# nLab CW complex

### Context

#### Topology

topology

algebraic topology

# CW complexes

## Idea

A CW-complex is a nice topological space which is or can be built up inductively, by a process of attaching $n$-dimensional disks ${D}^{n}$ along their boundary spheres ${S}^{n.1}$ for all $n\in ℕ$: a cell complex built from the basic topological cells ${S}^{n-1}↪{D}^{n}$.

Being, therefore, essentially combinatorial objects, CW complexes are the principal objects of interest in algebraic topology; in fact, most spaces of interest to algebraic topologists are homotopy equivalent to CW-complexes. Notably the geometric realization of every simplicial set, hence also of every groupoid, 2-groupoid, etc., is a CW complex.

Also, CW complexes are the cofibrant objects in the standard model structure on topological spaces. This means in particular that every (Hausdorff) topological space is weakly homotopy equivalent to a CW-complex (but need not be strongly homotopy equivalent to one). Since every topological space is a fibrant object in this model category structure, this means that the full subcategory of Top on the CW-complexes is a category of “homotopically very good representatives” of homotopy types. See at homotopy theory and homotopy hypothesis for more on this.

## Definition

A CW-complex is a topological space $X$ equipped with a sequence of spaces and continuous maps

(1)$\varnothing ={X}_{-1}↪{X}_{0}↪{X}_{1}↪\dots ↪{X}_{n}↪\dots$\varnothing = X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \ldots \hookrightarrow X_n \hookrightarrow \ldots

and a cocone making $X$ into its colimit (in Top, or else in the category of compactly generated spaces or one of many other nice categories of spaces) where each space ${X}_{n}$ (called the $n$-skeleton of $X$) is the result of attaching copies of the $n$-disk ${D}^{n}=\left\{x\in {ℝ}^{n}:\mid \mid x\mid \mid \le 1\right\}$ along their boundaries ${S}^{n-1}=\partial {D}^{n}$ to ${X}_{n-1}$. Specifically, ${X}_{-1}$ is the empty space, and each ${X}_{n}$ is a pushout in Top of a diagram of the form

${X}_{n-1}\stackrel{\left({f}_{i}\right)}{←}\coprod _{i\in I}{S}_{i}^{n-1}\stackrel{\coprod _{i}{j}_{i}}{\to }\coprod _{i\in I}{D}_{i}^{n}$X_{n-1} \stackrel{(f_i)}{\leftarrow} \coprod_{i \in I} S_{i}^{n-1} \stackrel{\coprod_i j_i}{\to} \coprod_{i \in I} D_{i}^n

where $I$ is some index set, each ${j}_{i}:{S}_{i}^{n-1}\to {D}_{i}^{n}$ is the boundary inclusion of a copy of ${D}^{n}$, and ${f}_{i}:{S}_{i}^{n-1}\to {X}_{n-1}$ is a continuous map, often called an attaching map. The coprojections ${X}_{n-1}\to {X}_{n}$ of these pushouts give the arrows on which diagram (1) is based.

A relative CW-complex $\left(X,A\right)$ is defined as above, except ${X}_{-1}=A$ is allowed to be any space.

A finite CW-complex is one which admits a presentation in which there are only finitely many attaching maps, and similarly a countable CW-complex is one which admits a presentation with countably many attaching maps.

Formally this means that (relative) CW-complexes are (relative) cell complexes with respect to the generating cofibrations in the standard model structure on topological spaces.

Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called m-cofibrant spaces, is a convenient category of spaces for algebraic topology.

## Properties

### Local contractibility

###### Proposition

A CW-complex is a locally contractible topological space.

For instance (Hatcher, prop. A.4).

### Subcomplexes

###### Proposition

For $X$ a CW complex, the inclusion $X\prime ↪X$ of any subcomplex has an open neighbourhood in $X$ which is a deformation retract of $X\prime$. In particular such an inclusion is a good pair in the sense of relative homology.

For instance (Hatcher, prop. A.5).

### Singular homology

We discuss aspects of the singular homology ${H}_{n}\left(-\right):$ Top $\to$ Ab of CW-complexes. See also at cellular homology of CW-complexes.

Let $X$ be a CW-complex and write

${X}_{0}↪{X}_{1}↪{X}_{2}↪\cdots ↪X$X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X

for its filtered topological space-structure with ${X}_{n+1}$ the topological space obtained from ${X}_{n}$ by gluing on $\left(n+1\right)$-cells. For $n\in ℕ$ write $\mathrm{nCells}\in \mathrm{Set}$ for the set of $n$-cells of $X$.

###### Proposition

The relative singular homology of the filtering degrees is

${H}_{n}\left({X}_{k},{X}_{k-1}\right)\simeq \left\{\begin{array}{cc}ℤ\left[\mathrm{nCells}\right]& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}k=n\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule{thinmathspace}{0ex}},$H_n(X_k , X_{k-1}) \simeq \left\{ \array{ \mathbb{Z}[nCells] & if\; k = n \\ 0 & otherwise } \right. \,,

where $ℤ\left[\mathrm{nCells}\right]$ denotes the free abelian group on the set of $n$-cells.

The proof is spelled out at Relative singular homology - Of CW complexes.

###### Proposition

With $k,n\in ℕ$ we have

$\left(k>n\right)⇒\left({H}_{k}\left({X}_{n}\right)\simeq 0\right)\phantom{\rule{thinmathspace}{0ex}}.$(k \gt n) \Rightarrow (H_k(X_n) \simeq 0) \,.

In particular if $X$ is a CW-complex of finite dimension $\mathrm{dim}X$ (the maximum degree of cells), then

$\left(k>\mathrm{dim}X\right)⇒\left({H}_{k}\left(X\right)\simeq 0\right).$(k \gt dim X) \Rightarrow (H_k(X) \simeq 0).

Moreover, for $k the inclusion

${H}_{k}\left({X}_{n}\right)\stackrel{\simeq }{\to }{H}_{k}\left(X\right)$H_k(X_n) \stackrel{\simeq}{\to} H_k(X)

is an isomorphism and for $k=n$ we have an isomorphism

$\mathrm{image}\left({H}_{n}\left({X}_{n}\right)\to {H}_{n}\left(X\right)\right)\simeq {H}_{n}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$image(H_n(X_n) \to H_n(X)) \simeq H_n(X) \,.

This is mostly for instance in (Hatcher, lemma 2.34 b),c)).

###### Proof

By the long exact sequence in relative homology, discussed at Relative homology – long exact sequences, we have an exact sequence

${H}_{k+1}\left({X}_{n},{X}_{n-1}\right)\to {H}_{k}\left({X}_{n-1}\right)\to {H}_{k}\left({X}_{n}\right)\to {H}_{k}\left({X}_{n},{X}_{n-1}\right)\phantom{\rule{thinmathspace}{0ex}}.$H_{k+1}(X_n , X_{n-1}) \to H_k(X_{n-1}) \to H_k(X_n) \to H_k(X_n, X_{n-1}) \,.

Now by prop. 3 the leftmost and rightmost homology groups here vanish when $k\ne n$ and $k\ne n-1$ and hence exactness implies that

${H}_{k}\left({X}_{n-1}\right)\stackrel{\simeq }{\to }{H}_{k}\left({X}_{n}\right)$H_k(X_{n-1}) \stackrel{\simeq}{\to} H_k(X_n)

is an isomorphism for $k\ne n,n-1$. This implies the first claims by induction on $n$.

Finally for the last claim use that the above exact sequence gives

${H}_{n-1+1}\left({X}_{n},{X}_{n-1}\right)\to {H}_{n-1}\left({X}_{n-1}\right)\to {H}_{n-1}\left({X}_{n}\right)\to 0$H_{n-1+1}(X_n , X_{n-1}) \to H_{n-1}(X_{n-1}) \to H_{n-1}(X_n) \to 0

and hence that with the above the map ${H}_{n-1}\left({X}_{n-1}\right)\to {H}_{n-1}\left(X\right)$ is surjective.

## References

Basic textbook accounts include

Original articles include

• John Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280.

• John Milnor, The geometric realization of a semi-simplicial complex, Annals of Mathematics, 2nd Ser., 65, n. 2. (Mar., 1957), pp. 357-362; pdf