Handle decomposition
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In mathematics, a handle decomposition of an n-manifold M is a representation of that manifold as an exhaustion
where each Mi is obtained from Mi − 1 by attaching a ni-handle. Handle decompositions are never unique.
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[edit] Preliminaries
A handle is a ball attached to a manifold along part or all of the ball's boundary.
For example, starting with a three-dimensional ball B, one can attach another three-ball D to it as follows: identify two disjoint two-dimensional balls in the boundary of D with two disjoint two-balls in the boundary of B, and form the adjunction space. The result is actually a solid torus.
In that example, a three-dimensional one-handle was attached along the product of a 0-sphere and a 2-ball. In general, an n-dimensional k-handle is attached to an n-manifold along the product of a (k − 1)-sphere and an (n − k)-ball, forming a new manifold. Here k is called the index of the handle.
Therefore, a k-handle H is topologically an n-ball but geometrically it is the product of two balls: a k-dimensional core K, whose boundary is the gluing sphere; and an (n − k)-dimensional co-core C, whose boundary is the waist sphere.
For instance, a three dimensional 1-handle is the product of a segment and a disk.
The boundary of the handle
is
The boundary is broken up into two parts, the gluing tube
and the "waist tube"
For instance, the boundary of the previous 3-handle consists of a "gluing tube" which is a disjoint union of two disks, and a "waist tube" which is a cylinder.
[edit] Addition of handles
Adding an n-handle to an n-manifold means attaching the gluing tube of the handle to the boundary of the manifold. In mathematical terms, one says that the gluing tube is identified with a portion of the boundary of the manifold. More generally, the gluing tube can be identified with an appropriate (n-1)-dimensional submanifold of a handlebody.
A handle whose core is a point has no "gluing tube" and so can be "attached" to any handlebody, resulting in the addition of one disconnected component.
As an example, it is possible to view a 3-sphere as a 3-ball (0-handle attached to the empty set) with a 3-handle attached along the entire 2-sphere boundary.
[edit] Morse theoretic viewpoint
Smooth handle decompositions correspond to Morse functions on the smooth manifold. Each handle corresponds to a critical point of the Morse function and the index of the critical point corresponds to a handle of that index being attached.
[edit] Connection to Heegaard splittings
A closed 3-manifold admits a Heegaard splitting. This splitting can be thought of as being obtained by a specific handle decomposition where we add handles in order of increasing index. In other words we start with all 0-handles; add all 1-handles (getting a handlebody); add all 2-handles; and then add all 3-handles. The 2-handles and 3-handles form the other handlebody of the splitting.
For a given pair of handles of different indices, it may be possible to switch the order of gluing. By doing this we obtain a generalized Heegaard splitting.
[edit] Connection to surgery
Attaching a handle to a manifold with boundary produces surgery on its boundary. For instance, in the example above, adding a 1-handle to a 3-dimensional manifold replaces a pair of disks with a cylinder. To perform surgery on a link L embedded in a closed 3-manifold M, glue a 2-handle to the tubular neighbourhood of the link so that the 1-sphere glues to the curve determined by the framing, and plug the resulting 2-sphere boundary component which a 3-ball.
