Editing CW complex

{{Short description|Type of topological space}}
In [[mathematics]], and specifically in [[topology]], a '''CW complex''' (also '''cellular complex''' or '''cell complex''') is a [[topological space]] that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both [[topological manifold|manifolds]] and [[simplicial complex]]es and has particular significance for [[algebraic topology]].<ref>{{cite book|last=Hatcher|first=Allen|author-link=Allen Hatcher|title=Algebraic topology|publisher=[[Cambridge University Press]]|year=2002|isbn=0-521-79540-0}} This textbook defines CW complexes in the first chapter and uses them throughout; includes an appendix on the topology of CW complexes. A free electronic version is available on the [http://pi.math.cornell.edu/~hatcher/ author's homepage].</ref> It was initially introduced by [[J. H. C. Whitehead]] to meet the needs of [[homotopy theory]].<ref name=":2">{{cite journal|last=Whitehead|first=J. H. C.|author-link=J. H. C. Whitehead|year=1949a|title=Combinatorial homotopy. I.|url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-3.P1/Combinatorial-homotopy-I/bams/1183513543.pdf|journal=[[Bulletin of the American Mathematical Society]]|volume=55|issue=5|pages=213&ndash;245|doi=10.1090/S0002-9904-1949-09175-9|mr=0030759|doi-access=free}} (open access)</ref>
CW complexes have better [[category theory|categorical]] properties than [[simplicial complex]]es, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). 

The C in CW stands for "closure-finite", and the W for "weak" topology.<ref name=":2" />

==Definition==

=== CW complex ===
A '''CW complex''' is constructed by taking the union of a sequence of topological spaces <math display="block">\emptyset = X_{-1} \subset X_0 \subset X_1 \subset \cdots</math> such that each <math>X_k</math> is obtained from <math>X_{k-1}</math> by gluing copies of k-cells <math>(e^k_\alpha)_\alpha</math>, each homeomorphic to the open <math>k</math>-[[ball (mathematics)|ball]] <math>B^k</math>, to <math>X_{k-1}</math> by continuous gluing maps <math>g^k_\alpha: \partial e^k_\alpha \to X_{k-1}</math>. The maps are also called [[attaching map]]s. Thus as a set, <math>X_k = X_{k-1} \sqcup_{\alpha} e^k_\alpha</math>.

Each <math>X_k</math> is called the '''k-skeleton''' of the complex.

The topology of <math>X = \cup_{k} X_k</math> is '''weak topology''': a subset <math>U\subset X</math> is open [[iff]] <math>U\cap X_k</math> is open for each k-skeleton <math>X_k</math>.

In the language of category theory, the topology on <math>X</math> is the [[direct limit]] of the diagram <math display="block">X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots</math>The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:

{{Math theorem
| name = Theorem
| note = 
| math_statement = A [[Hausdorff space]] ''X'' is homeomorphic to a CW complex iff there exists a [[partition of a set|partition]] of ''X'' into "open cells" <math>e^k_\alpha</math>, each with a corresponding closure (or "closed cell") <math>\bar{e}^k_\alpha := cl_X(e^k_\alpha)</math> that satisfies:

* For each <math>e^k_\alpha</math>, there exists a [[Continuous function#Continuous functions between topological spaces|continuous surjection]] <math>g_\alpha^k: D^k \to \bar{e}^k_\alpha</math> from the <math>k</math>-dimensional closed ball such that
** The restriction to the open ball <math>g_\alpha^k: B^k\to e^k_\alpha </math> is a [[homeomorphism]].
** (closure-finiteness) The image of the boundary <math>g^k_\alpha(\partial D^k) </math> is covered by a finite number of closed cells, each having cell dimension less than k.
* (weak topology) A subset of ''X'' is [[Closed set|closed]] if and only if it meets each closed cell in a closed set.
}}

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 space|discrete topology]]).
* A 1-''dimensional CW complex'' is constructed by taking the [[Disjoint union (topology)|disjoint union]] of a 0-dimensional CW complex with one or more copies of the [[unit interval]]. For each copy, there is a map that "[[Gluing (topology)|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 (topology)|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 <math>k<n</math>) with one or more copies of the [[Ball (mathematics)|''n''-dimensional ball]]. For each copy, there is a map that "glues" its boundary (the <math>(n-1)</math>-dimensional [[N-sphere|sphere]]) to elements of the <math>k</math>-dimensional complex. The topology of the CW complex is the [[quotient topology]] 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 <math>\cup_k X_k</math> 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 [[Loop (graph theory)|loopless]] graph is represented by a regular 1-dimensional CW-complex. A [[graph embedding|closed 2-cell graph embedding]] on a [[surface]] is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every [[k-vertex-connected graph|2-connected graph]] is the 1-skeleton of a regular CW-complex on the [[3-sphere|3-dimensional sphere]].<ref>{{cite conference|title=The 3-Sphere Regular Cellulation Conjecture|first=Sergio|last=De Agostino|conference=International Workshop on Combinatorial Algorithms|year=2016 |url=https://twiki.di.uniroma1.it/pub/Users/SergioDeAgostino/DeAgostino.pdf}}</ref>

===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.<ref>{{cite book |last1=Davis |first1=James F. |title=Lecture Notes in Algebraic Topology |last2=Kirk |first2=Paul |date=2001 |publisher=American Mathematical Society |location=Providence, R.I.}}</ref><ref>{{Cite web |title=CW complex in nLab |url=https://ncatlab.org/nlab/show/CW+complex}}</ref><ref>{{Cite web |title=CW-complex - Encyclopedia of Mathematics |url=https://www.encyclopediaofmath.org/index.php/CW-complex}}</ref>

== Examples ==

=== 0-dimensional CW complexes ===
Every [[Discrete space|discrete topological space]] is a 0-dimensional CW complex.

=== 1-dimensional CW complexes ===
Some examples of 1-dimensional CW complexes are:<ref name=":1">Archived at [https://ghostarchive.org/varchive/youtube/20211212/HjiooyBH6es Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20201211210326/https://www.youtube.com/watch?v=HjiooyBH6es&gl=US&hl=en Wayback Machine]{{cbignore}}: {{Cite web |last=channel |first=Animated Math |date=2020 |title=1.3 Introduction to Algebraic Topology. Examples of CW Complexes. |url=https://www.youtube.com/watch?v=HjiooyBH6es&t=25s |website=Youtube}}{{cbignore}}</ref>

* '''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.''' Given a [[multigraph|graph]], a 1-dimensional CW complex can be constructed in which the 0-cells are the vertices and the 1-cells are the edges of the graph. The endpoints of each edge are identified with the incident vertices to it. This realization of a combinatorial graph as a topological space is sometimes called a '''topological graph'''.
**[[Trivalent graph|3-regular graph]]s can be considered as ''[[Generic property|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 [[discrete two-point space|two-point space]] to ''X'', <math>f : \{0,1\} \to X</math>.  This map can be perturbed to be disjoint from the 0-skeleton of ''X'' if and only if <math>f(0)</math> and <math>f(1)</math> are not 0-valence vertices of ''X''.
* The ''standard CW structure'' on the real numbers has as 0-skeleton the integers <math>\mathbb Z</math> and as 1-cells the intervals <math>\{ [n,n+1] : n \in \mathbb Z\}</math>. Similarly, the standard CW structure on <math>\mathbb R^n</math> has cubical cells that are products of the 0 and 1-cells from <math>\mathbb R</math>. This is the standard ''[[Integer lattice|cubic lattice]]'' cell structure on <math>\mathbb R^n</math>.

=== Finite-dimensional CW complexes ===
Some examples of finite-dimensional CW complexes are:<ref name=":1" />
* '''An [[n-sphere|''n''-dimensional sphere]]'''. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell <math>D^{n}</math> is attached by the constant mapping from its boundary <math>S^{n-1}</math> to the single 0-cell. An alternative cell decomposition has one (''n''-1)-dimensional sphere (the "[[equator]]") and two ''n''-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere").  Inductively, this gives <math>S^n</math> a CW decomposition with two cells in every dimension k such that <math>0 \leq k \leq n</math>.
* '''The ''n''-dimensional real [[projective space]].''' It admits a CW structure with one cell in each dimension.
* The terminology for a generic 2-dimensional CW complex is a '''shadow'''.<ref>{{cite book |last=Turaev |first=V. G. |title=Quantum invariants of knots and 3-manifolds |date=1994 |publisher=Walter de Gruyter & Co. |isbn=9783110435221 |series=De Gruyter Studies in Mathematics |volume=18 |location=Berlin}}</ref>
* A [[polyhedron]] is naturally a CW complex.
*[[Grassmannian]] manifolds admit a CW structure called '''Schubert cells'''.
*[[Differentiable manifold]]s, algebraic and projective [[algebraic variety|varieties]] have the [[homotopy type]] of CW complexes.
* The [[Alexandroff extension|one-point compactification]] of a cusped [[hyperbolic 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, such as [[SnapPea]].

=== Infinite-dimensional CW complexes ===

* The infinite dimensional sphere <math>S^\infty:=\mathrm{colim}_{n\to\infty}S^n</math>. It admits a CW-structure with 2 cells in each dimension which are assembled in a way such that the <math>n</math>-skeleton is precisely given by the <math>n</math>-sphere.
* The infinite dimensional projective spaces <math>\mathbb{RP}^\infty</math>, <math>\mathbb{CP}^\infty</math> and <math>\mathbb{HP}^\infty</math>. <math>\mathbb{RP}^\infty</math> has one cell in every dimension, <math>\mathbb{CP}^\infty</math>, has one cell in every even dimension and <math>\mathbb{HP}^\infty</math> has one cell in every dimension divisible by 4. The respective skeletons are then given by <math>\mathbb{RP}^n</math>, <math>\mathbb{CP}^n</math> (2n-skeleton) and <math>\mathbb{HP}^n</math> (4n-skeleton).

=== Non CW-complexes ===

* An infinite-dimensional [[Hilbert space]] is not a CW complex: it is a [[Baire space]] 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]] <math>\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subseteq \mathbb C</math> is homotopy equivalent to a CW complex (the point) but it does not admit a CW decomposition, since it is not [[Contractible space#Locally contractible spaces|locally contractible]].
* The [[Hawaiian earring]] has no CW decomposition, because it is not locally contractible at origin. It is also not homotopy equivalent to a CW complex, because it has no good open cover.

== Properties ==
* CW complexes are locally contractible.<ref>{{Cite book |last=Hatcher |first=Allen |title=Algebraic topology |publisher=[[Cambridge University Press]] |year=2002 |isbn=0-521-79540-0 |pages=522}} Proposition A.4</ref>
* If a space is [[homotopy equivalent]] to a CW complex, then it has a good open cover.<ref>{{Cite journal |last=Milnor |first=John |date=February 1959 |title=On Spaces Having the Homotopy Type of a CW-Complex |url=http://dx.doi.org/10.2307/1993204 |journal=Transactions of the American Mathematical Society |volume=90 |issue=2 |pages=272–280 |doi=10.2307/1993204 |jstor=1993204 |issn=0002-9947|url-access=subscription }}</ref> A good open cover is an open cover, such that every nonempty finite intersection is contractible.
* CW complexes are [[paracompact]]. Finite CW complexes are [[compact space|compact]]. A compact subspace of a CW complex is always contained in a finite subcomplex.<ref>[[Allen Hatcher|Hatcher, Allen]], ''Algebraic topology'', Cambridge University Press (2002). {{ISBN|0-521-79540-0}}. A free electronic version is available on the [http://pi.math.cornell.edu/~hatcher/ author's homepage]</ref><ref>[[Allen Hatcher|Hatcher, Allen]], ''Vector bundles and K-theory'', preliminary version available on the [http://pi.math.cornell.edu/~hatcher/ author's homepage]</ref>
* 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]] of a CW complex is also a CW complex.<ref>{{Cite book |last=Hatcher |first=Allen |title=Algebraic topology |publisher=[[Cambridge University Press]] |year=2002 |isbn=0-521-79540-0 |pages=529}} Exercise 1</ref>
* 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]]. The underlying set of ''X'' × ''Y'' is then the [[Cartesian product]] of ''X'' and ''Y'', as expected. In addition, the weak topology on this set often agrees with the more familiar [[product topology]] on ''X'' × ''Y'', for example if either ''X'' or ''Y'' is finite.  However, the weak topology can be [[comparison of topologies|finer]] than the product topology, for example if neither ''X'' nor ''Y'' is [[locally compact space|locally compact]].  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]]s agrees with the weak topology and therefore defines a CW complex.
* Let ''X'' and ''Y'' be CW complexes.  Then the [[function spaces]] Hom(''X'',''Y'') (with the [[compact-open topology]]) are ''not'' CW complexes in general. If ''X'' is finite then Hom(''X'',''Y'') is homotopy equivalent to a CW complex by a theorem of [[John Milnor]] (1959).<ref name="milnor">{{cite journal |last1=Milnor |first1=John |author-link=John Milnor |year=1959 |title=On spaces having the homotopy type of a CW-complex |journal=Trans. Amer. Math. Soc. |volume=90 |issue=2 |pages=272–280 |doi=10.1090/s0002-9947-1959-0100267-4 |jstor=1993204 |doi-access=free}}</ref>  Note that ''X'' and ''Y'' are [[compactly generated Hausdorff space]]s, so Hom(''X'',''Y'') is often taken with the [[compactly generated space|compactly generated]] variant of the compact-open topology; the above statements remain true.<ref>{{cite web |title=Compactly Generated Spaces |url=http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |access-date=2012-08-26 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303174529/http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |url-status=dead }}</ref>
* [[Cellular approximation theorem]]

==Homology and cohomology of CW complexes==

[[Singular homology]] and [[singular cohomology|cohomology]] of CW complexes is readily computable via [[cellular homology]].  Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as a [[homology theory]]. To compute an [[cohomology#Generalized cohomology theories|extraordinary (co)homology theory]] for a CW complex, the [[Atiyah–Hirzebruch spectral sequence]] is the analogue of cellular homology.

Some examples:

* For the sphere, <math>S^n,</math> take the cell decomposition with two cells: a single 0-cell and a single ''n''-cell. The cellular homology [[chain complex]] <math>C_*</math> and homology are given by:
::<math>C_k = \begin{cases} \Z & k \in \{0,n\} \\ 0 & k \notin \{0,n\} \end{cases} \quad H_k =  \begin{cases} \Z & k \in \{0,n\} \\ 0 & k \notin \{0,n\} \end{cases}</math> 
:since all the differentials are zero.

:Alternatively, if we use the equatorial decomposition with two cells in every dimension 
::<math>C_k = \begin{cases} \Z^2 & 0 \leqslant k \leqslant n \\ 0 & \text{otherwise} \end{cases}</math> 
:and the differentials are matrices of the form <math>\left ( \begin{smallmatrix} 1 & -1 \\ 1 & -1\end{smallmatrix} \right ).</math> This gives the same homology computation above, as the chain complex is exact at all terms except <math>C_0</math> and <math>C_n.</math>

* For <math>\mathbb{P}^n(\Complex)</math> we get similarly
::<math>H^k \left (\mathbb{P}^n(\Complex) \right ) = \begin{cases} \Z & 0\leqslant k\leqslant 2n, \text{ even}\\ 0  & \text{otherwise}\end{cases}</math>

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 (discrete mathematics)|graph]]. Now consider a maximal [[Tree (graph theory)|forest]] ''F'' in this graph.  Since it is a collection of trees, and trees are contractible, consider the space <math>X/{\sim}</math> where the equivalence relation is generated by <math>x \sim y</math> if they are contained in a common tree in the maximal forest ''F''.  The quotient map <math>X \to X/{\sim}</math> is a homotopy equivalence.  Moreover, <math>X/{\sim}</math> naturally inherits a CW structure, with cells corresponding to the cells of <math>X</math> that are not contained in ''F''. In particular, the 1-skeleton of <math>X/{\sim}</math> 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 <math>X^1</math> consists of a single point?  The answer is yes.  The first step is to observe that <math>X^1</math> and the attaching maps to construct <math>X^2</math> from <math>X^1</math> form a [[Presentation of a group|group presentation]]. The [[Tietze transformations|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 <math>X^1</math>.  If we let <math>\tilde X</math> be the corresponding CW complex <math>\tilde X = X \cup e^1 \cup e^2</math> then there is a homotopy equivalence <math>\tilde X \to X</math> 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 <math>\tilde X = X \cup e^2 \cup e^3</math> where the new ''3''-cell has an attaching map that consists of the new 2-cell and remainder mapping into <math>X^2</math>.  A similar slide gives a homotopy-equivalence <math>\tilde X \to X</math>.

If a CW complex ''X'' is [[N-connected space|''n''-connected]] one can find a homotopy-equivalent CW complex <math>\tilde X</math> whose ''n''-skeleton <math>X^n</math> consists of a single point.  The argument for <math>n \geq 2</math> is similar to the <math>n=1</math> case, only one replaces Tietze moves for the [[fundamental group]] presentation by [[elementary matrix]] operations for the presentation matrices for <math>H_n(X;\mathbb Z)</math> (using the presentation matrices coming from [[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]]s is actually used).<ref>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 {{SpringerEOM| title=CW-complex | id=CW-complex | oldid=15603 | first=D.O. | last=Baladze }}</ref> Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the [[representable functor]]s on the homotopy category have a simple characterisation (the [[Brown representability theorem]]).

See also: [[Milnor's theorem on Kan complexes]]

==See also==
*[[Abstract cell complex]]
*The notion of CW complex has an adaptation to [[differentiable manifold|smooth manifolds]] called a [[handle decomposition]], which is closely related to [[surgery theory]].

== References ==

===Notes===
{{Reflist}}

===General references ===
{{Refbegin<!--too few as yet for: |2-->}}
*{{cite book|first1=A. T.|last1=Lundell|first2=S.|last2=Weingram|title=The topology of CW complexes|publisher=[[Van Nostrand (publisher)|Van Nostrand]] University Series in Higher Mathematics|year=1970|isbn=0-442-04910-2}}
* {{cite book|first1= R.|last1= Brown|first2=P.J. | last2= Higgins| first3= R. | last3= Sivera|title= Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids| publisher= [[European Mathematical Society]] Tracts in Mathematics Vol 15| year=2011| isbn=978-3-03719-083-8 }} More details on the [http://groupoids.org.uk/nonab-a-t.html] first author's home page]
{{Refend}}

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