Inmathematics, and specifically intopology, aCW complex (alsocellular complex orcell complex) is atopological space that is built by gluing together topological balls (so-calledcells) of different dimensions in specific ways. It generalizes bothmanifolds andsimplicial complexes and has particular significance foralgebraic topology.^[1] It was initially introduced byJ. H. C. Whitehead to meet the needs ofhomotopy theory.^[2]CW complexes have bettercategorical properties thansimplicial complexes, 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]

ACW complex is constructed by taking the union of a sequence of topological spaces$${\displaystyle \mathrm{\varnothing }=X_{-1}\subset X_0\subset X_1\subset \cdots }$$ such that each${\displaystyle X_k}$ is obtained from${\displaystyle X_{k-1}}$ by gluing copies of k-cells${\displaystyle (e_{\alpha }^k{)}_{\alpha }}$, each homeomorphic to the open${\displaystyle k}$-ball${\displaystyle B^k}$, to${\displaystyle X_{k-1}}$ by continuous gluing maps${\displaystyle g_{\alpha }^k:\mathrm{\partial }{e}_{\alpha }^k\to X_{k-1}}$. The maps are also calledattaching maps. Thus as a set,${\displaystyle X_k=X_{k-1}{\bigsqcup }_{\alpha }{e}_{\alpha }^k}$.

Each${\displaystyle X_k}$ is called thek-skeleton of the complex.

The topology of${\displaystyle X={\cup }_k{X}_k}$ isweak topology: a subset${\displaystyle U\subset X}$ is openiff${\displaystyle U\cap X_k}$ is open for each k-skeleton${\displaystyle X_k}$.

In the language of category theory, the topology on${\displaystyle X}$ is thedirect limit of the diagram$${\displaystyle X_{-1}\hookrightarrow X_0\hookrightarrow X_1\hookrightarrow \cdots }$$The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:

This partition ofX is also called acellulation.

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 thediscrete topology).
• A 1-dimensional CW complex is constructed by taking thedisjoint union of a 0-dimensional CW complex with one or more copies of theunit interval. 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 thequotient space defined by these gluing maps.
• In general, ann-dimensional CW complex is constructed by taking the disjoint union of ak-dimensional CW complex (for some) with one or more copies of then-dimensional ball. For each copy, there is a map that "glues" its boundary (the${\displaystyle (n-1)}$-dimensionalsphere) to elements of the${\displaystyle k}$-dimensional complex. The topology of the CW complex is thequotient topology defined by these gluing maps.
• Aninfinite-dimensional CW complex can be constructed by repeating the above process countably many times. Since the topology of the union${\displaystyle {\cup }_k{X}_k}$ 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.

Aregular CW complex is a CW complex whose gluing maps are homeomorphisms. Accordingly, the partition ofX is also called aregular cellulation.

Aloopless graph is represented by a regular 1-dimensional CW-complex. Aclosed 2-cell graph embedding on asurface is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every2-connected graph is the 1-skeleton of a regular CW-complex on the3-dimensional sphere.^[3]

Roughly speaking, arelative 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]

Everydiscrete topological space is a 0-dimensional CW complex.

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

• An interval. It can be constructed from two points (x andy), and the 1-dimensional ballB (an interval), such that one endpoint ofB is glued tox and the other is glued toy. The two pointsx andy are the 0-cells; the interior ofB 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 pointx and the 1-dimensional ballB, such thatboth endpoints ofB are glued tox. Alternatively, it can be constructed from two pointsx andy and two 1-dimensional ballsA andB, such that the endpoints ofA are glued tox andy, and the endpoints ofB are glued tox andy too.
• A graph. Given agraph, 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 atopological graph.
  • 3-regular graphs can be considered asgeneric 1-dimensional CW complexes. Specifically, ifX is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from atwo-point space toX,${\displaystyle f:\{0,1\}\to X}$. This map can be perturbed to be disjoint from the 0-skeleton ofX if and only if${\displaystyle f(0)}$ and${\displaystyle f(1)}$ are not 0-valence vertices ofX.
• Thestandard CW structure on the real numbers has as 0-skeleton the integers${\displaystyle \mathbb{Z}}$ and as 1-cells the intervals${\displaystyle \{[n,n+1]:n\in \mathbb{Z}\}}$. Similarly, the standard CW structure on${\displaystyle {\mathbb{R}}^n}$ has cubical cells that are products of the 0 and 1-cells from${\displaystyle \mathbb{R}}$. This is the standardcubic lattice cell structure on${\displaystyle {\mathbb{R}}^n}$.

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

• Ann-dimensional sphere. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell${\displaystyle D^n}$ is attached by the constant mapping from its boundary${\displaystyle S^{n-1}}$ to the single 0-cell. An alternative cell decomposition has one (n-1)-dimensional sphere (the "equator") and twon-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere"). Inductively, this gives${\displaystyle S^n}$ a CW decomposition with two cells in every dimension k such that${\displaystyle 0\le k\le n}$.
• Then-dimensional realprojective space. It admits a CW structure with one cell in each dimension.
• The terminology for a generic 2-dimensional CW complex is ashadow.^[8]
• Apolyhedron is naturally a CW complex.
• Grassmannian manifolds admit a CW structure calledSchubert cells.
• Differentiable manifolds, algebraic and projectivevarieties have thehomotopy type of CW complexes.
• Theone-point compactification of a cuspedhyperbolic manifold has a canonical CW decomposition with only one 0-cell (the compactification point) called theEpstein–Penner Decomposition. Such cell decompositions are frequently calledideal polyhedral decompositions and are used in popular computer software, such asSnapPea.

• The infinite dimensional sphere${\displaystyle S^{\mathrm{\infty }}:={\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}{S}^n}$. It admits a CW-structure with 2 cells in each dimension which are assembled in a way such that the${\displaystyle n}$-skeleton is precisely given by the${\displaystyle n}$-sphere.
• The infinite dimensional projective spaces${\displaystyle {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}}$,${\displaystyle {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}}$ and${\displaystyle {\mathbb{H}\mathbb{P}}^{\mathrm{\infty }}}$.${\displaystyle {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}}$ has one cell in every dimension,${\displaystyle {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}}$, has one cell in every even dimension and${\displaystyle {\mathbb{H}\mathbb{P}}^{\mathrm{\infty }}}$ has one cell in every dimension divisible by 4. The respective skeletons are then given by${\displaystyle {\mathbb{R}\mathbb{P}}^n}$,${\displaystyle {\mathbb{C}\mathbb{P}}^n}$ (2n-skeleton) and${\displaystyle {\mathbb{H}\mathbb{P}}^n}$ (4n-skeleton).

• An infinite-dimensionalHilbert space is not a CW complex: it is aBaire space and therefore cannot be written as a countable union ofn-skeletons, each of which being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.
• Thehedgehog space${\displaystyle \{r{e}^{2\pi i\theta }:0\le r\le 1,\theta \in \mathbb{Q}\}\subseteq \mathbb{C}}$ is homotopy equivalent to a CW complex (the point) but it does not admit a CW decomposition, since it is notlocally contractible.
• TheHawaiian 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.

• CW complexes are locally contractible.^[9]
• If a space ishomotopy 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 areparacompact. Finite CW complexes arecompact. A compact subspace of a CW complex is always contained in a finite subcomplex.^[11][12]
• CW complexes satisfy theWhitehead theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups.
• Acovering 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, ifX andY are CW complexes, then one can form a CW complexX ×Y in which each cell is a product of a cell inX and a cell inY, endowed with theweak topology. The underlying set ofX ×Y is then theCartesian product ofX andY, as expected. In addition, the weak topology on this set often agrees with the more familiarproduct topology onX ×Y, for example if eitherX orY is finite. However, the weak topology can befiner than the product topology, for example if neitherX norY islocally compact. In this unfavorable case, the productX ×Y in the product topology isnot a CW complex. On the other hand, the product ofX andY in the category ofcompactly generated spaces agrees with the weak topology and therefore defines a CW complex.
• LetX andY be CW complexes. Then thefunction spaces Hom(X,Y) (with thecompact-open topology) arenot CW complexes in general. IfX is finite then Hom(X,Y) is homotopy equivalent to a CW complex by a theorem ofJohn Milnor (1959).^[14] Note thatX andY arecompactly generated Hausdorff spaces, so Hom(X,Y) is often taken with thecompactly generated variant of the compact-open topology; the above statements remain true.^[15]
• Cellular approximation theorem

Singular homology andcohomology of CW complexes is readily computable viacellular homology. Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as ahomology theory. To compute anextraordinary (co)homology theory for a CW complex, theAtiyah–Hirzebruch spectral sequence is the analogue of cellular homology.

Some examples:

• For the sphere,${\displaystyle S^n,}$ take the cell decomposition with two cells: a single 0-cell and a singlen-cell. The cellular homologychain complex${\displaystyle C_{\ast }}$ and homology are given by:

since all the differentials are zero.

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

and the differentials are matrices of the form This gives the same homology computation above, as the chain complex is exact at all terms except${\displaystyle C_0}$ and${\displaystyle C_n.}$

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

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

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

Consider, for example, an arbitrary CW complex. Its 1-skeleton can be fairly complicated, being an arbitrarygraph. Now consider a maximalforestF in this graph. Since it is a collection of trees, and trees are contractible, consider the space${\displaystyle X/\sim }$ where the equivalence relation is generated by${\displaystyle x\sim y}$ if they are contained in a common tree in the maximal forestF. The quotient map${\displaystyle X\to X/\sim }$ is a homotopy equivalence. Moreover,${\displaystyle X/\sim }$ naturally inherits a CW structure, with cells corresponding to the cells of${\displaystyle X}$ that are not contained inF. In particular, the 1-skeleton of${\displaystyle X/\sim }$ 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—assumeX is a simply-connected CW complex whose 0-skeleton consists of a point. Can we, through suitable modifications, replaceX by a homotopy-equivalent CW complex where${\displaystyle X^1}$ consists of a single point? The answer is yes. The first step is to observe that${\displaystyle X^1}$ and the attaching maps to construct${\displaystyle X^2}$ from${\displaystyle X^1}$ form agroup presentation. TheTietze 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 thetrivial 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${\displaystyle X^1}$. If we let${\displaystyle \tilde{X}}$ be the corresponding CW complex${\displaystyle \tilde{X}=X\cup e^1\cup e^2}$ then there is a homotopy equivalence${\displaystyle \tilde{X}\to X}$ given by sliding the new 2-cell intoX.

2) Adding/removing a relation. The act of adding a relation is similar, only one is replacingX by${\displaystyle \tilde{X}=X\cup e^2\cup e^3}$ where the new3-cell has an attaching map that consists of the new 2-cell and remainder mapping into${\displaystyle X^2}$. A similar slide gives a homotopy-equivalence${\displaystyle \tilde{X}\to X}$.

If a CW complexX isn-connected one can find a homotopy-equivalent CW complex${\displaystyle \tilde{X}}$ whosen-skeleton${\displaystyle X^n}$ consists of a single point. The argument for${\displaystyle n\ge 2}$ is similar to the${\displaystyle n=1}$ case, only one replaces Tietze moves for thefundamental group presentation byelementary matrix operations for the presentation matrices for${\displaystyle H_n(X;\mathbb{Z})}$ (using the presentation matrices coming fromcellular 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.

Thehomotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate forthe homotopy category (for technical reasons the version forpointed spaces is actually used).^[16] Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that therepresentable functors on the homotopy category have a simple characterisation (theBrown representability theorem).

• Abstract cell complex
• The notion of CW complex has an adaptation tosmooth manifolds called ahandle decomposition, which is closely related tosurgery theory.

• Algebraic topology
• Homotopy theory
• Topological spaces

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