- 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

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Handle (mathematics)**— In topology, a branch of mathematics, a handle is just a topological ball; it is called a handle because of the context in which it is discussed, of which there are two: handle decompositions and handlebodies.A handle is a subset of a manifold… … Wikipedia**Handle decompositions of 3-manifolds**— allows you to simplify the original 3 manifold into pieces which are easier to study. An important method is decompose into handlebodies is the Heegaard splitting, which give us a decomposition in two handlebodies of equal genus … Wikipedia**Manifold decomposition**— In topology, a branch of mathematics, a manifold M may be decomposed or split by 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… … Wikipedia**Open book decomposition**— In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3 manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a… … Wikipedia**Singular value decomposition**— Visualization of the SVD of a 2 dimensional, real shearing matrix M. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the action of M, which distorts the disk to an ellipse. The SVD decomposes M into… … Wikipedia**Kirby calculus**— In mathematics, the Kirby calculus in geometric topology is a method for modifying framed links in the 3 sphere using a finite set of moves, the Kirby moves. It is named for Robion Kirby. Using four dimensional Cerf theory, he proved that if M… … Wikipedia**CW complex**— In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still… … Wikipedia**List of mathematics articles (H)**— NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… … Wikipedia**Mazur manifold**— In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4 dimensional manifold which is not diffeomorphic to the standard 4 ball. The boundary of a Mazur manifold is necessarily a homology 3 sphere.… … Wikipedia**Cobordism**— A cobordism (W;M,N). In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint… … Wikipedia