CW complex

<h2 id="definition">Definition</h2>
<h3>CW complex</h3>
A CW complex is constructed by taking the union of a sequence of topological spaces 
 
 
 
 ∅
 =
 
 X
 
 −
 1
 
 
 ⊂
 
 X
 
 0
 
 
 ⊂
 
 X
 
 1
 
 
 ⊂
 ⋯
 
 
 {\displaystyle \emptyset =X_{-1}\subset X_{0}\subset X_{1}\subset \cdots }
 
 such that each 
 
 
 
 
 X
 
 k
 
 
 
 
 {\displaystyle X_{k}}
 
 is obtained from 
 
 
 
 
 X
 
 k
 −
 1
 
 
 
 
 {\displaystyle X_{k-1}}
 
 by gluing copies of k-cells 
 
 
 
 (
 
 e
 
 α
 
 
 k
 
 
 
 )
 
 α
 
 
 
 
 {\displaystyle (e_{\alpha }^{k})_{\alpha }}
 
, each homeomorphic to the open 
 
 
 
 k
 
 
 {\displaystyle k}
 
-<a href="/facts/Ball_(mathematics)/f9vpMVMp">ball</a> 
 
 
 
 
 B
 
 k
 
 
 
 
 {\displaystyle B^{k}}
 
, to 
 
 
 
 
 X
 
 k
 −
 1
 
 
 
 
 {\displaystyle X_{k-1}}
 
 by continuous gluing maps 
 
 
 
 
 g
 
 α
 
 
 k
 
 
 :
 ∂
 
 e
 
 α
 
 
 k
 
 
 →
 
 X
 
 k
 −
 1
 
 
 
 
 {\displaystyle g_{\alpha }^{k}:\partial e_{\alpha }^{k}\to X_{k-1}}
 
. The maps are also called <a href="/facts/Attaching_map/qxczNFKG">attaching maps</a>. Thus as a set, 
 
 
 
 
 X
 
 k
 
 
 =
 
 X
 
 k
 −
 1
 
 
 
 ⊔
 
 α
 
 
 
 e
 
 α
 
 
 k
 
 
 
 
 {\displaystyle X_{k}=X_{k-1}\sqcup _{\alpha }e_{\alpha }^{k}}
 
.
Each 
 
 
 
 
 X
 
 k
 
 
 
 
 {\displaystyle X_{k}}
 
 is called the k-skeleton of the complex.
The topology of 
 
 
 
 X
 =
 
 ∪
 
 k
 
 
 
 X
 
 k
 
 
 
 
 {\displaystyle X=\cup _{k}X_{k}}
 
 is a weak topology: a subset 
 
 
 
 U
 ⊂
 X
 
 
 {\displaystyle U\subset X}
 
 is open <a href="/facts/Iff/bYSxGJ66">iff</a> 
 
 
 
 U
 ∩
 
 X
 
 k
 
 
 
 
 {\displaystyle U\cap X_{k}}
 
 is open for each k-skeleton 
 
 
 
 
 X
 
 k
 
 
 
 
 {\displaystyle X_{k}}
 
.
In the language of category theory, the topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the <a href="/facts/Direct_limit/V3JuHnYK">direct limit</a> of the diagram 
 
 
 
 
 X
 
 −
 1
 
 
 ↪
 
 X
 
 0
 
 
 ↪
 
 X
 
 1
 
 
 ↪
 ⋯
 
 
 {\displaystyle X_{-1}\hookrightarrow X_{0}\hookrightarrow X_{1}\hookrightarrow \cdots }
 
The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:

Theorem—A <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> X is homeomorphic to a CW complex iff there exists a <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> of X into "open cells" 
 
 
 
 
 e
 
 α
 
 
 k
 
 
 
 
 {\displaystyle e_{\alpha }^{k}}
 
, each with a corresponding closure (or "closed cell") 
 
 
 
 
 
 
 
 e
 ¯
 
 
 
 
 α
 
 
 k
 
 
 :=
 c
 
 l
 
 X
 
 
 (
 
 e
 
 α
 
 
 k
 
 
 )
 
 
 {\displaystyle {\bar {e}}_{\alpha }^{k}:=cl_{X}(e_{\alpha }^{k})}
 
 that satisfies:

<ul><li>For each 
 
 
 
 
 e
 
 α
 
 
 k
 
 
 
 
 {\displaystyle e_{\alpha }^{k}}
 
, there exists a <a href="/facts/Continuous_function/sKbl02pB">continuous surjection</a> 
 
 
 
 
 g
 
 α
 
 
 k
 
 
 :
 
 D
 
 k
 
 
 →
 
 
 
 
 e
 ¯
 
 
 
 
 α
 
 
 k
 
 
 
 
 {\displaystyle g_{\alpha }^{k}:D^{k}\to {\bar {e}}_{\alpha }^{k}}
 
 from the 
 
 
 
 k
 
 
 {\displaystyle k}
 
-dimensional closed ball such that
<ul><li>The restriction to the open ball 
 
 
 
 
 g
 
 α
 
 
 k
 
 
 :
 
 B
 
 k
 
 
 →
 
 e
 
 α
 
 
 k
 
 
 
 
 {\displaystyle g_{\alpha }^{k}:B^{k}\to e_{\alpha }^{k}}
 
 is a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>.</li>
<li>(closure-finiteness) The image of the boundary 
 
 
 
 
 g
 
 α
 
 
 k
 
 
 (
 ∂
 
 D
 
 k
 
 
 )
 
 
 {\displaystyle g_{\alpha }^{k}(\partial D^{k})}
 
 is covered by a finite number of closed cells, each having cell dimension less than k.</li></ul></li>
<li>(weak topology) A subset of X is <a href="/facts/Closed_set/upYnRTUj">closed</a> if and only if it meets each closed cell in a closed set.</li></ul>

This partition of X is also called a cellulation.

<h4>The construction, in words</h4>
The CW complex construction is a straightforward generalization of the following process:

<ul><li>A 0-dimensional CW complex is just a set of zero or more discrete points (with the <a href="/facts/Discrete_space/hpvXA23u">discrete topology</a>).</li>
<li>A 1-dimensional CW complex is constructed by taking the <a href="/facts/Disjoint_union_(topology)/4SLSRbLt">disjoint union</a> of a 0-dimensional CW complex with one or more copies of the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a>. For each copy, there is a map that "<a href="/facts/Gluing_(topology)/muE0qQVC">glues</a>" 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 <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient space</a> defined by these gluing maps.</li>
<li>In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex (for some 
 
 
 
 k
 <
 n
 
 
 {\displaystyle k<n}
 
) with one or more copies of the <a href="/facts/Ball_(mathematics)/f9vpMVMp">n-dimensional ball</a>. For each copy, there is a map that "glues" its boundary (the 
 
 
 
 (
 n
 −
 1
 )
 
 
 {\displaystyle (n-1)}
 
-dimensional <a href="/facts/N-sphere/bFu2S7E2">sphere</a>) to elements of the 
 
 
 
 k
 
 
 {\displaystyle k}
 
-dimensional complex. The topology of the CW complex is the <a href="/facts/Quotient_topology/muE0qQVC">quotient topology</a> defined by these gluing maps.</li>
<li>An infinite-dimensional CW complex can be constructed by repeating the above process countably many times. Since the topology of the union 
 
 
 
 
 ∪
 
 k
 
 
 
 X
 
 k
 
 
 
 
 {\displaystyle \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.</li></ul>
<h3>Regular CW complexes</h3>
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 <a href="/facts/Loop_(graph_theory)/jCuevIad">loopless</a> graph is represented by a regular 1-dimensional CW-complex. A <a href="/facts/Graph_embedding/IyBA3ttM">closed 2-cell graph embedding</a> on a <a href="/facts/Surface/0XA8ce89">surface</a> is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every <a href="/facts/K-vertex-connected_graph/EST9uf8W">2-connected graph</a> is the 1-skeleton of a regular CW-complex on the <a href="/facts/3-sphere/u67nImpY">3-dimensional sphere</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h3>Relative CW complexes</h3>
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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="examples">Examples</h2>
<h3>0-dimensional CW complexes</h3>
Every <a href="/facts/Discrete_space/hpvXA23u">discrete topological space</a> is a 0-dimensional CW complex.

<h3>1-dimensional CW complexes</h3>
Some examples of 1-dimensional CW complexes are:<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<ul><li>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.</li>
<li>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.</li>
<li>A graph. Given a <a href="/facts/Multigraph/udmS0Cp0">graph</a>, 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.
<ul><li><a href="/facts/Trivalent_graph/6M32WUIJ">3-regular graphs</a> can be considered as <a href="/facts/Generic_property/Uz0FcCBW">generic</a> 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 <a href="/facts/Discrete_two-point_space/kdTMqJoW">two-point space</a> to X, 
 
 
 
 f
 :
 {
 0
 ,
 1
 }
 →
 X
 
 
 {\displaystyle f:\{0,1\}\to X}
 
. This map can be perturbed to be disjoint from the 0-skeleton of X if and only if 
 
 
 
 f
 (
 0
 )
 
 
 {\displaystyle f(0)}
 
 and 
 
 
 
 f
 (
 1
 )
 
 
 {\displaystyle f(1)}
 
 are not 0-valence vertices of X.</li></ul></li>
<li>The standard CW structure on the real numbers has as 0-skeleton the integers 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 and as 1-cells the intervals 
 
 
 
 {
 [
 n
 ,
 n
 +
 1
 ]
 :
 n
 ∈
 
 Z
 
 }
 
 
 {\displaystyle \{[n,n+1]:n\in \mathbb {Z} \}}
 
. Similarly, the standard CW structure on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 has cubical cells that are products of the 0 and 1-cells from 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
. This is the standard <a href="/facts/Integer_lattice/h4FP4pjm">cubic lattice</a> cell structure on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
.</li></ul>
<h3>Finite-dimensional CW complexes</h3>
Some examples of finite-dimensional CW complexes are:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<ul><li>An <a href="/facts/N-sphere/bFu2S7E2">n-dimensional sphere</a>. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell 
 
 
 
 
 D
 
 n
 
 
 
 
 {\displaystyle D^{n}}
 
 is attached by the constant mapping from its boundary 
 
 
 
 
 S
 
 n
 −
 1
 
 
 
 
 {\displaystyle S^{n-1}}
 
 to the single 0-cell. An alternative cell decomposition has one (n-1)-dimensional sphere (the "<a href="/facts/Equator/CjaxbVaN">equator</a>") and two n-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere"). Inductively, this gives 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S^{n}}
 
 a CW decomposition with two cells in every dimension k such that 
 
 
 
 0
 ≤
 k
 ≤
 n
 
 
 {\displaystyle 0\leq k\leq n}
 
.</li>
<li>The n-dimensional real <a href="/facts/Projective_space/7MSpA9cM">projective space</a>. It admits a CW structure with one cell in each dimension.</li>
<li>The terminology for a generic 2-dimensional CW complex is a shadow.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>A <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> is naturally a CW complex.</li>
<li><a href="/facts/Grassmannian/QcmgLRq8">Grassmannian</a> manifolds admit a CW structure called Schubert cells.</li>
<li><a href="/facts/Differentiable_manifold/aSnbXKcV">Differentiable manifolds</a>, algebraic and projective <a href="/facts/Algebraic_variety/Vk7f97vn">varieties</a> have the <a href="/facts/Homotopy_type/1nWrPxdj">homotopy type</a> of CW complexes.</li>
<li>The <a href="/facts/Alexandroff_extension/d3Xc0QKu">one-point compactification</a> of a cusped <a href="/facts/Hyperbolic_manifold/eUBhz9uW">hyperbolic manifold</a> 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 <a href="/facts/SnapPea/hx3QbENv">SnapPea</a>.</li></ul>
<h3>Infinite-dimensional CW complexes</h3>
<ul><li>The infinite dimensional sphere 
 
 
 
 
 S
 
 ∞
 
 
 :=
 
 
 c
 o
 l
 i
 m
 
 
 n
 →
 ∞
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S^{\infty }:=\mathrm {colim} _{n\to \infty }S^{n}}
 
. It admits a CW-structure with 2 cells in each dimension which are assembled in a way such that the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-skeleton is precisely given by the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-sphere.</li>
<li>The infinite dimensional projective spaces 
 
 
 
 
 
 R
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {RP} ^{\infty }}
 
, 
 
 
 
 
 
 C
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {CP} ^{\infty }}
 
 and 
 
 
 
 
 
 H
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {HP} ^{\infty }}
 
. 
 
 
 
 
 
 R
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {RP} ^{\infty }}
 
 has one cell in every dimension, 
 
 
 
 
 
 C
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {CP} ^{\infty }}
 
, has one cell in every even dimension and 
 
 
 
 
 
 H
 P
 
 
 ∞
 
 
 
 
 {\displaystyle \mathbb {HP} ^{\infty }}
 
 has one cell in every dimension divisible by 4. The respective skeletons are then given by 
 
 
 
 
 
 R
 P
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {RP} ^{n}}
 
, 
 
 
 
 
 
 C
 P
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {CP} ^{n}}
 
 (2n-skeleton) and 
 
 
 
 
 
 H
 P
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {HP} ^{n}}
 
 (4n-skeleton).</li></ul>
<h3>Non CW-complexes</h3>
<ul><li>An infinite-dimensional <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> is not a CW complex: it is a <a href="/facts/Baire_space/HToekdT8">Baire space</a> and therefore cannot be written as a countable union of n-skeletons, each of them being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.</li>
<li>The <a href="/facts/Hedgehog_space/cu2tXXwA">hedgehog space</a> 
 
 
 
 {
 r
 
 e
 
 2
 π
 i
 θ
 
 
 :
 0
 ≤
 r
 ≤
 1
 ,
 θ
 ∈
 
 Q
 
 }
 ⊆
 
 C
 
 
 
 {\displaystyle \{re^{2\pi i\theta }:0\leq r\leq 1,\theta \in \mathbb {Q} \}\subseteq \mathbb {C} }
 
 is homotopy equivalent to a CW complex (the point) but it does not admit a CW decomposition, since it is not <a href="/facts/Contractible_space/REXYj8uS">locally contractible</a>.</li>
<li>The <a href="/facts/Hawaiian_earring/3D52nchD">Hawaiian earring</a> 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.</li></ul>
<h2 id="properties">Properties</h2>
<ul><li>CW complexes are locally contractible.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li>
<li>If a space is <a href="/facts/Homotopy_equivalent/1nWrPxdj">homotopy equivalent</a> to a CW complex, then it has a good open cover.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> A good open cover is an open cover, such that every nonempty finite intersection is contractible.</li>
<li>CW complexes are <a href="/facts/Paracompact/rjTrvF7j">paracompact</a>. Finite CW complexes are <a href="/facts/Compact_space/d0cgXJH7">compact</a>. A compact subspace of a CW complex is always contained in a finite subcomplex.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li>
<li>CW complexes satisfy the <a href="/facts/Whitehead_theorem/43UKropG">Whitehead theorem</a>: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups.</li>
<li>A <a href="/facts/Covering_space/bDRGasl7">covering space</a> of a CW complex is also a CW complex.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>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 <a href="/facts/Weak_topology/XJASE9Up">weak topology</a>. The underlying set of X × Y is then the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of X and Y, as expected. In addition, the weak topology on this set often agrees with the more familiar <a href="/facts/Product_topology/0hg02IaX">product topology</a> on X × Y, for example if either X or Y is finite. However, the weak topology can be <a href="/facts/Comparison_of_topologies/5Q9F2fjf">finer</a> than the product topology, for example if neither X nor Y is <a href="/facts/Locally_compact_space/hesalT7q">locally compact</a>. 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 <a href="/facts/Compactly_generated_space/2A5umYGa">compactly generated spaces</a> agrees with the weak topology and therefore defines a CW complex.</li>
<li>Let X and Y be CW complexes. Then the <a href="/facts/Function_spaces/lLUE2R1t">function spaces</a> Hom(X,Y) (with the <a href="/facts/Compact-open_topology/qMGsW8gk">compact-open topology</a>) 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 <a href="/facts/John_Milnor/LTf24paN">John Milnor</a> (1959).<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> Note that X and Y are <a href="/facts/Compactly_generated_Hausdorff_space/2A5umYGa">compactly generated Hausdorff spaces</a>, so Hom(X,Y) is often taken with the <a href="/facts/Compactly_generated_space/2A5umYGa">compactly generated</a> variant of the compact-open topology; the above statements remain true.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a></li>
<li><a href="/facts/Cellular_approximation_theorem/KYLg2rwL">Cellular approximation theorem</a></li></ul>
<h2 id="homology-and-cohomology-of-cw-complexes">Homology and cohomology of CW complexes</h2>
<a href="/facts/Singular_homology/6c2AHr85">Singular homology</a> and <a href="/facts/Singular_cohomology/J7FJzTNh">cohomology</a> of CW complexes is readily computable via <a href="/facts/Cellular_homology/B4BfgxtM">cellular homology</a>. Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as a <a href="/facts/Homology_theory/VFJmryEW">homology theory</a>. To compute an <a href="/facts/Cohomology/J7FJzTNh">extraordinary (co)homology theory</a> for a CW complex, the <a href="/facts/Atiyah%25E2%2580%2593Hirzebruch_spectral_sequence/2cwxuanD">Atiyah–Hirzebruch spectral sequence</a> is the analogue of cellular homology.
Some examples:

<ul><li>For the sphere, 
 
 
 
 
 S
 
 n
 
 
 ,
 
 
 {\displaystyle S^{n},}
 
 take the cell decomposition with two cells: a single 0-cell and a single n-cell. The cellular homology <a href="/facts/Chain_complex/YAlchtCT">chain complex</a> 
 
 
 
 
 C
 
 ∗
 
 
 
 
 {\displaystyle C_{*}}
 
 and homology are given by:</li></ul>

C
          
            k
          
        
        =
        
          
            {
            
              
                
                  
                    Z
                  
                
                
                  k
                  ∈
                  {
                  0
                  ,
                  n
                  }
                
              
              
                
                  0
                
                
                  k
                  ∉
                  {
                  0
                  ,
                  n
                  }
                
              
            
            
          
        
        
        
          H
          
            k
          
        
        =
        
          
            {
            
              
                
                  
                    Z
                  
                
                
                  k
                  ∈
                  {
                  0
                  ,
                  n
                  }
                
              
              
                
                  0
                
                
                  k
                  ∉
                  {
                  0
                  ,
                  n
                  }
                
              
            
            
          
        
      
    
    {\displaystyle C_{k}={\begin{cases}\mathbb {Z} &k\in \{0,n\}\\0&k\notin \{0,n\}\end{cases}}\quad H_{k}={\begin{cases}\mathbb {Z} &k\in \{0,n\}\\0&k\notin \{0,n\}\end{cases}}}

since all the differentials are zero.
Alternatively, if we use the equatorial decomposition with two cells in every dimension

C
          
            k
          
        
        =
        
          
            {
            
              
                
                  
                    
                      Z
                    
                    
                      2
                    
                  
                
                
                  0
                  ⩽
                  k
                  ⩽
                  n
                
              
              
                
                  0
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\displaystyle C_{k}={\begin{cases}\mathbb {Z} ^{2}&0\leqslant k\leqslant n\\0&{\text{otherwise}}\end{cases}}}

and the differentials are matrices of the form 
  
    
      
        
          (
          
            
              
                
                  
                    1
                  
                  
                    −
                    1
                  
                
                
                  
                    1
                  
                  
                    −
                    1
                  
                
              
            
          
          )
        
        .
      
    
    {\displaystyle \left({\begin{smallmatrix}1&-1\\1&-1\end{smallmatrix}}\right).}
  
 This gives the same homology computation above, as the chain complex is exact at all terms except 
  
    
      
        
          C
          
            0
          
        
      
    
    {\displaystyle C_{0}}
  
 and 
  
    
      
        
          C
          
            n
          
        
        .
      
    
    {\displaystyle C_{n}.}

<ul><li>For 
 
 
 
 
 
 P
 
 
 n
 
 
 (
 
 C
 
 )
 
 
 {\displaystyle \mathbb {P} ^{n}(\mathbb {C} )}
 
 we get similarly</li></ul>

H
          
            k
          
        
        
          (
          
            
              
                P
              
              
                n
              
            
            (
            
              C
            
            )
          
          )
        
        =
        
          
            {
            
              
                
                  
                    Z
                  
                
                
                  0
                  ⩽
                  k
                  ⩽
                  2
                  n
                  ,
                  
                     even
                  
                
              
              
                
                  0
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\displaystyle H^{k}\left(\mathbb {P} ^{n}(\mathbb {C} )\right)={\begin{cases}\mathbb {Z} &0\leqslant k\leqslant 2n,{\text{ even}}\\0&{\text{otherwise}}\end{cases}}}

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.

<h2 id="modification-of-cw-structures">Modification of CW structures</h2>
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 <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a>. Now consider a maximal <a href="/facts/Tree_(graph_theory)/TCWk4Joy">forest</a> F in this graph. Since it is a collection of trees, and trees are contractible, consider the space 
 
 
 
 X
 
 /
 
 
 ∼
 
 
 
 {\displaystyle X/{\sim }}
 
 where the equivalence relation is generated by 
 
 
 
 x
 ∼
 y
 
 
 {\displaystyle x\sim y}
 
 if they are contained in a common tree in the maximal forest F. The quotient map 
 
 
 
 X
 →
 X
 
 /
 
 
 ∼
 
 
 
 {\displaystyle X\to X/{\sim }}
 
 is a homotopy equivalence. Moreover, 
 
 
 
 X
 
 /
 
 
 ∼
 
 
 
 {\displaystyle X/{\sim }}
 
 naturally inherits a CW structure, with cells corresponding to the cells of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that are not contained in F. In particular, the 1-skeleton of 
 
 
 
 X
 
 /
 
 
 ∼
 
 
 
 {\displaystyle X/{\sim }}
 
 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
 
 
 
 
 {\displaystyle X^{1}}
 
 consists of a single point? The answer is yes. The first step is to observe that 
 
 
 
 
 X
 
 1
 
 
 
 
 {\displaystyle X^{1}}
 
 and the attaching maps to construct 
 
 
 
 
 X
 
 2
 
 
 
 
 {\displaystyle X^{2}}
 
 from 
 
 
 
 
 X
 
 1
 
 
 
 
 {\displaystyle X^{1}}
 
 form a <a href="/facts/Presentation_of_a_group/f9emPUoI">group presentation</a>. The <a href="/facts/Tietze_transformations/whL6MwF1">Tietze theorem</a> 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 <a href="/facts/Trivial_group/XPjQkQFm">trivial group</a>. 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
 
 
 
 
 {\displaystyle X^{1}}
 
. If we let 
 
 
 
 
 
 
 X
 ~
 
 
 
 
 
 {\displaystyle {\tilde {X}}}
 
 be the corresponding CW complex 
 
 
 
 
 
 
 X
 ~
 
 
 
 =
 X
 ∪
 
 e
 
 1
 
 
 ∪
 
 e
 
 2
 
 
 
 
 {\displaystyle {\tilde {X}}=X\cup e^{1}\cup e^{2}}
 
 then there is a homotopy equivalence 
 
 
 
 
 
 
 X
 ~
 
 
 
 →
 X
 
 
 {\displaystyle {\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 
 
 
 
 
 
 
 X
 ~
 
 
 
 =
 X
 ∪
 
 e
 
 2
 
 
 ∪
 
 e
 
 3
 
 
 
 
 {\displaystyle {\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
 
 
 
 
 {\displaystyle X^{2}}
 
. A similar slide gives a homotopy-equivalence 
 
 
 
 
 
 
 X
 ~
 
 
 
 →
 X
 
 
 {\displaystyle {\tilde {X}}\to X}
 
.
If a CW complex X is <a href="/facts/N-connected_space/dbCgWgu3">n-connected</a> one can find a homotopy-equivalent CW complex 
 
 
 
 
 
 
 X
 ~
 
 
 
 
 
 {\displaystyle {\tilde {X}}}
 
 whose n-skeleton 
 
 
 
 
 X
 
 n
 
 
 
 
 {\displaystyle X^{n}}
 
 consists of a single point. The argument for 
 
 
 
 n
 ≥
 2
 
 
 {\displaystyle n\geq 2}
 
 is similar to the 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
 case, only one replaces Tietze moves for the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> presentation by <a href="/facts/Elementary_matrix/coFCElwg">elementary matrix</a> operations for the presentation matrices for 
 
 
 
 
 H
 
 n
 
 
 (
 X
 ;
 
 Z
 
 )
 
 
 {\displaystyle H_{n}(X;\mathbb {Z} )}
 
 (using the presentation matrices coming from <a href="/facts/Cellular_homology/B4BfgxtM">cellular homology</a>. i.e.: one can similarly realize elementary matrix operations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.

<h2 id="the-homotopy-category">'The' homotopy category</h2>
See also: <a href="/facts/Milnor%2527s_theorem_on_Kan_complexes/9v7KbyeV">Milnor's theorem on Kan complexes</a>
The <a href="/facts/Homotopy_category/JaEnPakj">homotopy category</a> 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 <a href="/facts/Pointed_space/V4DTgRCd">pointed spaces</a> is actually used).<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the <a href="/facts/Representable_functor/dILwN4oj">representable functors</a> on the homotopy category have a simple characterisation (the <a href="/facts/Brown_representability_theorem/YcpTQdBs">Brown representability theorem</a>).

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Abstract_cell_complex/Dx2RHWF3">Abstract cell complex</a></li>
<li>The notion of CW complex has an adaptation to <a href="/facts/Differentiable_manifold/aSnbXKcV">smooth manifolds</a> called a <a href="/facts/Handle_decomposition/NFd5F0zl">handle decomposition</a>, which is closely related to <a href="/facts/Surgery_theory/5Ip2RAGW">surgery theory</a>.</li></ul>

<h3>Notes</h3>

<h3>General references</h3>

<ul><li>Lundell, A. T.; Weingram, S. (1970). The topology of CW complexes. <a href="/facts/Van_Nostrand_(publisher)/SBWcDeMm">Van Nostrand</a> University Series in Higher Mathematics. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-442-04910-2.</li>
<li>Brown, R.; Higgins, P.J.; Sivera, R. (2011). Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids. <a href="/facts/European_Mathematical_Society/B3h7b672">European Mathematical Society</a> Tracts in Mathematics Vol 15. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-03719-083-8. More details on the <a href="http://groupoids.org.uk/nonab-a-t.html">[1]</a> first author's home page]</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. 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 author's homepage. <a href="0-521-79540-0" target="_blank">0-521-79540-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Whitehead, J. H. C. (1949a). "Combinatorial homotopy. I." (PDF). Bulletin of the American Mathematical Society. 55 (5): 213–245. doi:10.1090/S0002-9904-1949-09175-9. MR 0030759. (open access) <a href="/wiki/J._H._C._Whitehead" target="_blank">/wiki/J._H._C._Whitehead</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Whitehead, J. H. C. (1949a). "Combinatorial homotopy. I." (PDF). Bulletin of the American Mathematical Society. 55 (5): 213–245. doi:10.1090/S0002-9904-1949-09175-9. MR 0030759. (open access) <a href="/wiki/J._H._C._Whitehead" target="_blank">/wiki/J._H._C._Whitehead</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">De Agostino, Sergio (2016). The 3-Sphere Regular Cellulation Conjecture (PDF). International Workshop on Combinatorial Algorithms. <a href="https://twiki.di.uniroma1.it/pub/Users/SergioDeAgostino/DeAgostino.pdf" target="_blank">https://twiki.di.uniroma1.it/pub/Users/SergioDeAgostino/DeAgostino.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology. Providence, R.I.: American Mathematical Society. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">"CW complex in nLab". <a href="https://ncatlab.org/nlab/show/CW+complex" target="_blank">https://ncatlab.org/nlab/show/CW+complex</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"CW-complex - Encyclopedia of Mathematics". <a href="https://www.encyclopediaofmath.org/index.php/CW-complex" target="_blank">https://www.encyclopediaofmath.org/index.php/CW-complex</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Archived at Ghostarchive and the Wayback Machine: channel, Animated Math (2020). "1.3 Introduction to Algebraic Topology. Examples of CW Complexes". Youtube. <a href="https://ghostarchive.org/varchive/youtube/20211212/HjiooyBH6es" target="_blank">https://ghostarchive.org/varchive/youtube/20211212/HjiooyBH6es</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Archived at Ghostarchive and the Wayback Machine: channel, Animated Math (2020). "1.3 Introduction to Algebraic Topology. Examples of CW Complexes". Youtube. <a href="https://ghostarchive.org/varchive/youtube/20211212/HjiooyBH6es" target="_blank">https://ghostarchive.org/varchive/youtube/20211212/HjiooyBH6es</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Turaev, V. G. (1994). Quantum invariants of knots and 3-manifolds. De Gruyter Studies in Mathematics. Vol. 18. Berlin: Walter de Gruyter & Co. ISBN 9783110435221. <a href="9783110435221" target="_blank">9783110435221</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. p. 522. ISBN 0-521-79540-0. Proposition A.4 <a href="0-521-79540-0" target="_blank">0-521-79540-0</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Milnor, John (February 1959). "On Spaces Having the Homotopy Type of a CW-Complex". Transactions of the American Mathematical Society. 90 (2): 272–280. doi:10.2307/1993204. ISSN 0002-9947. JSTOR 1993204. <a href="https://dx.doi.org/10.2307/1993204" target="_blank">https://dx.doi.org/10.2307/1993204</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Hatcher, Allen, Algebraic topology, Cambridge University Press (2002). ISBN 0-521-79540-0. A free electronic version is available on the author's homepage <a href="/wiki/Allen_Hatcher" target="_blank">/wiki/Allen_Hatcher</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the author's homepage <a href="/wiki/Allen_Hatcher" target="_blank">/wiki/Allen_Hatcher</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. p. 529. ISBN 0-521-79540-0. Exercise 1 <a href="0-521-79540-0" target="_blank">0-521-79540-0</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Milnor, John (1959). "On spaces having the homotopy type of a CW-complex". Trans. Amer. Math. Soc. 90 (2): 272–280. doi:10.1090/s0002-9947-1959-0100267-4. JSTOR 1993204. <a href="/wiki/John_Milnor" target="_blank">/wiki/John_Milnor</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">"Compactly Generated Spaces" (PDF). Archived from the original (PDF) on 2016-03-03. Retrieved 2012-08-26. <a href="https://web.archive.org/web/20160303174529/http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf" target="_blank">https://web.archive.org/web/20160303174529/http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">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 Baladze, D.O. (2001) [1994], "CW-complex", Encyclopedia of Mathematics, EMS Press <a href="https://www.encyclopediaofmath.org/index.php?title=CW-complex&oldid=15603" target="_blank">https://www.encyclopediaofmath.org/index.php?title=CW-complex&oldid=15603</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
</ol>

CW complex open-in-new

CW complex