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CW complex

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

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Extra stuff, structure, properties

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topological homotopy theory

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homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

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Homotopy groups

Basic facts

Theorems

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.

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.

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.

Remark

(origin of the “CW” 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

In the following let Top be the category of topological spaces, or any of its variants, convenient category of topological spaces.

Definition

(spheres and disks)

For nn \in \mathbb{N} write

Notice that

  • S 1=S^{-1} = \emptyset;

  • S 0=**S^0 = \ast \sqcup \ast.

Definition

(single cell attachment)

For XX any topological space and for 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. 1):

Let

ϕ:S n1X \phi \;\colon\; S^{n-1} \longrightarrow X

be a continuous function, then the attaching space

X ϕD nTop 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 n1 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 n1D 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 n1S^{n-1}. The result is the (n+1)(n+1)-sphere:

S n1 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. 2 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 i1ϕ iX} iI\{S^{n_i-1} \overset{\phi_i}{\longrightarrow} X\}_{i \in I} (as in def. 2), 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) iI:iIS n i1X. (\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:

iIS n i1 (ϕ i) iI X (po) iID n X (ϕ i) iI(iID n). \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 &\longrightarrow& X \cup_{(\phi_i)_{i \in I}} \left(\underset{i \in I}{\sqcup} D^n\right) } \,.

Apart from attaching a set of cells all at once to a fixed base space, we may “attach cells to cells” in that after forming a given cell attachment, then we further attach cells to the resulting attaching space, and ever so on:

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:XY f \colon X \longrightarrow Y

and a sequential diagram of topological space of the form

X=X 0X 1X 2X 3 X = X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow X_3 \hookrightarrow \cdots

such that

  1. each X kX k+1X_k \hookrightarrow X_{k+1} is exhibited as a cell attachment according to def. 3, hence presented by a pushout diagram of the form

    iIS n i1 (ϕ i) iI X k (po) iID 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=kX kY = \underset{k\in \mathbb{N}}{\cup} X_k is the union of all these cell attachments, and f:XYf \colon X \to Y is the canonical inclusion; or stated more abstractly: the map f:XYf \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=limX \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 is the (k+1)(k+1)-st cell attachment X kX 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:

iIS k (ϕ i) iI X k (po) iID 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.

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.

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 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

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limitsAAAA\phantom{AAAA}colimits
\, point space\,\, empty space \,
\, product topological space \,\, disjoint union topological space \,
\, topological subspace \,\, quotient topological space \,
\, fiber space \,\, space attachment \,
\, mapping cocylinder, mapping cocone \,\, mapping cylinder, mapping cone, mapping telescope \,
\, cell complex, CW-complex \,

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 May 29, 2017 04:48:11 by Urs Schreiber (94.220.14.70)