nLab
CW complex

Context

Topology

Homotopy theory

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 nD^n along their boundary (n-1)-spheres S n1S^{n-1} for all nn \in \mathbb{N}: a cell complex built from the basic topological cells S n1D nS^{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 nn-disk.)

Definition

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

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

and a cocone making XX 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 nX_n (called the nn-skeleton of XX) is the result of attaching copies of the nn-disk D n={x n:||x||1}D^n = \{x \in \mathbb{R}^n: ||x|| \leq 1\} along their boundaries S n1=D nS^{n-1} = \partial D^n to X n1X_{n-1}. Specifically, X 1X_{-1} is the empty space, and each X nX_n is a pushout in Top of a diagram of the form

X n1(f i) iIS i n1 ij i iID i nX_{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 II is some index set, each j i:S i n1D i nj_i: S_{i}^{n-1} \to D_{i}^n is the boundary inclusion of a copy of D nD^n, and f i:S i n1X n1f_i: S_{i}^{n-1} \to X_{n-1} is a continuous map, often called an attaching map. The coprojections X n1X nX_{n-1} \to X_n of these pushouts give the arrows on which diagram (1) is based.

A relative CW-complex (X,A)(X, A) is defined as above, except X 1=AX_{-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 nX_n is obtained from X n1X_{n-1} by attaching cells of dimension nn, instead of cells of arbitrary dimension.

A cellular map between CW-complexes XX and YY is a continuous function f:XYf\colon X \to Y such that f(X n)Y nf(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 AXA \hookrightarrow X is an inclusion of CW-complexes, then the quotient X/AX/A is naturally itself a CW-complex, such that the quotient map XX/AX \to X/A is cellular.

If XX is a CW-complex and KK is a finite CW-complex, then the product topological space X×KX \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 1XS^1 \wedge X of a pointed CW-complex XX is itself a CW-complex.

Proposition

For XX and YY CW-complexes with attaching maps {ϕ α}\{\phi_\alpha\} and {Ψ β}\{\Psi_\beta\}, then the k-ification (X×Y) c(X \times Y)_c of their product topological space X×YX \times Y (hence their Cartesian product in the category of compactly generated topological spaces) is again a CW-complex with attaching maps {Φ α×Ψ β}\{\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×Y) cX×Y (X\times Y)_c \simeq X \times Y

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

(Hatcher, theorem A.6)

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 XX is a colimit in Top over attachments of standard n-disks D n iD^{n_i} (its cells), by the characterization of colimits in TopTop (prop.) a subset of XX is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the nn-disks are compact, this implies one direction: if a subset AA of XX intersected with all compact subsets is closed, then AA 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 YY has the homotopy type of a CW complex and XX is a finite CW complex, then the mapping space Y XY^X with the compact-open topology has the homotopy type of a CW complex.

(Milnor 59)

Subcomplexes

Proposition

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

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

Remark

For AXA \hookrightarrow X the inclusion of a subcomplex into a CW complex, then the pair (X,A)(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():H_n(-) \colon Top \to Ab of CW-complexes. See also at cellular homology of CW-complexes.

Let XX be a CW-complex and write

X 0X 1X 2X X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X

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

Proposition

The relative singular homology of the filtering degrees is

H n(X k,X k1){[nCells] ifk=n 0 otherwise, H_n(X_k , X_{k-1}) \simeq \left\{ \array{ \mathbb{Z}[nCells] & if\; k = n \\ 0 & otherwise } \right. \,,

where [nCells]\mathbb{Z}[nCells] denotes the free abelian group on the set of nn-cells.

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

Proposition

With k,nk,n \in \mathbb{N} we have

(k>n)(H k(X n)0). (k \gt n) \Rightarrow (H_k(X_n) \simeq 0) \,.

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

(k>dimX)(H k(X)0). (k \gt dim X) \Rightarrow (H_k(X) \simeq 0).

Moreover, for k<nk \lt n the inclusion

H k(X n)H k(X) H_k(X_n) \stackrel{\simeq}{\to} H_k(X)

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

image(H n(X n)H n(X))H n(X). 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 n1)H k(X n1)H k(X n)H k(X n,X n1). 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 knk \neq n and kn1k \neq n-1 and hence exactness implies that

H k(X n1)H k(X n) H_k(X_{n-1}) \stackrel{\simeq}{\to} H_k(X_n)

is an isomorphism for kn,n1k \neq n,n-1. This implies the first claims by induction on nn.

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

H n1+1(X n,X n1)H n1(X n1)H n1(X n)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 n1(X n1)H n1(X)H_{n-1}(X_{n-1}) \to H_{n-1}(X) is surjective.

Examples

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

See also

An inconclusive discussion here about what parts of the definition a CW complex should be properties and what parts should be structure.

Revised on July 8, 2016 03:58:12 by Urs Schreiber (131.220.184.222)