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CW approximation

Context

Homotopy theory

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

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 lemma 1 iteratively for nn \in \mathbb{N} to produce a sequence with cocone of the form

A f 0 X 0 f 2 X 1 f ϕ 0 ϕ 1 X, \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 nf_n is a relative CW-complex adding cells exactly of dimension nn, 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 0 X 0 f 1 X 1 X^ ϕ X. \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 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 i\phi_i, which is (i>n)(i \gt n)-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 24, 2016 16:54:44 by Urs Schreiber (131.220.184.222)