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\begin{document}

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\section*{CW approximation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{for_topological_spaces}{For topological spaces}\dotfill \pageref*{for_topological_spaces} \linebreak
\noindent\hyperlink{for_sequential_topological_spectra}{For sequential topological spectra}\dotfill \pageref*{for_sequential_topological_spectra} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 \textbf{CW approximation} to $X$.

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 $X$ and further cells to kill off spurious homotopy groups introduced this way (e.g. \hyperlink{Hatcher}{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 \emph{functorial} CW approximation (by \href{geometric+realization#mono}{this proposition}) via

\begin{displaymath}
X \mapsto {\vert Sing X\vert}
\end{displaymath}
([[geometric realization]] of the [[singular simplicial complex]] of $X$) with the [[adjunction counit]]

\begin{displaymath}
{\vert Sing X\vert}
  \overset{\in W}{\longrightarrow}
  X
\end{displaymath}
a weak homotopy equivalence.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\hypertarget{for_topological_spaces}{}\subsubsection*{{For topological spaces}}\label{for_topological_spaces}

\begin{prop}
\label{nConnectedCWApproximationOfContinuousFunction}\hypertarget{nConnectedCWApproximationOfContinuousFunction}{}
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.

\begin{displaymath}
\itexarray{
    A &\overset{f}{\longrightarrow}& X
    \\
    {}^{\mathllap{\hat f}}\downarrow & \nearrow_{\mathrlap{\phi}}
    \\
    \hat X
  }
\end{displaymath}
such that $\phi$ is [[n-connected continuous function|n-connected]].

Moreover:

\begin{itemize}%
\item if $f$ itself is [[n-connected continuous function|k-connected]], then the relative CW-complex $\hat f$ may be chosen to have cells only of [[dimension]] $k + 1 \leq dim \leq n$.


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



\end{itemize}
\end{prop}
(\hyperlink{tomDieck08}{tomDieck 08, theorem 8.6.1})

\begin{prop}
\label{CWApproximationForContinuousFunctions}\hypertarget{CWApproximationForContinuousFunctions}{}
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]]

\begin{displaymath}
\itexarray{   
     A && \overset{f}{\longrightarrow} && X
     \\
     & {}_{\mathllap{\hat f}}\searrow && \nearrow_{\mathrlap{{\phi} \atop {\in WHE}}}
     \\
     && \hat X
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Apply lemma \ref{nConnectedCWApproximationOfContinuousFunction} iteratively for $n \in \mathbb{N}$ to produce a sequence with [[cocone]] of the form

\begin{displaymath}
\itexarray{
    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
  }
  \,,
\end{displaymath}
where each $f_n$ is a [[relative CW-complex]] adding cells exactly of dimension $n$, and where $\phi_n$ in [[n-connected continuous function|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

\begin{displaymath}
\itexarray{
    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
  }
  \,.
\end{displaymath}
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 \href{Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes}{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 $\alpha$.

\end{proof}
\hypertarget{for_sequential_topological_spectra}{}\subsubsection*{{For sequential topological spectra}}\label{for_sequential_topological_spectra}

\begin{prop}
\label{CWApproximationForSequentialSpectra}\hypertarget{CWApproximationForSequentialSpectra}{}
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]].

\end{prop}
\begin{proof}
First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via prop. \ref{CWApproximationForContinuousFunctions}. 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

\begin{displaymath}
\Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1}
  \,.
\end{displaymath}
Applying prop. \ref{CWApproximationForContinuousFunctions} to this function factors it as

\begin{displaymath}
\Sigma X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1}
  \,.
\end{displaymath}
Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:

\begin{displaymath}
\itexarray{  
    \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}
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Whitehead's theorem]]


\item \href{CW-spectrum#CWApproximation}{CW-approximation for sequential spectra}



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Tammo tom Dieck]], section 8.6 of \emph{Algebraic topology}, EMS (2008)


\item [[Allen Hatcher]], \emph{Algebraic topology}, Cambridge Univ. Press 2002; Chapter 4, \href{http://www.math.cornell.edu/~hatcher/AT/AT-CWapprox.pdf}{Section 4.1 pdf}



\end{itemize}
[[!redirects CW-approximation]]

[[!redirects CW approximations]] [[!redirects CW-approximations]]



\end{document}