Handle decomposition - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a handle decomposition of an m-manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

M is a union

where each

is obtained from

by the attaching of

-handles. A handle decomposition is to a manifold what a CW-decomposition

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 retains a combinatorial naturethat allows for...

is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory

Morse theory

In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...

. The modification of handle structures is closely linked to Cerf theory

Cerf theory

In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functionsf:M \to \mathbb R...

Motivation

Consider the standard CW-decomposition of the n-sphere, with one zero cell and a single n-cell. From the point of view of smooth manifolds, this is a degenerate decomposition of the sphere, as there is no natural way to see the smooth structure of

from the eyes of this decomposition -- in particular the smooth structure near the 0-cell depends on the behavior of the characteristic map

in a neighbourhood of

.

The problem with CW-decompositions is that the attaching maps for cells do not live in the world of smooth maps between manifolds. The germinal insight to correct this defect is the tubular neighbourhood theorem

Tubular neighborhood

In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve...

. Given a point p in a manifold M, its closed tubular neighbourhood

is diffeomorphic to

, thus we have decomposed M into the disjoint union of

and

glued along their common boundary. The vital issue here is that the gluing map is a diffeomorphism. Similarly, take a smooth embedded arc in

, its tubular neighbourhood is diffeomorphic to

. This allows us to write

as the union of three manifolds, glued along parts of their boundaries: 1)

and 3) the complement of the open tubular neighbourhood of the arc in

. Notice all the gluing maps are smooth maps—in particular when we glue

the equivalence relation is generated by the embedding of

, which is smooth by the tubular neighbourhood theorem

Tubular neighborhood

.

Handle decompositions are an invention of Stephen Smale. In his original formulation, the process of attaching a j-handle to an m-manifold M assumes that one has a smooth embedding of

. Let

. The manifold

(in words, M union a j-handle along f ) refers to the disjoint union of

and

with the identification of

with its image in

, i.e.:

where the equivalence relation

Quotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

is generated by

for all

.

One says a manifold N is obtained from M by attaching j-handles, if M union finitely many j-handles is diffeomorphic to N. The definition of a handle decomposition is then as in the introduction. Thus, a manifold has a handle decomposition with only 0-handles if it is diffeomorphic to a disjoint union of balls. A connected manifold containing handles of only two types (ie: 0-handles and j-handles for some fixed j) is called a handlebody

Handlebody

In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds...

Terminology

When forming M union a j-handle

is known as the attaching sphere.

is sometimes called the framing of the attaching sphere, since it gives trivialization

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

of its normal bundle

Normal bundle

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding .-Riemannian manifold:...

is the belt sphere of the handle

.

A manifold obtained by attaching g k-handles to the disc

is an (m,k)-handlebody of genus g .

Cobordism presentations

A handle presentation of a cobordism consists of a cobordism W where

and an ascending union

where M is m-dimensional, W is m+1-dimensional,

is diffeomorphic to

and

is obtained from

by the attachment of i-handles. Whereas handle decompositions are the analogue for manifolds what cell decompositions are to topological spaces, handle presentations of cobordisms are to manifolds with boundary what relative cell decompositions are for pairs of spaces.

Morse theoretic viewpoint

Given a Morse function

Morse theory

on a compact boundaryless manifold M, such that the critical points

Critical point (mathematics)

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...

of f satisfy

, and provided

,

then for all j,

is diffeomorphic to

where I(j) is the index of the critical point

. The index I(j) refers to the dimension of the maximal subspace of the tangent space

where the Hessian

Hessian matrix

In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...

is negative definite.

Provided the indices satisfy

this is a handle decomposition of M, moreover, every manifold has such Morse functions, so they have handle decompositions. Similarly, given a cobordism

with

and a function

which is Morse on the interior and constant on the boundary and satisfying the increasing index property, there is an induced handle presentation of the cobordism W.

When f is a Morse function on M, -f is also a Morse function. The corresponding handle decomposition / presentation is called the dual decomposition.

Some major theorems and observations

A Heegaard splitting
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.-Definitions:...

of a closed, orientable 3-manifold is a decomposition of a 3-manifold into the union of two (3,1)-handlebodies along their common boundary, called the Heegaard splitting surface. Heegaard splittings arise for 3-manifolds in several natural ways: given a handle decomposition of a 3-manifold, the union of the 0 and 1-handles is a (3,1)-handlebody, and the union of the 3 and 2-handles is also a (3,1)-handlebody (from the point of view of the dual decomposition), thus a Heegaard splitting. If the 3-manifold has a triangulation
Triangulation (topology)
In mathematics, topology generalizes the notion of triangulation in a natural way as follows:A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h:K\to X....

T, there is an induced Heegaard splitting where the first (3,1)-handlebody is a regular neighbourhood of the 1-skeleton , and the other (3,1)-handlebody is a regular neighbourhood of the dual 1-skeleton
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...

.

When attaching two handles in succession , it is possible to switch the order of attachment, provided , i.e.: this manifold is diffeomorphic to a manifold of the form for suitable attaching maps.

The boundary of is diffeomorphic to surgered along the framed sphere . This is the primary link between surgery
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...

, handles and Morse functions.

As a consequence, an m-manifold M is the boundary of an m+1-manifold W if and only if M can be obtained from by surgery on a collection of framed links in . For example, it's known that every 3-manifold bounds a 4-manifold (similarly oriented and spin 3-manifolds bound oriented and spin 4-manifolds respectively) due to René Thom's work on cobordism
Cobordism
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 union is the boundary of a manifold one dimension higher. The name comes...

. Thus every 3-manifold can be obtained via surgery on framed links in the 3-sphere. In the oriented case, it's conventional to reduce this framed link to a framed embedding of a disjoint union of circles.

The H-cobordism Theorem
H-cobordism
A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...

is proven by simplifying handle decompositions of smooth manifolds.

Motivation

Terminology

Cobordism presentations

Morse theoretic viewpoint

Some major theorems and observations

See also

General references