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CW complex

CW complexes

Idea

A CW-complex is a nice topological space which is or can be built up inductively, by a process of attaching nn-dimensional disks D nD^n along their boundary spheres S n.1S^{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 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 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:x1}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 (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 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).

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. 3 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 November 13, 2013 09:22:35 by Urs Schreiber (82.169.114.243)