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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{CW complex} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{cw_complexes}{}\section*{{CW complexes}}\label{cw_complexes} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{closure_properties}{Closure properties}\dotfill \pageref*{closure_properties} \linebreak \noindent\hyperlink{LocalContractibility}{Local contractibility}\dotfill \pageref*{LocalContractibility} \linebreak \noindent\hyperlink{compactness_properties}{Compactness properties}\dotfill \pageref*{compactness_properties} \linebreak \noindent\hyperlink{up_to_homotopy_equivalence}{Up to homotopy equivalence}\dotfill \pageref*{up_to_homotopy_equivalence} \linebreak \noindent\hyperlink{Subcomplexes}{Subcomplexes}\dotfill \pageref*{Subcomplexes} \linebreak \noindent\hyperlink{fibrations}{Fibrations}\dotfill \pageref*{fibrations} \linebreak \noindent\hyperlink{SingularHomology}{Singular homology}\dotfill \pageref*{SingularHomology} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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} A \textbf{CW-complex} is a [[nice topological space|nice]] [[topological space]] which is, or can be, built up inductively, by a process of [[attaching space|attaching]] [[n-disks]] $D^n$ along their [[boundary]] [[n-spheres|(n-1)-spheres]] $S^{n-1}$ for all $n \in \mathbb{N}$: a [[cell complex]] built from the basic topological cells $S^{n-1} \hookrightarrow D^n$. 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 equivalence|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 [[nice category of spaces|convenient category of spaces]] for [[algebraic topology]]. Also, CW complexes are among the [[cofibrant objects]] in the [[classical model structure on topological spaces]]. In fact, \emph{every} topological space is \emph{[[weak homotopy equivalence|weakly homotopy equivalent]]} to a CW-complex (but need not be [[homotopy equivalence|strongly homotopy equivalent]] to one). See also at \emph{[[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 \emph{[[homotopy theory]]} and \emph{[[homotopy hypothesis]]} for more on this. \begin{remark} \label{}\hypertarget{}{} \textbf{(origin of the ``CW'' terminology)} The terminology ``CW-complex'' goes back to [[John Henry Constantine Whitehead]] (and see the discussion in \hyperlink{HatcherTopologyOfCellComplexes}{Hatcher, ``Topology of cell complexes'', p. 520}). To quote from the original paper, which was ``an address delivered before the Princeton Meeting of the (American Mathematical) Society on November 2, 1946'', Whitehead states: \begin{quote}% In this presentation we abandon [[simplicial complexes]] in favor of [[cell complexes]]. This first part consists of geometrical preliminaries, including some elementary propositions concerning what we call closure finite complexes with weak topology, abbreviated to CW-complexes, \ldots{} \end{quote} Thus the ``CW'' stands for the following two properties shared by any CW-complex: \begin{itemize}% \item \textbf{C} = ``closure finiteness'': a [[compact subset]] of a CW-complex intersects the [[interior]] of only finitely many cells (\href{classical+model+structure+on+topological+spaces#CompactSubsetsAreSmallInCellComplexes}{prop.}), hence in particular so does the closure of any cell. \item \textbf{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. \end{itemize} (Whitehead called the [[interior]] of the [[n-disks]] the ``cells'', so that their closure of each cell is the corresponding $n$-disk.) \end{remark} $\backslash$linebreak \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} In the following let [[Top]] be the [[category]] of [[topological spaces]], or any of its variants, [[convenient category of topological spaces]]. \begin{defn} \label{SpheresAndDisks}\hypertarget{SpheresAndDisks}{} \textbf{(spheres and disks)} For $n \in \mathbb{N}$ write \begin{itemize}% \item $D^n \in Top$ for the [[n-disk]], for instance realized (up to [[homeomorphism]]) as the [[closed ball|closed unit ball]] in the $n$-dimensional [[Euclidean space]] $\mathbb{R}^n$ and equipped with the induced [[subspace topology]]; \item $S^{n-1} \in Top$ for the [[n-sphere|(n-1)-sphere]], for instance realized as the [[boundary]] of the [[n-disk]], also equipped with the corresponding [[subspace topology]]; \item $i_n \;\colon\; S^{n-1} \hookrightarrow D^n$ for the [[continuous function]] that exhibits this [[boundary]] inclusion. We also call these functions the \emph{generating cofibrations} (of the [[classical model structure on topological spaces]]). \end{itemize} Notice that \begin{itemize}% \item $S^{-1} = \emptyset$; \item $S^0 = \ast \sqcup \ast$. \end{itemize} \end{defn} \begin{defn} \label{SingleCellAttachment}\hypertarget{SingleCellAttachment}{} \textbf{(single cell attachment)} For $X$ any [[topological space]] and for $n \in \mathbb{N}$, then an \textbf{$n$-cell [[attaching space|attachment]]} to $X$ is the result of gluing an [[n-disk]] to $X$, along a prescribed image of its bounding [[n-sphere|(n-1)-sphere]] (def. \ref{SpheresAndDisks}): Let \begin{displaymath} \phi \;\colon\; S^{n-1} \longrightarrow X \end{displaymath} be a [[continuous function]], then the \emph{[[attaching space]]} \begin{displaymath} X \cup_\phi D^n \,\in Top \end{displaymath} is the topological space which is the [[pushout]] of the boundary inclusion of the $n$-sphere along $\phi$, hence the universal space that makes the following [[commuting diagram|diagram commute]] \begin{displaymath} \itexarray{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& X \cup_\phi D^n } \,. \end{displaymath} \end{defn} \begin{example} \label{}\hypertarget{}{} If we take the defining boundary inclusion $\iota_n \colon S^{n-1} \to D^n$ itself as an attaching map, then we are gluing two $n$-disks to each other along their common boundary $S^{n-1}$. The result is the $(n+1)$-sphere: \begin{displaymath} \itexarray{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. \end{displaymath} (graphics from Ueno-Shiga-Morita 95) \end{example} \begin{example} \label{}\hypertarget{}{} A single cell [[attaching space|attachment]] of a 0-cell, according to def. \ref{SingleCellAttachment} is the same as forming the [[disjoint union space]] $X \sqcup \ast$ with the [[point]] space $\ast$: \begin{displaymath} \itexarray{ (S^{-1} = \emptyset) &\overset{\exists !}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ (D^0 = \ast) &\longrightarrow& X \sqcup \ast } \,. \end{displaymath} In particular if we start with the [[empty topological space]] $X = \emptyset$ itself, then by [[attaching space|attaching]] 0-cells we obtain a [[discrete topological space]]. To this then we may attach higher dimensional cells. \end{example} \begin{defn} \label{CellAttachments}\hypertarget{CellAttachments}{} \textbf{(attaching many cells at once)} If we have a [[set]] of attaching maps $\{S^{n_i-1} \overset{\phi_i}{\longrightarrow} X\}_{i \in I}$ (as in def. \ref{SingleCellAttachment}), all to the same space $X$, we may think of these as one single continuous function out of the [[disjoint union space]] of their [[domain]] spheres \begin{displaymath} (\phi_i)_{i \in I} \;\colon\; \underset{i \in I}{\sqcup} S^{n_i-1} \longrightarrow X \,. \end{displaymath} Then the result of attaching \emph{all} the corresponding $n$-cells to $X$ is the pushout of the corresponding [[disjoint union]] of boundary inclusions: \begin{displaymath} \itexarray{ \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) } \,. \end{displaymath} \end{defn} 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: \begin{defn} \label{RelativeCellComplexes}\hypertarget{RelativeCellComplexes}{} \textbf{([[relative cell complexes]])} Let $X$ be a topological space, then a \emph{topological [[relative cell complex]]} of countable height based on $X$ is a [[continuous function]] \begin{displaymath} f \colon X \longrightarrow Y \end{displaymath} and a [[sequential diagram]] of [[topological space]] of the form \begin{displaymath} X = X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow X_3 \hookrightarrow \cdots \end{displaymath} such that \begin{enumerate}% \item each $X_k \hookrightarrow X_{k+1}$ is exhibited as a cell attachment according to def. \ref{CellAttachments}, hence presented by a [[pushout]] diagram of the form \begin{displaymath} \itexarray{ \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} } \,. \end{displaymath} \item $Y = \underset{k\in \mathbb{N}}{\cup} X_k$ is the [[union]] of all these cell attachments, and $f \colon X \to Y$ is the canonical inclusion; or stated more abstractly: the map $f \colon X \to Y$ is the inclusion of the first component of the diagram into its [[colimit|colimiting cocone]] $\underset{\longrightarrow}{\lim}_k X_k$: \begin{displaymath} \itexarray{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet } \end{displaymath} \end{enumerate} If here $X = \emptyset$ is the [[empty space]] then the result is a map $\emptyset \hookrightarrow Y$, which is equivalently just a space $Y$ built form ``attaching cells to nothing''. This is then called just a \emph{topological [[cell complex]]} of countable hight. Finally, a topological (relative) cell complex of countable hight is called a \textbf{CW-complex} if the $(k+1)$-st cell attachment $X_k \to X_{k+1}$ is entirely by $(k+1)$-cells, hence exhibited specifically by a pushout of the following form: \begin{displaymath} \itexarray{ \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} } \,. \end{displaymath} A \emph{[[finite CW-complex]]} is one which admits a presentation in which there are only finitely many attaching maps, and similarly a \emph{countable CW-complex} is one which admits a presentation with countably many attaching maps. Given a CW-complex, then $X_n$ is also called its $n$-[[skeleton]]. \end{defn} A \textbf{cellular map} between CW-complexes $X$ and $Y$ is a [[continuous function]] $f\colon X \to Y$ such that $f(X_n) \subset Y_n$. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{closure_properties}{}\subsubsection*{{Closure properties}}\label{closure_properties} If $A \hookrightarrow X$ is an inclusion of CW-complexes, then the quotient $X/A$ is naturally itself a CW-complex, such that the quotient map $X \to X/A$ is cellular. If $X$ is a CW-complex and $K$ is a [[finite CW-complex]], then the [[product topological space]] $X \times K$ 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 topological space|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]] $S^1 \wedge X$ of a pointed CW-complex $X$ is itself a CW-complex. \begin{prop} \label{ClosureOfCWComplexesUnderCartesianProduct}\hypertarget{ClosureOfCWComplexesUnderCartesianProduct}{} For $X$ and $Y$ [[CW-complexes]] with attaching maps $\{\phi_\alpha\}$ and $\{\Psi_\beta\}$, then the [[compactly generated topological space|k-ification]] $(X \times Y)_c$ of their [[product topological space]] $X \times Y$ (hence their Cartesian product in the category of [[compactly generated topological spaces]]) is again a CW-complex with attaching maps $\{\Phi_\alpha \times \Psi_\beta\}$. 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 \begin{displaymath} (X\times Y)_c \simeq X \times Y \end{displaymath} and so then the [[product topological space]] $X \times Y$ itself has CW-complex structure. \end{prop} (\hyperlink{Hatcher}{Hatcher, theorem A.6}) \hypertarget{LocalContractibility}{}\subsubsection*{{Local contractibility}}\label{LocalContractibility} \begin{prop} \label{}\hypertarget{}{} A CW-complex is a [[locally contractible topological space]]. \end{prop} For instance (\hyperlink{Hatcher}{Hatcher, prop. A.4}). \hypertarget{compactness_properties}{}\subsubsection*{{Compactness properties}}\label{compactness_properties} \begin{prop} \label{}\hypertarget{}{} \textbf{([[CW-complexes are paracompact Hausdorff spaces]])} Every CW-complex is a \begin{enumerate}% \item a [[normal topological space]], in particular a [[Hausdorff space]], \item a [[paracompact topological space]]. \end{enumerate} \end{prop} \begin{prop} \label{CWComplexIsCompactlyGenerated}\hypertarget{CWComplexIsCompactlyGenerated}{} Every CW-complex is a [[compactly generated topological space]]. \end{prop} \begin{proof} Since a CW-complex $X$ is a [[colimit]] in [[Top]] over attachments of standard [[n-disks]] $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ (\href{Top#DescriptionOfLimitsAndColimitsInTop}{prop.}) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ 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. \end{proof} \hypertarget{up_to_homotopy_equivalence}{}\subsubsection*{{Up to homotopy equivalence}}\label{up_to_homotopy_equivalence} \begin{theorem} \label{}\hypertarget{}{} Every CW complex is homotopy equivalent to (the \href{simplicial+complex#geometric_realisations_and_polyhedra}{realization} of) a [[simplicial complex]]. \end{theorem} See \hyperlink{Gray}{Gray}, Corollary 16.44 (p. 149) and Corollary 21.15 (p. 206). For more see at \emph{[[CW approximation]]}. \begin{cor} \label{}\hypertarget{}{} Every CW complex is homotopy equivalent to a space that admits a [[good open cover]]. \end{cor} \begin{theorem} \label{}\hypertarget{}{} If $Y$ has the [[homotopy type]] of a CW complex and $X$ is a [[finite CW complex]], then the [[mapping space]] $Y^X$ with the [[compact-open topology]] has the homotopy type of a CW complex. \end{theorem} (\hyperlink{Milnor59}{Milnor 59}) \hypertarget{Subcomplexes}{}\subsubsection*{{Subcomplexes}}\label{Subcomplexes} \begin{prop} \label{}\hypertarget{}{} For $X$ a CW complex, the inclusion $X' \hookrightarrow X$ of any subcomplex has an [[open neighbourhood]] in $X$ which is a [[deformation retract]] of $X'$. In particular such an inclusion is a \emph{\href{relative+homology#GoodPair}{good pair}} in the sense of [[relative homology]]. \end{prop} For instance (\hyperlink{Hatcher}{Hatcher, prop. A.5}). \begin{remark} \label{}\hypertarget{}{} For $A \hookrightarrow X$ the inclusion of a subcomplex into a CW complex, then the pair $(X,A)$ is often called a \emph{[[CW-pair]]}. This appears notably in the [[axioms]] for [[generalized (Eilenberg-Steenrod) cohomology]]. \end{remark} e.g. (\hyperlink{AGP02}{AGP 02, def. 5.1.11}) \hypertarget{fibrations}{}\subsubsection*{{Fibrations}}\label{fibrations} Fibrations between CW-complexes also behave particularly well: [[a Serre fibration between CW-complexes is a Hurewicz fibration]]. \hypertarget{SingularHomology}{}\subsubsection*{{Singular homology}}\label{SingularHomology} We discuss aspects of the [[singular homology]] $H_n(-) \colon$ [[Top]] $\to$ [[Ab]] of CW-complexes. See also at \emph{[[cellular homology]] of CW-complexes}. Let $X$ be a CW-complex and write \begin{displaymath} X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X \end{displaymath} for its [[filtered topological space]]-structure with $X_{n+1}$ the topological space obtained from $X_n$ by gluing on $(n+1)$-cells. For $n \in \mathbb{N}$ write $nCells \in Set$ for the set of $n$-cells of $X$. \begin{prop} \label{SkeletalRelativeSingularHomologyOfCW}\hypertarget{SkeletalRelativeSingularHomologyOfCW}{} The [[relative singular homology]] of the filtering degrees is \begin{displaymath} H_n(X_k , X_{k-1}) \simeq \left\{ \itexarray{ \mathbb{Z}[nCells] & if\; k = n \\ 0 & otherwise } \right. \,, \end{displaymath} where $\mathbb{Z}[nCells]$ denotes the [[free abelian group]] on the set of $n$-cells. \end{prop} The proof is spelled out at \emph{\href{relative+homology#RelativeHomologyOfCWComplexes}{Relative singular homology - Of CW complexes}}. \begin{prop} \label{RelativeHomologyOfFilterStep}\hypertarget{RelativeHomologyOfFilterStep}{} With $k,n \in \mathbb{N}$ we have \begin{displaymath} (k \gt n) \Rightarrow (H_k(X_n) \simeq 0) \,. \end{displaymath} In particular if $X$ is a CW-complex of [[finite number|finite]] [[dimension of a CW-complex]] $dim X$ (the maximum degree of cells), then \begin{displaymath} (k \gt dim X) \Rightarrow (H_k(X) \simeq 0). \end{displaymath} Moreover, for $k \lt n$ the inclusion \begin{displaymath} H_k(X_n) \stackrel{\simeq}{\to} H_k(X) \end{displaymath} is an [[isomorphism]] and for $k = n$ we have an isomorphism \begin{displaymath} image(H_n(X_n) \to H_n(X)) \simeq H_n(X) \,. \end{displaymath} \end{prop} This is mostly for instance in (\hyperlink{Hatcher}{Hatcher, lemma 2.34 b),c)}). \begin{proof} By the [[long exact sequence]] in [[relative homology]], discussed at \emph{\href{relative+homology#LongExactSequences}{Relative homology -- long exact sequences}}, we have an [[exact sequence]] \begin{displaymath} 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}) \,. \end{displaymath} Now by prop. \ref{SkeletalRelativeSingularHomologyOfCW} the leftmost and rightmost homology groups here vanish when $k \neq n$ and $k \neq n-1$ and hence exactness implies that \begin{displaymath} H_k(X_{n-1}) \stackrel{\simeq}{\to} H_k(X_n) \end{displaymath} is an [[isomorphism]] for $k \neq n,n-1$. This implies the first claims by [[induction]] on $n$. Finally for the last claim use that the above exact sequence gives \begin{displaymath} H_{n-1+1}(X_n , X_{n-1}) \to H_{n-1}(X_{n-1}) \to H_{n-1}(X_n) \to 0 \end{displaymath} and hence that with the above the map $H_{n-1}(X_{n-1}) \to H_{n-1}(X)$ is surjective. \end{proof} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item The [[geometric realization]] of any locally finite [[simplicial set]] is a CW-complex (\hyperlink{Milnor57}{Milnor 57}). \begin{itemize}% \item In particular in the context of the [[homotopy hypothesis]] the [[Quillen equivalence]] between [[∞-groupoid]]s and [[nice topological space]]s maps each [[∞-groupoid]] to a CW-complex. \end{itemize} \item any [[compact space|noncompact]] [[smooth manifold]] of [[dimension]] $n$ is [[homotopy equivalence|homotopy equivalent]] to an $(n-1)$-dimensional CW-complex. (\hyperlink{NapierRamachandran}{Napier-Ramachandran}). \item Any undirected [[graph]] (loops and/or multiple edges allowed) has a geometric realization as a 1-dimensional CW complex. \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[dimension of a CW-complex]] \item [[cell complex]] \item [[CW approximation]] \item [[quasi-finite CW-complex]] \item [[cellular homology]] \item [[simplicial set]], [[geometric realization]] \item [[CW-spectrum]] \item [[G-CW complex]] \end{itemize} [[!include universal constructions of topological spaces -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The introduction of the term is contained in \begin{itemize}% \item J. H. C. Whitehead, \emph{Combinatorial homotopy I} , Bull. Amer. Math. Soc, 55, (1949), 213–245. \end{itemize} Basic textbook accounts include \begin{itemize}% \item Brayton Gray, \emph{Homotopy Theory: An Introduction to Algebraic Topology}, Academic Press, New York (1975). \item [[George Whitehead]], chapter II of \emph{Elements of homotopy theory}, 1978 \item [[Peter May]], \emph{[[A Concise Course in Algebraic Topology]]}, U. Chicago Press (1999) \item [[Allen Hatcher]], \emph{\href{http://www.math.cornell.edu/~hatcher/AT/ATpage.html}{Algebraic Topology}}, 2002 \item [[Allen Hatcher]], \emph{Topology of cell complexes} (\href{https://www.math.cornell.edu/~hatcher/AT/ATapp.pdf}{pdf}) in \emph{Algebraic Topology} \item [[Allen Hatcher]], \emph{Vector bundles \& K-theory} (\href{https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web}) \item [[Rudolf Fritsch]], Renzo A. Piccinini, \emph{Cellular structures in topology}, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (\href{https://epub.ub.uni-muenchen.de/4493/1/4493.pdf}{pdf}) \item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 5.1 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf}) \end{itemize} Original articles include \begin{itemize}% \item [[John Milnor]], \emph{On spaces having the homotopy type of a CW-complex}, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280. \item [[John Milnor]], \emph{The geometric realization of a semi-simplicial complex}, Annals of Mathematics, 2nd Ser., \textbf{65}, n. 2. (Mar., 1957), pp. 357-362; \href{http://www.math.binghamton.edu/dwyer/Milnor-GeomReal.pdf}{pdf} \end{itemize} See also \begin{itemize}% \item Terrance Napier, Mohan Ramachandran, \emph{\href{http://www.unige.ch/math/EnsMath/EM_en/}{Elementary Construction of Exhausting Subsolutions of Elliptic Operators}} L'Enseignement Math\'e{}matique, t. 50 (2004), p. 367 - 389. \end{itemize} An inconclusive discussion \href{http://nforum.mathforge.org/discussion/4135/simplicial-homology/?Focus=33785#Comment_33785}{here} about what parts of the definition a CW complex should be [[stuff, structure, property|properties]] and what parts should be structure. [[!redirects CW complex]] [[!redirects CW-complex]] [[!redirects CW complexes]] [[!redirects CW-complexes]] [[!redirects relative CW-complex]] [[!redirects relative CW-complexes]] \end{document}