nLab
CW approximation

Context

Homotopy theory

Topology

Contents

Idea

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

Such CW-approximation may be constructed case-by-case by iteratively attaching (starting from the empty space) cells for each representative of a homotopy group of XX 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 QuillenTop_{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 QuillenSing||sSet QuillenTop_{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|SingX| X \mapsto {\vert Sing X\vert}

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

|SingX|WX {\vert Sing X\vert} \overset{\in W}{\longrightarrow} X

a weak homotopy equivalence.

Statement

For topological spaces

Proposition

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

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

such that ϕ\phi is n-connected.

Moreover:

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

  • if AA is already a CW-complex, then f^:AX\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:AXf \colon A \longrightarrow X out of a CW-complex AA, there exists a relative CW-complex f^:AX^\hat f \colon A \longrightarrow \hat X that factors ff followed by a weak homotopy equivalence

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

Apply prop. 1 iteratively for all nn to produce a sequence with cocone of the form

A f 1 X 1 f 2 X 2 f ϕ 1 ϕ 2 X, \array{ A &\overset{f_1}{\longrightarrow}& X_1 &\overset{f_2}{\longrightarrow}& X_2 &\longrightarrow & \cdots \\ &{}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{\phi_1}} & \swarrow_{\mathrlap{\phi_2}} & \cdots \\ && X } \,,

where each f if_i is a relative CW-complex and where ϕ n\phi_n in n-connected.

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

By the universal property of the colimit this factors ff as

A f 1 X 1 f 2 X 2 X^ ϕ X. \array{ A &\overset{f_1}{\longrightarrow}& X_1 &\overset{f_2}{\longrightarrow}& X_2 &\longrightarrow & \cdots \\ &{}_{\mathllap{}}\searrow & \downarrow^{\mathrlap{}} & \swarrow_{\mathrlap{}} & \cdots \\ && \hat X \\ && \downarrow^{\mathrlap{\phi}} \\ && X } \,.

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

For sequential topological spectra

Proposition

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

Proof

First let X^ 0X 0\hat X_0 \longrightarrow X_0 be a CW-approximation of the component space in degree 0, via prop. 2. Then proceed by induction: suppose that for nn \in \mathbb{N} a CW-approximation ϕ kn:X^ knX kn\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

ΣX^ nΣϕ nΣX nσ nX n+1. \Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1} \,.

Applying prop. 2 to this function factors it as

ΣX nX^ n+1ϕ n+1X n+1. \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:

ΣX^ n Σϕ n ΣX n incl σ n X^ n+1 ϕ n+1 X n+1. \array{ \Sigma \hat X_n &\overset{\Sigma \phi_n}{\longrightarrow}& \Sigma X_n \\ {}^{incl}\downarrow && \downarrow^{\mathrlap{\sigma_n}} \\ \hat X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1} } \,.

References

Revised on May 4, 2016 14:42:36 by Urs Schreiber (131.220.184.222)