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A CW complex (also called cellular complex or cell complex) is a kind of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that is particularly important in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. It was introduced by J. H. C. Whitehead (open access) to meet the needs of
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
. This class of spaces is broader and has some better categorical properties than
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
es, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology.


Definition


CW complex

A CW complex is constructed by taking the union of a sequence of topological spaces\emptyset = X_ \subset X_0 \subset X_1 \subset \cdotssuch that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to D^k, to X_ by continuous gluing maps g^k_\alpha: \partial e^k_\alpha \to X_. The maps are also called attaching maps. Each X_k is called the k-skeleton of the complex. The topology of X = \cup_ X_k is weak topology: a subset U\subset X is open iff U\cap e^k_\alpha is open for each cell e^k_\alpha. In the language of category theory, the topology on X is the
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of the diagram X_ \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdotsThe name "CW" stands for "closure-finite weak topology", which is explained by the following theorem: This partition of ''X'' is also called a cellulation.


The construction, in words

The CW complex construction is a straightforward generalization of the following process: * A 0-''dimensional CW complex'' is just a set of zero or more discrete points (with the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
). * A 1-''dimensional CW complex'' is constructed by taking the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of a 0-dimensional CW complex with one or more copies of the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
. For each copy, there is a map that " glues" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the quotient space defined by these gluing maps. * In general, an ''n-dimensional CW complex'' is constructed by taking the disjoint union of a ''k''-dimensional CW complex (for some k) with one or more copies of the ''n''-dimensional ball. For each copy, there is a map that "glues" its boundary (the (n-1)-dimensional
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
) to elements of the k-dimensional complex. The topology of the CW complex is the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
defined by these gluing maps. * An ''infinite-dimensional CW complex'' can be constructed by repeating the above process countably many times. Since the topology of the union \cup_k X_k is indeterminate, one takes the direct limit topology, since the diagram is highly suggestive of a direct limit. This turns out to have great technical benefits.


Regular CW complexes

A regular CW complex is a CW complex whose gluing maps are homeomorphisms. Accordingly, the partition of ''X'' is also called a regular cellulation. A loopless graph is a regular 1-dimensional CW-complex. A closed 2-cell graph embedding on a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every 2-connected graph is the 1-skeleton of a regular CW-complex on the 3-dimensional sphere.


Relative CW complexes

Roughly speaking, a ''relative CW complex'' differs from a CW complex in that we allow it to have one extra building block that does not necessarily possess a cellular structure. This extra-block can be treated as a (-1)-dimensional cell in the former definition.


Examples


0-dimensional CW complexes

Every discrete topological space is a 0-dimensional CW complex.


1-dimensional CW complexes

Some examples of 1-dimensional CW complexes are:Archived a
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and th
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* An interval. It can be constructed from two points (''x'' and ''y''), and the 1-dimensional ball ''B'' (an interval), such that one endpoint of ''B'' is glued to ''x'' and the other is glued to ''y''. The two points ''x'' and ''y'' are the 0-cells; the interior of ''B'' is the 1-cell. Alternatively, it can be constructed just from a single interval, with no 0-cells. * A circle. It can be constructed from a single point ''x'' and the 1-dimensional ball ''B'', such that ''both'' endpoints of ''B'' are glued to ''x''. Alternatively, it can be constructed from two points ''x'' and ''y'' and two 1-dimensional balls ''A'' and ''B'', such that the endpoints of ''A'' are glued to ''x'' and ''y'', and the endpoints of ''B'' are glued to ''x'' and ''y'' too. * A
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
. It is a 1-dimensional CW complex in which the 0-cells are the vertices and the 1-cells are the edges. The endpoints of each edge are identified with the vertices adjacent to it. ** 3-regular graphs can be considered as ''
generic Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
'' 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 to ''X'', f : \ \to X. This map can be perturbed to be disjoint from the 0-skeleton of ''X'' if and only if f(0) and f(1) are not 0-valence vertices of ''X''. * The ''standard CW structure'' on the real numbers has as 0-skeleton the integers \mathbb Z and as 1-cells the intervals \. Similarly, the standard CW structure on \mathbb R^n has cubical cells that are products of the 0 and 1-cells from \mathbb R. This is the standard '' cubic lattice'' cell structure on \mathbb R^n.


Finite-dimensional CW complexes

Some examples of finite-dimensional CW complexes are: * An ''n''-dimensional sphere. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell D^ is attached by the constant mapping from its boundary S^ to the single 0-cell. An alternative cell decomposition has one (''n''-1)-dimensional sphere (the "
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
") and two ''n''-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere"). Inductively, this gives S^n a CW decomposition with two cells in every dimension k such that 0 \leq k \leq n. * The ''n''-dimensional real
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generall ...
. It admits a CW structure with one cell in each dimension. * The terminology for a generic 2-dimensional CW complex is a shadow. * A
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...
is naturally a CW complex. *
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
manifolds admit a CW structure called Schubert cells. *
Differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s, algebraic and projective varieties have the homotopy-type of CW complexes. * The
one-point compactification In the 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 after the Russian mathematician Pavel Ale ...
of a cusped
hyperbolic manifold In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, resp ...
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, such as
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 ...
.


Infinite-dimensional CW complexes


Non CW-complexes

* An infinite-dimensional
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is not a CW complex: it is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
and therefore cannot be written as a countable union of ''n''-skeletons, each of which being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces. * The hedgehog space \ \subseteq \mathbb C is homotopic to a CW complex (the point) but it does not admit a CW decomposition, since it is not locally contractible. * The
Hawaiian earring In mathematics, the Hawaiian earring \mathbb is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac,0\right) and radius \tfrac for n = 1, 2, 3, \ldots endowed with the subspace topology: ...
is not homotopic to a CW complex. It has no CW decomposition, because it is not locally contractible at origin. It is not homotopic to a CW complex, because it has no good open cover.


Properties

* CW complexes are locally contractible (Hatcher, prop. A.4). * If a space is homotopic to a CW complex, then it has a good open cover. A good open cover is an open cover, such that every nonempty finite intersection is contractible. * CW complexes are
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norma ...
. Finite CW complexes are
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. A compact subspace of a CW complex is always contained in a finite subcomplex. * CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups. * A
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of a CW complex is also a CW complex. * The product of two CW complexes can be made into a CW complex. Specifically, if ''X'' and ''Y'' are CW complexes, then one can form a CW complex ''X'' × ''Y'' in which each cell is a product of a cell in ''X'' and a cell in ''Y'', endowed with the
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
. The underlying set of ''X'' × ''Y'' is then the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
of ''X'' and ''Y'', as expected. In addition, the weak topology on this set often agrees with the more familiar
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. This topology differs from another, perhaps more natural-seemi ...
on ''X'' × ''Y'', for example if either ''X'' or ''Y'' is finite. However, the weak topology can be finer than the product topology, for example if neither ''X'' nor ''Y'' is
locally compact 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. More precisely, it is a topological space in which ...
. In this unfavorable case, the product ''X'' × ''Y'' in the product topology is ''not'' a CW complex. On the other hand, the product of ''X'' and ''Y'' in the category of
compactly generated space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subsp ...
s agrees with the weak topology and therefore defines a CW complex. * Let ''X'' and ''Y'' be CW complexes. Then the
function spaces In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vec ...
Hom(''X'',''Y'') (with the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory an ...
) are ''not'' CW complexes in general. If ''X'' is finite then Hom(''X'',''Y'') is
homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
to a CW complex by a theorem of
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
(1959). Note that ''X'' and ''Y'' are
compactly generated Hausdorff space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subspa ...
s, so Hom(''X'',''Y'') is often taken with the compactly generated variant of the compact-open topology; the above statements remain true.


Homology and cohomology of CW complexes

Singular homology and cohomology of CW complexes is readily computable via
cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
. Moreover, in the category of CW complexes and cellular maps,
cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
can be interpreted as a
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...
. To compute an extraordinary (co)homology theory for a CW complex, the
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, i ...
is the analogue of
cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
. Some examples: * For the sphere, S^n, take the cell decomposition with two cells: a single 0-cell and a single ''n''-cell. The cellular homology chain complex C_* and homology are given by: ::C_k = \begin \Z & k \in \ \\ 0 & k \notin \ \end \quad H_k = \begin \Z & k \in \ \\ 0 & k \notin \ \end :since all the differentials are zero. :Alternatively, if we use the equatorial decomposition with two cells in every dimension ::C_k = \begin \Z^2 & 0 \leqslant k \leqslant n \\ 0 & \text \end :and the differentials are matrices of the form \left ( \begin 1 & -1 \\ 1 & -1\end \right ). This gives the same homology computation above, as the chain complex is exact at all terms except C_0 and C_n. * For \mathbb^n(\Complex) we get similarly ::H^k \left (\mathbb^n(\Complex) \right ) = \begin \Z & 0\leqslant k\leqslant 2n, \text\\ 0 & \text\end 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 that has a ''simpler'' CW decomposition. Consider, for example, an arbitrary CW complex. Its 1-skeleton can be fairly complicated, being an arbitrary
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
. Now consider a maximal
forest A forest is an area of land dominated by trees. Hundreds of definitions of forest are used throughout the world, incorporating factors such as tree density, tree height, land use, legal standing, and ecological function. The United Nations' ...
''F'' in this graph. Since it is a collection of trees, and trees are contractible, consider the space X/ where the equivalence relation is generated by x \sim y if they are contained in a common tree in the maximal forest ''F''. The quotient map X \to X/ is a homotopy equivalence. Moreover, X/ naturally inherits a CW structure, with cells corresponding to the cells of X that are not contained in ''F''. In particular, the 1-skeleton of X/ 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 X^1 consists of a single point? The answer is yes. The first step is to observe that X^1 and the attaching maps to construct X^2 from X^1 form a
group presentation In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises 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 ...
. The Tietze theorem 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 X^1. If we let \tilde X be the corresponding CW complex \tilde X = X \cup e^1 \cup e^2 then there is a homotopy equivalence \tilde X \to X 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 \tilde X = X \cup e^2 \cup e^3 where the new ''3''-cell has an attaching map that consists of the new 2-cell and remainder mapping into X^2. A similar slide gives a homotopy-equivalence \tilde X \to X. If a CW complex ''X'' is ''n''-connected one can find a homotopy-equivalent CW complex \tilde X whose ''n''-skeleton X^n consists of a single point. The argument for n \geq 2 is similar to the n=1 case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for H_n(X;\mathbb Z) (using the presentation matrices coming from
cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
. 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 In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed b ...
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 In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains ...
s is actually used).For example, the opinion "The class of CW complexes (or the class of spaces of the same homotopy type as a CW complex) is the most suitable class of topological spaces in relation to homotopy theory" appears in Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets an ...
s on the homotopy category have a simple characterisation (the Brown representability theorem).


See also

* Abstract cell complex *The notion of CW complex has an adaptation to
smooth manifolds In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
called a
handle decomposition In mathematics, a handle decomposition of an ''m''-manifold ''M'' is a union \emptyset = M_ \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_ \subset M_m = M where each M_i is obtained from M_ by the attaching of i-handles. A handle dec ...
, which is closely related to
surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while A ...
.


References


Notes


General references

* * More details on th

first author's home page] {{DEFAULTSORT:Cw Complex Algebraic topology Homotopy theory Topological spaces