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.
For explanations of cut along and union along the boundary, see WHERE???.
| 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 | 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 |