Handle Decomposition
- A closed, smooth 4-manifold is usually described by a handle decomposition.
- A 0-handle is just a ball, and the attaching map is disjoint union.
- A 1-handle is attached along two disjoint 3-balls.
- A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between handle decompositions on 4-manifolds, and knot theory in 3-manifolds.
- A pair of handles with index differing by 1, whose cores link each other in a sufficiently simple way can be cancelled without changing the underlying manifold. Similarly, such a cancelling pair can be created.
Two different smooth handlebody decompositions of a smooth 4-manifold are related by a finite sequence of isotopies of the attaching maps, and the creation/cancellation of handle pairs.
Read more about this topic: Kirby Calculus
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