Manifold decomposition
In topology, a branch of mathematics, the decomposition of a manifold M is 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.
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 | For more information |
|---|---|---|---|---|---|
| Jaco-Shalen/Johansson 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 | See JSJ decomposition |
| Prime decomposition | Any manifold. (But only for 3-manifolds is there a theorem that there will be a unique prime decomposition.) | Cut along embedded spheres; then union by the trivial homeomorphism along the resultant boundaries with disjoint balls. | Prime manifolds | Connected sum | See Connected sum |
| Heegaard splitting | Closed, orientable 3-manifolds | (?) | Two handlebodies of equal genus | Union along the boundary by some homeomorphism | See Heegaard splitting |
| Handle decomposition | Any manifold (?) | (?) | Balls (called handles) | Union along a subset of the boundaries. Note that the handles must generally be added in a specific order. | See Handle decomposition |