# CW complex

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Type of topological space

In [mathematics](/source/Mathematics), and specifically in [topology](/source/Topology), a **CW complex** (also **cellular complex** or **cell complex**) is a [topological space](/source/Topological_space) that is built by gluing together topological balls (so-called *cells*) of different dimensions in specific ways. The notion generalizes both [manifolds](/source/Topological_manifold) and [simplicial complexes](/source/Simplicial_complex) and has particular significance for [algebraic topology](/source/Algebraic_topology).[1] It was initially introduced by [J. H. C. Whitehead](/source/J._H._C._Whitehead) to meet the needs of [homotopy theory](/source/Homotopy_theory).[2] CW complexes have better [categorical](/source/Category_theory) properties than [simplicial complexes](/source/Simplicial_complex), 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.[2]

## Definition

### CW complex

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 ∈ J {\displaystyle (e_{\alpha }^{k})_{\alpha \in J}} , each homeomorphic to the open unit [ball](/source/Ball_(mathematics)) B k {\displaystyle B^{k}} in k {\displaystyle k} -dimensional [Euclidean space](/source/Euclidean_space), 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 {\bar {e}}_{\alpha }^{k}\to X_{k-1}} where e ¯ k ≅ B ¯ k {\displaystyle {\bar {e}}^{k}\cong {\bar {B}}^{k}} the closed unit ball. The maps are also called [attaching maps](/source/Attaching_map). Thus as a set, X k = X k − 1 ⊔ α e α k {\displaystyle X_{k}=X_{k-1}\sqcup _{\alpha }e_{\alpha }^{k}} .

To be precise, X k {\displaystyle X_{k}} is homeomorphic to ( X k − 1 ⊔ ( J × e ¯ k ) ) / ∼ {\displaystyle (X_{k-1}\sqcup (J\times {\overline {e}}^{k}))/\sim } , (here e ¯ k {\displaystyle {\overline {e}}^{k}} is the closed k {\displaystyle k} -disk in R k {\displaystyle \mathbb {R} ^{k}} ) where we equip J {\displaystyle J} with the discrete topology and where ∼ {\displaystyle \sim } is the equivalence relation generated by ( a , x ) ∼ g a k ( x ) {\displaystyle (a,x)\sim g_{a}^{k}(x)} for a ∈ J {\displaystyle a\in J} and x ∈ ∂ e k {\displaystyle x\in \partial e^{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 [iff](/source/Iff) 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](/source/Category_theory), the topology on X {\displaystyle X} is the [direct limit](/source/Direct_limit) of the [diagram](/source/Diagram_(category_theory)) 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 [Hausdorff space](/source/Hausdorff_space) *X* is [homeomorphic](/source/Homeomorphic) to a CW complex iff there exists a [partition](/source/Partition_of_a_set) 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:

- For each e α k {\displaystyle e_{\alpha }^{k}} , there exists a [continuous surjection](/source/Continuous_function#Continuous_functions_between_topological_spaces) 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 - 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 homeomorphism. - (closure-finiteness) The image of the [boundary](/source/Boundary_(topology)) 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.

- (weak topology) A subset of *X* is [closed](/source/Closed_set) 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 topology](/source/Discrete_space)).

- A 1-*dimensional CW complex* is constructed by taking the [disjoint union](/source/Disjoint_union_(topology)) of a 0-dimensional CW complex with one or more copies of the [unit interval](/source/Unit_interval). For each copy, there is a map that "[glues](/source/Gluing_(topology))" 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](/source/Quotient_space_(topology)) 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 < n {\displaystyle k<n} ) with one or more copies of the [*n*-dimensional ball](/source/Ball_(mathematics)). For each copy, there is a map that "glues" its boundary (the ( n − 1 ) {\displaystyle (n-1)} -dimensional [sphere](/source/N-sphere)) to elements of the k {\displaystyle k} -dimensional complex. The topology of the CW complex is the [quotient topology](/source/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 ∪ k X k {\displaystyle \cup _{k}X_{k}} is indeterminate, one takes the [direct limit](/source/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](/source/Loop_(graph_theory)) graph is represented by a regular 1-dimensional CW-complex. A [closed 2-cell graph embedding](/source/Graph_embedding) on a [surface](/source/Surface) is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every [2-connected graph](/source/K-vertex-connected_graph) is the 1-skeleton of a regular CW-complex on the [3-dimensional sphere](/source/3-sphere).[3]

### 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.[4][5][6]

## Examples

### 0-dimensional CW complexes

Every [discrete topological space](/source/Discrete_space) is a 0-dimensional CW complex.

### 1-dimensional CW complexes

Some examples of 1-dimensional CW complexes are:[7]

- **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 [graph](/source/Multigraph), 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**. - [3-regular graphs](/source/Trivalent_graph) can be considered as *[generic](/source/Generic_property)* 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](/source/Discrete_two-point_space) 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*.

- 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 *[cubic lattice](/source/Integer_lattice)* cell structure on R n {\displaystyle \mathbb {R} ^{n}} .

### Finite-dimensional CW complexes

Some examples of finite-dimensional CW complexes are:[7]

- **An [*n*-dimensional sphere](/source/N-sphere)**. 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 "[equator](/source/Equator)") 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} .

- **The *n*-dimensional real [projective space](/source/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**.[8]

- A [polyhedron](/source/Polyhedron) is naturally a CW complex.

- [Grassmannian](/source/Grassmannian) manifolds admit a CW structure called **Schubert cells**.

- [Differentiable manifolds](/source/Differentiable_manifold), algebraic and projective [varieties](/source/Algebraic_variety) have the [homotopy type](/source/Homotopy_type) of CW complexes.

- The [one-point compactification](/source/Alexandroff_extension) of a cusped [hyperbolic manifold](/source/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](/source/SnapPea).

### Infinite-dimensional CW complexes

- The [infinite-dimensional sphere](/source/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.

- 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).

### Non CW-complexes

- An infinite-dimensional [Hilbert space](/source/Hilbert_space) is not a CW complex: it is a [Baire space](/source/Baire_space) 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.

- The [hedgehog space](/source/Hedgehog_space) { 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 [locally contractible](/source/Contractible_space#Locally_contractible_spaces).

- The [Hawaiian earring](/source/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.[9]

- If a space is [homotopy equivalent](/source/Homotopy_equivalent) to a CW complex, then it has a good open cover.[10] A good open cover is an open cover, such that every nonempty finite intersection is contractible.

- CW complexes are [paracompact](/source/Paracompact). Finite CW complexes are [compact](/source/Compact_space). A compact subspace of a CW complex is always contained in a finite subcomplex.[11][12]

- CW complexes satisfy the [Whitehead theorem](/source/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](/source/Covering_space) of a CW complex is also a CW complex.[13]

- 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](/source/Weak_topology). The underlying set of *X* × *Y* is then the [Cartesian product](/source/Cartesian_product) of *X* and *Y*, as expected. In addition, the weak topology on this set often agrees with the more familiar [product topology](/source/Product_topology) on *X* × *Y*, for example if either *X* or *Y* is finite. However, the weak topology can be [finer](/source/Comparison_of_topologies) than the product topology, for example if neither *X* nor *Y* is [locally compact](/source/Locally_compact_space). 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 spaces](/source/Compactly_generated_space) agrees with the weak topology and therefore defines a CW complex.

- Let *X* and *Y* be CW complexes. Then the [function spaces](/source/Function_spaces) Hom(*X*,*Y*) (with the [compact-open topology](/source/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](/source/John_Milnor) (1959).[14] Note that *X* and *Y* are [compactly generated Hausdorff spaces](/source/Compactly_generated_Hausdorff_space), so Hom(*X*,*Y*) is often taken with the [compactly generated](/source/Compactly_generated_space) variant of the compact-open topology; the above statements remain true.[15]

- [Cellular approximation theorem](/source/Cellular_approximation_theorem)

## Homology and cohomology of CW complexes

[Singular homology](/source/Singular_homology) and [cohomology](/source/Singular_cohomology) of CW complexes is readily computable via [cellular homology](/source/Cellular_homology). Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as a [homology theory](/source/Homology_theory). To compute an [extraordinary (co)homology theory](/source/Cohomology#Generalized_cohomology_theories) for a CW complex, the [Atiyah–Hirzebruch spectral sequence](/source/Atiyah%E2%80%93Hirzebruch_spectral_sequence) is the analogue of cellular homology.

Some examples:

- 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 [chain complex](/source/Chain_complex) C ∗ {\displaystyle C_{*}} and homology are given by:

- - 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}.}

- For P n ( C ) {\displaystyle \mathbb {P} ^{n}(\mathbb {C} )} we get similarly

- - 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.

## 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](/source/Graph_(discrete_mathematics)). Now consider a maximal [forest](/source/Tree_(graph_theory)) *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 [group presentation](/source/Presentation_of_a_group). The [Tietze theorem](/source/Tietze_transformations) 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](/source/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 {\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 [*n*-connected](/source/N-connected_space) 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 [fundamental group](/source/Fundamental_group) presentation by [elementary matrix](/source/Elementary_matrix) operations for the presentation matrices for H n ( X ; Z ) {\displaystyle H_{n}(X;\mathbb {Z} )} (using the presentation matrices coming from [cellular homology](/source/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

See also: [Milnor's theorem on Kan complexes](/source/Milnor's_theorem_on_Kan_complexes)

The [homotopy category](/source/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 spaces](/source/Pointed_space) is actually used).[16] Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the [representable functors](/source/Representable_functor) on the homotopy category have a simple characterisation (the [Brown representability theorem](/source/Brown_representability_theorem)).

## See also

- [Abstract cell complex](/source/Abstract_cell_complex)

- The notion of CW complex has an adaptation to [smooth manifolds](/source/Differentiable_manifold) called a [handle decomposition](/source/Handle_decomposition), which is closely related to [surgery theory](/source/Surgery_theory).

## References

### Notes

1. **[^](#cite_ref-1)** [Hatcher, Allen](/source/Allen_Hatcher) (2002). *Algebraic topology*. [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [0-521-79540-0](https://en.wikipedia.org/wiki/Special:BookSources/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](https://pi.math.cornell.edu/~hatcher/).

1. ^ [***a***](#cite_ref-:2_2-0) [***b***](#cite_ref-:2_2-1) [Whitehead, J. H. C.](/source/J._H._C._Whitehead) (1949a). ["Combinatorial homotopy. I."](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-3.P1/Combinatorial-homotopy-I/bams/1183513543.pdf) (PDF). *[Bulletin of the American Mathematical Society](/source/Bulletin_of_the_American_Mathematical_Society)*. **55** (5): 213–245. [doi](/source/Doi_(identifier)):[10.1090/S0002-9904-1949-09175-9](https://doi.org/10.1090%2FS0002-9904-1949-09175-9). [MR](/source/MR_(identifier)) [0030759](https://mathscinet.ams.org/mathscinet-getitem?mr=0030759). (open access)

1. **[^](#cite_ref-3)** De Agostino, Sergio (2016). [*The 3-Sphere Regular Cellulation Conjecture*](https://twiki.di.uniroma1.it/pub/Users/SergioDeAgostino/DeAgostino.pdf) (PDF). International Workshop on Combinatorial Algorithms.

1. **[^](#cite_ref-4)** Davis, James F.; Kirk, Paul (2001). *Lecture Notes in Algebraic Topology*. Providence, R.I.: American Mathematical Society.

1. **[^](#cite_ref-5)** ["CW complex in nLab"](https://ncatlab.org/nlab/show/CW+complex).

1. **[^](#cite_ref-6)** ["CW-complex - Encyclopedia of Mathematics"](https://www.encyclopediaofmath.org/index.php/CW-complex).

1. ^ [***a***](#cite_ref-:1_7-0) [***b***](#cite_ref-:1_7-1) Archived at [Ghostarchive](https://ghostarchive.org/varchive/youtube/20211212/HjiooyBH6es) and the [Wayback Machine](https://web.archive.org/web/20201211210326/https://www.youtube.com/watch?v=HjiooyBH6es&gl=US&hl=en): channel, Animated Math (2020). ["1.3 Introduction to Algebraic Topology. Examples of CW Complexes"](https://www.youtube.com/watch?v=HjiooyBH6es&t=25s). *Youtube*.

1. **[^](#cite_ref-8)** Turaev, V. G. (1994). *Quantum invariants of knots and 3-manifolds*. De Gruyter Studies in Mathematics. Vol. 18. Berlin: Walter de Gruyter & Co. [ISBN](/source/ISBN_(identifier)) [9783110435221](https://en.wikipedia.org/wiki/Special:BookSources/9783110435221).

1. **[^](#cite_ref-9)** Hatcher, Allen (2002). *Algebraic topology*. [Cambridge University Press](/source/Cambridge_University_Press). p. 522. [ISBN](/source/ISBN_(identifier)) [0-521-79540-0](https://en.wikipedia.org/wiki/Special:BookSources/0-521-79540-0). Proposition A.4

1. **[^](#cite_ref-10)** Milnor, John (February 1959). ["On Spaces Having the Homotopy Type of a CW-Complex"](https://dx.doi.org/10.2307/1993204). *Transactions of the American Mathematical Society*. **90** (2): 272–280. [doi](/source/Doi_(identifier)):[10.2307/1993204](https://doi.org/10.2307%2F1993204). [ISSN](/source/ISSN_(identifier)) [0002-9947](https://search.worldcat.org/issn/0002-9947). [JSTOR](/source/JSTOR_(identifier)) [1993204](https://www.jstor.org/stable/1993204).

1. **[^](#cite_ref-11)** [Hatcher, Allen](/source/Allen_Hatcher), *Algebraic topology*, Cambridge University Press (2002). [ISBN](/source/ISBN_(identifier)) [0-521-79540-0](https://en.wikipedia.org/wiki/Special:BookSources/0-521-79540-0). A free electronic version is available on the [author's homepage](https://pi.math.cornell.edu/~hatcher/)

1. **[^](#cite_ref-12)** [Hatcher, Allen](/source/Allen_Hatcher), *Vector bundles and K-theory*, preliminary version available on the [author's homepage](https://pi.math.cornell.edu/~hatcher/)

1. **[^](#cite_ref-13)** Hatcher, Allen (2002). *Algebraic topology*. [Cambridge University Press](/source/Cambridge_University_Press). p. 529. [ISBN](/source/ISBN_(identifier)) [0-521-79540-0](https://en.wikipedia.org/wiki/Special:BookSources/0-521-79540-0). Exercise 1

1. **[^](#cite_ref-milnor_14-0)** [Milnor, John](/source/John_Milnor) (1959). ["On spaces having the homotopy type of a CW-complex"](https://doi.org/10.1090%2Fs0002-9947-1959-0100267-4). *Trans. Amer. Math. Soc*. **90** (2): 272–280. [doi](/source/Doi_(identifier)):[10.1090/s0002-9947-1959-0100267-4](https://doi.org/10.1090%2Fs0002-9947-1959-0100267-4). [JSTOR](/source/JSTOR_(identifier)) [1993204](https://www.jstor.org/stable/1993204).

1. **[^](#cite_ref-15)** ["Compactly Generated Spaces"](https://web.archive.org/web/20160303174529/http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf) (PDF). Archived from [the original](http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf) (PDF) on 2016-03-03. Retrieved 2012-08-26.

1. **[^](#cite_ref-16)** 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"](https://www.encyclopediaofmath.org/index.php?title=CW-complex&oldid=15603), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society)

### General references

- Lundell, A. T.; Weingram, S. (1970). *The topology of CW complexes*. [Van Nostrand](/source/Van_Nostrand_(publisher)) University Series in Higher Mathematics. [ISBN](/source/ISBN_(identifier)) [0-442-04910-2](https://en.wikipedia.org/wiki/Special:BookSources/0-442-04910-2).

- Brown, R.; Higgins, P.J.; Sivera, R. (2011). *Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids*. [European Mathematical Society](/source/European_Mathematical_Society) Tracts in Mathematics Vol 15. [ISBN](/source/ISBN_(identifier)) [978-3-03719-083-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-03719-083-8). More details on the [\[1\]](http://groupoids.org.uk/nonab-a-t.html) first author's home page]

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Adapted from the Wikipedia article [CW complex](https://en.wikipedia.org/wiki/CW_complex) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/CW_complex?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.