homeomorphism
Handle decompositionAny compact n-manifold Through Morse functions a handle is associated to each critical point.Balls Union along a subset of the boundaries. Note that the handles must generally be added in a specific order.
Haken hierarchyAny Haken manifoldCut along a sequence of incompressible surfaces3-balls
Disk decompositionCertain compact, orientable 3-manifoldsSuture the manifold, then cut along special surfaces 3-balls
Open book decompositionAny closed orientable 3-manifolda link and a family of 2-manifolds that share a boundary with that link
Trigenuscompact, closed 3-manifoldsSurgeriesthree orientable handlebodiesUnions along subsurfaces on boundaries of handlebodies