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-disks $D^n$ along their boundary (n-1)-spheres $S^{n-1}$ for all $n \in \mathbb{N}$: a cell complex built from the basic topological cells $S^{n-1} \hookrightarrow D^n$.

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 classical model structure on topological spaces. This means in particular that every topological space is weakly homotopy equivalent to a CW-complex (but need not be strongly homotopy equivalent to one). See also at CW-approximation. 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.

Terminology

The terminology “CW-complex” goes back to John Henry Constantine Whitehead (see Hatcher, “Topology of cell complexes”, p. 520). It stands for the following two properties shared by any CW complex:

• C = “closure finiteness”: a compact subset of a CW complex intersects the interior of only finitely many cells (prop.), hence in particular so does the closure of any cell.

• W = “weak topology”: Since a CW-complex is a colimit in Top over its cells, and as such equipped with the final topology of the cell inclusion maps, a subset of a CW-complex is open or closed precisely if its restriction to (the closure of) each cell is open or closed, respectively.

(Whitehead called the interior of the n-disks the “cells”, so that their closure of each cell is the corresponding $n$-disk.)

Definition

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

(1)$\varnothing = X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \ldots \hookrightarrow X_n \hookrightarrow \ldots$

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 $D^n = \{x \in \mathbb{R}^n: ||x|| \leq 1\}$ 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

$X_{n-1} \stackrel{(f_i)}{\leftarrow} \coprod_{i \in I} S_{i}^{n-1} \stackrel{\coprod_i j_i}{\to} \coprod_{i \in I} D_{i}^n$

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 $(X, A)$ 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 special (relative) cell complexes with respect to the generating cofibrations in the standard Quillen model structure on topological spaces: they are those cell complexes which are obtained from a countable transfinite composition of cell attachments, and where in addition the stage $X_n$ is obtained from $X_{n-1}$ by attaching cells of dimension $n$, instead of cells of arbitrary dimension.

A cellular map between CW-complexes $X$ and $Y$ is a continuous function $f\colon X \to Y$ such that $f(X_n) \subset Y_n$.

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

Closure properties

If $A \hookrightarrow X$ is an inclusion of CW-complexes, then the quotient $X/A$ is naturally itself a CW-complex, such that the quotient map $X \to X/A$ is cellular.

If $X$ is a CW-complex and $K$ is a finite CW-complex, then the product topological space $X \times K$ is naturally itself a CW-complex.

For example the suspension of a CW-complex itself carries the structure of a CW-complex.

Similarly for pointed CW-complexes: the smash product of a pointed CW-complex with a finite pointed CW-complex is a pointed CW-complex. For example the reduced suspension $S^1 \wedge X$ of a pointed CW-complex $X$ is itself a CW-complex.

Proposition

For $X$ and $Y$ CW-complexes with attaching maps $\{\phi_\alpha\}$ and $\{\Psi_\beta\}$, then the k-ification $(X \times Y)_c$ of their product topological space $X \times Y$ (hence their Cartesian product in the category of compactly generated topological spaces) is again a CW-complex with attaching maps $\{\Phi_\alpha \times \Psi_\beta\}$.

If either of the two CW-complexes is a locally compact topological space or if both are countable CW-complexes (have a countable set of cells) then

$(X\times Y)_c \simeq X \times Y$

and so then the product topological space $X \times Y$ itself has CW-complex structure.

Local contractibility

Proposition

A CW-complex is a locally contractible topological space.

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

Compactness properties

Proposition

Every CW-complex is a paracompact topological space.

See for instance (Hatcher K-theory, appendix of section 1.2). For a discussion that highlights which choice principles are involved, see (Fritsch-Piccinini 90, Theorem 1.3.5 (p. 29 and following)).

Proposition

Every CW-complex is a compactly generated topological space.

Proof

Since a CW-complex $X$ is a colimit in Top over attachments of standard n-disks $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ (prop.) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ is closed.

For the converse direction, since a CW-complex is a Hausdorff space and since compact subspaces of Hausdorff spaces are closed, the intersection of a closed subset with a compact subset is closed.

Up to homotopy equivalence

Theorem

Every CW complex is homotopy equivalent to (the realization of) a simplicial complex.

See Gray, Corollary 16.44 (p. 149) and Corollary 21.15 (p. 206). For more see at CW approximation.

Corollary

Every CW complex is homotopy equivalent to a space that admits a good open cover.

Theorem

If $Y$ has the homotopy type of a CW complex and $X$ is a finite CW complex, then the mapping space $Y^X$ with the compact-open topology has the homotopy type of a CW complex.

Subcomplexes

Proposition

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

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

Remark

For $A \hookrightarrow X$ the inclusion of a subcomplex into a CW complex, then the pair $(X,A)$ is often called a CW-pair. This appears notably in the axioms for generalized (Eilenberg-Steenrod) cohomology.

e.g. (AGP 02, def. 5.1.11)

Singular homology

We discuss aspects of the singular homology $H_n(-) \colon$ Top $\to$ Ab of CW-complexes. See also at cellular homology of CW-complexes.

Let $X$ be a CW-complex and write

$X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X$

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

Proposition

The relative singular homology of the filtering degrees is

$H_n(X_k , X_{k-1}) \simeq \left\{ \array{ \mathbb{Z}[nCells] & if\; k = n \\ 0 & otherwise } \right. \,,$

where $\mathbb{Z}[nCells]$ 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 $k,n \in \mathbb{N}$ we have

$(k \gt n) \Rightarrow (H_k(X_n) \simeq 0) \,.$

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

$(k \gt dim X) \Rightarrow (H_k(X) \simeq 0).$

Moreover, for $k \lt n$ the inclusion

$H_k(X_n) \stackrel{\simeq}{\to} H_k(X)$

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

$image(H_n(X_n) \to H_n(X)) \simeq H_n(X) \,.$

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

$H_{k+1}(X_n , X_{n-1}) \to H_k(X_{n-1}) \to H_k(X_n) \to H_k(X_n, X_{n-1}) \,.$

Now by prop. 6 the leftmost and rightmost homology groups here vanish when $k \neq n$ and $k \neq n-1$ and hence exactness implies that

$H_k(X_{n-1}) \stackrel{\simeq}{\to} H_k(X_n)$

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

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

$H_{n-1+1}(X_n , X_{n-1}) \to H_{n-1}(X_{n-1}) \to H_{n-1}(X_n) \to 0$

and hence that with the above the map $H_{n-1}(X_{n-1}) \to H_{n-1}(X)$ is surjective.

References

Basic textbook accounts include

• Brayton Gray, Homotopy Theory: An Introduction to Algebraic Topology, Academic Press, New York (1975).

• George Whitehead, chapter II of Elements of homotopy theory, 1978

• Peter May, A Concise Course in Algebraic Topology, U. Chicago Press (1999)

• Allen Hatcher, Topology of cell complexes (pdf) in Algebraic Topology

• Alan Hatcher, Vector bundles & K-theory (web)

• Rudolf Fritsch and Renzo A. Piccinini, Cellular structures in topology, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (pdf)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 5.1 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

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