CW complex in nLab

Context

Topology

Homotopy theory

• Idea
• Definition
• Properties
• Topological properties
• Products of CW-complexes
• Up to homotopy equivalence
• Subcomplexes
• Cellular approximation theorem
• Fibrations
• Singular homology
• Examples
• Related concepts
• References

Definition

(single cell attachment)

For XX any topological space and for n∈ℕn \in \mathbb{N}, then an nn-cell attachment to XX is the result of gluing an n-disk to XX, along a prescribed image of its bounding (n-1)-sphere (def. ):

Let

ϕ:S n−1⟶X \phi \;\colon\; S^{n-1} \longrightarrow X

be a continuous function, then the attaching space

X∪ ϕD n∈Top X \cup_\phi D^n \,\in Top

is the topological space which is the pushout of the boundary inclusion of the nn-sphere along ϕ\phi, hence the universal space that makes the following diagram commute

S n−1 ⟶ϕ X ι n↓ (po) ↓ D n ⟶ X∪ ϕD n. \array{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& X \cup_\phi D^n } \,.

Example

If we take the defining boundary inclusion ι n:S n−1→D n\iota_n \colon S^{n-1} \to D^n itself as an attaching map, then we are gluing two nn-disks to each other along their common boundary S n−1S^{n-1}. The result is the nn-sphere:

S n−1 ⟶i n D n i n↓ (po) ↓ D n ⟶ S n. \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,.

(graphics from Ueno-Shiga-Morita 95)

Example

A single cell attachment of a 0-cell, according to def. is the same as forming the disjoint union space X⊔*X \sqcup \ast with the point space *\ast:

(S −1=∅) ⟶∃! X ↓ (po) ↓ (D 0=*) ⟶ X⊔*. \array{ (S^{-1} = \emptyset) &\overset{\exists !}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ (D^0 = \ast) &\longrightarrow& X \sqcup \ast } \,.

In particular if we start with the empty topological space X=∅X = \emptyset itself, then by attaching 0-cells we obtain a discrete topological space. To this then we may attach higher dimensional cells.

Definition

(attaching many cells at once)

If we have a set of attaching maps {S n i−1⟶ϕ iX} i∈I\{S^{n_i-1} \overset{\phi_i}{\longrightarrow} X\}_{i \in I} (as in def. ), all to the same space XX, we may think of these as one single continuous function out of the disjoint union space of their domain spheres

(ϕ i) i∈I:⊔i∈IS n i−1⟶X. (\phi_i)_{i \in I} \;\colon\; \underset{i \in I}{\sqcup} S^{n_i-1} \longrightarrow X \,.

Then the result of attaching all the corresponding nn-cells to XX is the pushout of the corresponding disjoint union of boundary inclusions:

⊔i∈IS n i−1 ⟶(ϕ i) i∈I X ↓ (po) ↓ ⊔i∈ID n i ⟶ X∪ (ϕ i) i∈I(⊔i∈ID n i). \array{ \underset{i \in I}{\sqcup} S^{n_i - 1} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X \cup_{(\phi_i)_{i \in I}} \left(\underset{i \in I}{\sqcup} D^{n_i}\right) } \,.

Definition

(relative cell complexes)

Let XX be a topological space, then a topological relative cell complex of countable height based on XX is a continuous function

f:X⟶Y f \colon X \longrightarrow Y

and a sequential diagram of topological space of the form

X=X −1↪X 0↪X 1↪X 2↪⋯ X = X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots

such that

1. each X k↪X k+1X_k \hookrightarrow X_{k+1} is exhibited as a cell attachment according to def. , hence presented by a pushout diagram of the form

   ⊔i∈IS n i−1 ⟶(ϕ i) i∈I X k ↓ (po) ↓ ⊔i∈ID n i ⟶ X k+1. \array{ \underset{i \in I}{\sqcup} S^{n_i - 1} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X_k \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_{k+1} } \,.

2. Y=∪k∈ℕX kY = \underset{k\in \mathbb{N}}{\cup} X_k is the union of all these cell attachments, and f:X→Yf \colon X \to Y is the canonical inclusion; or stated more abstractly: the map f:X→Yf \colon X \to Y is the inclusion of the first component of the diagram into its colimiting cocone lim⟶ kX k\underset{\longrightarrow}{\lim}_k X_k:

   X=X 0 ⟶ X 1 ⟶ X 2 ⟶ ⋯ f↘ ↓ ↙ ⋯ Y=lim⟶X • \array{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet }

If here X=∅X = \emptyset is the empty space then the result is a map ∅↪Y\emptyset \hookrightarrow Y, which is equivalently just a space YY built form “attaching cells to nothing”. This is then called just a topological cell complex of countable hight.

Finally, a topological (relative) cell complex of countable hight is called a CW-complex if the (k+1)(k+1)-st cell attachment X k→X k+1X_k \to X_{k+1} is entirely by (k+1)(k+1)-cells, hence exhibited specifically by a pushout of the following form:

⊔i∈IS k ⟶(ϕ i) i∈I X k ↓ (po) ↓ ⊔i∈ID k+1 ⟶ X k+1. \array{ \underset{i \in I}{\sqcup} S^{k} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X_k \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{k+1} &\longrightarrow& X_{k+1} } \,.

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.

Given a CW-complex, then X nX_n is also called its nn-skeleton.

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.

Corollary

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

Proposition

The relative singular homology of the filtering degrees is

H n(X k,X k−1)≃{ℤ[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 ℤ[nCells]\mathbb{Z}[nCells] denotes the free abelian group on the set of nn-cells.

Proposition

With k,n∈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 of a CW-complex 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) \,.

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)→H k(X n−1)→H k(X n)→H k(X n,X n−1). 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. the leftmost and rightmost homology groups here vanish when k≠nk \neq n and k≠n−1k \neq n-1 and hence exactness implies that

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

is an isomorphism for k≠n,n−1k \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 n−1+1(X n,X n−1)→H n−1(X n−1)→H n−1(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 n−1(X n−1)→H n−1(X)H_{n-1}(X_{n-1}) \to H_{n-1}(X) is surjective.

examples of universal constructions of topological spaces: