nLab CW approximation

Idea

For every topological space $X$ there is a CW complex $Z$ and a weak homotopy equivalence $f \colon Z\to X$. Such a map $f \colon Z\to X$ is called a CW approximation to $X$.

Such CW-approximation may be constructed case-by-case by iteratively attaching cells] (starting from the [[empty space]) for each representative of a homotopy group of $X$ and further cells to kill off spurious homotopy groups introduced this way (e.g. Hatcher, p. 352-353).

In the classical model structure on topological spaces $Top_{Quillen}$, the cofibrant objects are the retracts of cell complexes, and hence CW approximations are in particular cofibrant replacements in this model structure.

The Quillen equivalence $Top_{Quillen} \stackrel{\overset{{\vert - \vert}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}} sSet_{Quillen}$ to the classical model structure on simplicial sets (“homotopy hypothesis”) yields a functorial CW approximation (by this proposition) via

$$X \mapsto {\vert Sing X\vert}$$

(geometric realization of the singular simplicial complex of $X$) with the adjunction counit

$${\vert Sing X\vert} \overset{\in W}{\longrightarrow} X$$

a weak homotopy equivalence.

Statement

For topological spaces

Proposition

Let $f \;\colon\; A \longrightarrow X$ be a continuous function between topological spaces. Then there exists for each $n \in \mathbb{N}$ a relative CW-complex $\hat f \colon A \hookrightarrow \hat X$ together with an extension $\phi \colon \hat X \to X$, i.e.

$$\array{ A &\overset{f}{\longrightarrow}& X \\ {}^{\mathllap{\hat f}}\downarrow & \nearrow_{\mathrlap{\phi}} \\ \hat X }$$

such that $\phi$ is n-connected.

Moreover:

• if $f$ itself is k-connected, then the relative CW-complex $\hat f$ may be chosen to have cells only of dimension $k + 1 \leq dim \leq n$.

• if $A$ is already a CW-complex, then $\hat f \colon A \to X$ may be chosen to be a subcomplex inclusion.

(tomDieck 08, theorem 8.6.1)

Proposition

For every continuous function $f \colon A \longrightarrow X$ out of a CW-complex $A$, there exists a relative CW-complex $\hat f \colon A \longrightarrow \hat X$ that factors $f$ followed by a weak homotopy equivalence

$$\array{ A && \overset{f}{\longrightarrow} && X \\ & {}_{\mathllap{\hat f}}\searrow && \nearrow_{\mathrlap{{\phi} \atop {\in WHE}}} \\ && \hat X } \,.$$

Proof

Apply lemma iteratively for $n \in \mathbb{N}$ to produce a sequence with cocone of the form

$$\array{ A &\overset{f_0}{\longrightarrow}& X_0 &\overset{f_2}{\longrightarrow}& X_1 &\longrightarrow & \cdots \\ &{}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{\phi_0}} & \swarrow_{\mathrlap{\phi_1}} & \cdots \\ && X } \,,$$

where each $f_n$ is a relative CW-complex adding cells exactly of dimension $n$, and where $\phi_n$ in n-connected.

Let then $\hat X$ be the colimit over the sequence (its transfinite composition) and $\hat f \colon A \to X$ the induced component map. By definition of relative CW-complexes, this $\hat f$ is itself a relative CW-complex.

By the universal property of the colimit this factors $f$ as

$$\array{ A &\overset{f_0}{\longrightarrow}& X_0 &\overset{f_1}{\longrightarrow}& X_1 &\longrightarrow & \cdots \\ &{}_{\mathllap{}}\searrow & \downarrow^{\mathrlap{}} & \swarrow_{\mathrlap{}} & \cdots \\ && \hat X \\ && \downarrow^{\mathrlap{\phi}} \\ && X } \,.$$

Finally to see that $\phi$ is a weak homotopy equivalence: since n-spheres are compact topological spaces, then every map $\alpha \colon S^n \to \hat X$ factors through a finite stage $i \in \mathbb{N}$ as $S^n \to X_i \to \hat X$ (by this lemma). By possibly including further into higher stages, we may choose $i \gt n$. But then the above says that further mapping along $\hat X \to X$ is the same as mapping along $\phi_i$, which is $(i \gt n)$-connected and hence an isomorphism on the homotopy class of $\alpha$.

For sequential topological spectra

Proposition

For $X$ any sequential spectrum in Top, then there exists a CW-spectrum $\hat X$ and a homomorphism $\phi \colon \hat X \to X$ which is degreewise a weak homotopy equivalence, hence in particular a stable weak homotopy equivalence.

Proof

First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via prop. . Then proceed by induction: suppose that for $n \in \mathbb{N}$ a CW-approximation $\phi_{k \leq n} \colon \hat X_{k \leq n} \to X_{k \leq n}$ has been found such that all the structure maps are respected. Consider then the continuous function

$$\Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1} \,.$$

Applying prop. to this function factors it as

$$\Sigma X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1} \,.$$

Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:

$$\array{ \Sigma \hat X_n &\overset{\Sigma \phi_n}{\longrightarrow}& \Sigma X_n \\ {}^{incl}\big\downarrow && \big\downarrow^{\mathrlap{\sigma_n}} \\ \hat X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1} } \,.$$

Related concepts

• in stable homotopy theory: CW-approximation for sequential spectra

• in equivariant homotopy theory: G-CW approximation

• Whitehead's theorem

• cellular approximation theorem

References

• Tammo tom Dieck, section 8.6 of: Algebraic topology, EMS (2008)

• Allen Hatcher, Algebraic topology, Cambridge Univ. Press 2002; Chapter 4, Section 4.1 pdf