Handle decomposition

In mathematics, a handle decomposition of an "n"-manifold "M" is a representation of that manifold as an exhaustion

:$M\_0\; subset\; M\_1\; subset\; dots\; subset\; M$

where each $M\_i$ is obtained from $M\_\{i-1\}$by attaching a $n\_i$-handle. Handle decompositions are never unique.

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

:$H\; =\; K\; imes\; C$

is

:$partial\{H\}\; =\; (partial\{K\}\; imes\; C)\; cup\; (K\; imes\; partial\{C\})\; !$

The boundary is broken up into two parts, the "gluing tube"

:$partial\{K\}\; imes\; C\; !$

and the "waist tube"

:$K\; imes\; partial\{C\}\; !$

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.

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.

Morse theoretic viewpoint

Smooth handle decompositions correspond to Morse functions on the smooth manifold. Each handle corresponds to acritical point of the Morse function and the index of the critical point corresponds to a handle of that index being attached.

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.

Connection to surgery

Attaching a handle to a manifold produces a 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. Given a framed link "L" in the 3-sphere, the result of performing an integral Dehn surgery appears as the boundary of the 4-ball with 2-handles attached via "L".

ee also

*Cobordism theory
*Handle (mathematics)
*Kirby calculus
*Manifold decomposition
*Morse theory
*CW complex
*Casson handle