CW complex - AbsoluteAstronomy.com

In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a CW complex is a type of topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

introduced by J. H. C. Whitehead

J. H. C. Whitehead

John Henry Constantine Whitehead FRS , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....

to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

properties than simplicial complex

Simplicial complex

In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...

es, but still retains a combinatorial nature
that allows for computation.

Formulation

Roughly speaking, a CW-complex is made of basic building blocks called cells. The precise definition prescribes how the cells may be topologically glued together. The C stands for "closure-finite", and the W for "weak topology".

An n-dimensional closed cell is a topological space that is homeomorphic to an n-dimensional closed ball. For example, a simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

is a closed cell, and more generally, a convex polytope

Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

is a closed cell. An n-dimensional open cell is a topological space that is homeomorphic to the open ball. A 0-dimensional open (and closed) cell is a singleton space.

A CW complex is a Hausdorff space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

X together with a partition

Partition of a set

In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...

of X into open cells (of varying dimension) that satisfies two additional properties:

For each n-dimensional open cell C in the set X, there exists a continuous map f from the n-dimensional closed ball to X such that
- the restriction of f from the interior of the n-dimensional ball is a homeomorphism
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
  onto the cell C, and
- the image of the boundary
  Boundary (topology)
  In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
  
  of the open ball, that is, under a homeomorphism
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
  , the boundary
  Boundary (topology)
  In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
  
  of the open cell C, intersects only finitely many other cells.
A subset of X is closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

if and only if it meets the closure of each cell in a closed set.

Inductive definition of CW-complexes

If the largest dimension of any of the cells is n, then the CW complex is said to have dimension n. If there is no bound to the cell dimensions then it is said to be infinite-dimensional. The n-skeleton

N-skeleton

In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions m ≤ n...

of a CW complex is the union of the cells whose dimension is at most n. If the union of a set of cells is closed, then this union is itself a CW complex, called a subcomplex. Thus the n-skeleton is the largest subcomplex of dimension n or less.

A CW complex is often constructed by defining its skeleta inductively. Begin by taking the 0-skeleton to be a discrete space

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...

. Next, attach 1-cells to the 0-skeleton. Here, the 1-cells are attached to points of the 0-skeleton via some continuous map from unit 0-sphere, that is,

. Define the 1-skeleton to be the identification space obtained from the union of the 0-skeleton, 1-cells, and the identification of points of boundary of 1-cells by assigning an identification mapping from the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

of the 1-cells into the 1-cells. In general, given the n-1-skeleton and a collection of (abstract) closed n-cells, as above, the n-cells are attached to the n-1-skeleton by some continuous mapping from

, and making an identification (equivalence relation) by specifying maps from the boundary of each n-cell into the n-1-skeleton. The n-skeleton is the identification space obtained from the union of the n-1-skeleton and the closed n-cells by identifying each point in the boundary of an n-cell with its image.

Up to isomorphism every n-dimensional complex can be obtained from its n-1 skeleton in this sense, and thus every finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensional complexes, with the understanding that the result of the infinite process is the direct limit

Direct limit

In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...

of the skeleta: a set is closed in X if and only if it meets each skeleton in a closed set.

Examples

The space has the homotopy-type of a CW-complex (it is contractible) but it does not admit a CW-decomposition, since it is not locally contractible.

The Hawaiian earring
Hawaiian earring
In mathematics, the Hawaiian earring H is the topological space defined by the union of circles in the Euclidean plane R2 with center and radius 1/n for n = 1, 2, 3, ......

is an example of a topological space that does not have the homotopy-type of a CW-complex.

The standard CW-structure on the real numbers has 0-skeleton the integers and as 1-cells the intervals . Similarly, the standard CW-structure on has cubical cells that are products of the 0 and 1-cells from . This is the standard cubical lattice cell structure on .

A polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

is naturally a CW-complex.

A graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

is a 1-dimensional CW-complex. Trivalent graphs can be considered as generic 1-dimensional CW-complexes. Specifically, if X is a 1-dimensional CW-complex, the attaching map for a 1-cell is a map from a two-point space
Discrete two-point space
In topology, a branch of mathematics, a discrete two-point space is the simplest example of a totally disconnected discrete space. The points can be denoted by the symbols 0 and 1....

to X, . This map can be perturbed to be disjoint from the 0-skeleton of X if and only if and are not 0-valence vertices of X.

The terminology for a generic 2-dimensional CW-complex is a shadow.

The n-dimensional sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

admits a CW-structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from to 0-cell. There is a popular alternative cell decomposition, since the equatorial inclusion has complement two balls: the upper and lower hemi-spheres. Inductively, this gives a CW-decomposition with two cells in every dimension k such that .

The n-dimensional real projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

admits a CW-structure with one cell in each dimension.

Grassmannian
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

manifolds admit a CW-structure called Schubert cells.

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....

s, algebraic and projective varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

have the homotopy-type of CW-complexes.

The one-point compactification
Alexandroff extension
In mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named for the Russian mathematician Pavel Alexandrov....

of a cusped hyperbolic manifold
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...

has a canonical CW-decomposition with only one 0-cell (the compactification point) called the Epstein-Penner Decomposition. Such cell decompositions are frequently called ideal polyhedral decompositions and are used in popular computer software, SnapPea
SnapPea
SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version as part of his doctoral thesis, supervised by William Thurston. The latest version is 3.0d3...

.

Homology and cohomology of CW-complexes

Singular homology and cohomology

Singular homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....

of CW-complexes is readily computable via cellular homology

Cellular homology

In mathematics, cellular homology in algebraic topology is a homology theory for CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.- Definition :...

. Moreover, in the category of CW-complexes and cellular maps, cellular homology

Cellular homology

can be interpreted as a homology theory

Homology theory

In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...

. To compute an extraordinary (co)homology theory for a CW-complex, the Atiyah-Hirzebruch spectral sequence is the analogue of cellular homology

Cellular homology

.

Some examples:

For the sphere , take the cell decomposition with two cells: a single 0-cell and a single n-cell. The cellular homology chain complex and homology are given by:

since all the differentials are zero.

Alternatively, if we use the equatorial decomposition with two cells in every dimension

and the differentials are matrices of the form

. This gives the same homology computation above, as the chain complex is exact at all terms
except

and

For we get similarly

Both of the above examples are particularly simple because the homology is determined by the number of cells—i.e.: the cellular attaching maps have no role in these computations. This is a very special phenomenon and is not indicative of the general case.

Modification of CW-structures

There is a technique, developed by Whitehead, for replacing a CW-complex with a homotopy-equivalent CW-complex which has a simpler CW-decomposition.

Consider, for example, an arbitrary CW-complex. Its 1-skeleton can be fairly complicated, being an arbitrary graph. Now consider a maximal forest F in this graph. Since it is a collection of trees, and trees are contractible, consider the space

where the equivalence relation is generated by

if they are contained in a common tree in the maximal forest F. The quotient map

is a homotopy equivalence. Moreover,

naturally inherits a CW-structure, with cells corresponding to the cells of

which are not contained in F. In particular, the 1-skeleton of

is a disjoint union of wedges of circles.

Another way of stating the above is that a connected CW-complex can be replaced by a homotopy-equivalent CW-complex whose 0-skeleton consists of a single point.

Consider climbing up the connectivity ladder—assume X is a simply-connected CW-complex whose 0-skeleton consists of a point. Can we, through suitable modifications, replace X by a homotopy-equivalent CW-complex where

consists of a single point? The answer is yes. The first step is to observe that

and the attaching maps to construct

from

form a group presentation

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...

. The Tietze theorem

Tietze transformations

In group theory, Tietze transformations are used to transform a given presentation of a group into another, often simpler presentation of the same group...

for group presentations states that there is a sequence of moves we can perform to reduce this group presentation to the trivial presentation of the trivial group. There are two Tietze moves:

1) Adding/removing a generator. Adding a generator, from the perspective of the CW-decomposition consists of adding a 1-cell and a 2-cell whose attaching map consists of the new 1-cell and the remainder of the attaching map is in

. If we let

be the corresponding CW-complex

then there is a homotopy-equivalence

given by sliding the new 2-cell into X.

2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing X by

where the new 3-cell has an attaching map that consists of the new 2-cell and remainder mapping into

. A similar slide gives a homotopy-equivalence

If a CW-complex X is n-connected one can find a homotopy-equivalent CW-complex

whose n-skeleton

consists of a single point. The argument for

is similar to the

case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for

(using the presentation matrices coming from cellular homology

Cellular homology

. i.e.: one can similarly realize elementary matrix operations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.

'The' homotopy category

The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category (for technical reasons the version for pointed space

Pointed space

In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...

s is actually used). Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the representable functor

Representable functor

In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...

s on the homotopy category have a simple characterisation (the Brown representability theorem).

Properties

CW-complexes are locally contractible.

CW-complexes satisfy the Whitehead theorem
Whitehead theorem
In homotopy theory , the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected and have the homotopy-type of CW complexes. This result was proved by J....

-- ie: a map between CW-complexes is a homotopy-equivalence if and only if it induces an isomorphism on all homotopy groups.

The product of two CW-complexes is a CW-complex. The weak topology on this product X×Y is the same as the more familiar product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...

on most spaces of interest, but can be finer
Comparison of topologies
In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set...

if X×Y has uncountably many cells and neither X nor Y is locally compact
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...

.

The function spaces Hom(X,Y) are not CW-complexes in general but are homotopy equivalent to CW-complexes by a theorem of John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...

(1959). Actual function spaces occur in the somewhat larger category of compactly generated Hausdorff spaces.

A covering space of a CW complex is also a CW complex.

CW-complexes are paracompact

Formulation

Inductive definition of CW-complexes

Examples

Homology and cohomology of CW-complexes

Modification of CW-structures

'The' homotopy category

Properties

See also

General references