- 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 asolid 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)-dimensionalsubmanifold 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 theempty 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 ahandlebody ); 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

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