+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Topology +--{: .hide} [[!include topology - contents]] =-- #### Homotopy theory +--{: .hide} [[!include homotopy - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea A **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, _every_ topological space is _[[weak homotopy equivalence|weakly homotopy equivalent]]_ to a CW-complex (but need not be [[homotopy equivalence|strongly homotopy equivalent]] to one). See also at _[[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 _[[homotopy theory]]_ and _[[homotopy hypothesis]]_ for more on this. +-- {: .num_remark} ###### Remark **(origin of the "CW" terminology)** The terminology "CW-complex" goes back to [[John Henry Constantine Whitehead]] (and see the discussion in [Hatcher, "Topology of cell complexes", p. 520](#HatcherTopologyOfCellComplexes)). 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: >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, ... Thus the "CW" stands for the following two properties shared by any CW-complex: * **C** = "closure finiteness": a [[compact subset]] of a CW-complex intersects the [[interior]] of only finitely many cells ([prop.](classical+model+structure+on+topological+spaces#CompactSubsetsAreSmallInCellComplexes)), hence in particular so does the closure of any cell. * **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. (Whitehead called the [[interior]] of the [[n-disks]] the "cells", so that their closure of each cell is the corresponding $n$-disk.) =-- \linebreak ## Definition {#Definition} In the following let [[Top]] be the [[category]] of [[topological spaces]], or any of its variants, [[convenient category of topological spaces]]. +-- {: .num_defn #SpheresAndDisks} ###### Definition **(spheres and disks)** For $n \in \mathbb{N}$ write * $D^n \in Top$ 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]]; * $S^{n-1} \in Top$ for the [[n-sphere|(n-1)-sphere]], for instance realized as the [[boundary]] of the [[n-disk]], also equipped with the corresponding [[subspace topology]]; * $i_n \;\colon\; S^{n-1} \hookrightarrow D^n$ for the [[continuous function]] that exhibits this [[boundary]] inclusion. We also call these functions the _generating cofibrations_ (of the [[classical model structure on topological spaces]]). Notice that * $S^{-1} = \emptyset$; * $S^0 = \ast \sqcup \ast$. =-- +-- {: .num_defn #SingleCellAttachment} ###### Definition **(single cell attachment)** For $X$ any [[topological space]] and for $n \in \mathbb{N}$, then an **$n$-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 $$ \phi \;\colon\; S^{n-1} \longrightarrow X $$ be a [[continuous function]], then the _[[attaching space]]_ $$ X \cup_\phi D^n \,\in Top $$ 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]] $$ \array{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& X \cup_\phi D^n } \,. $$ =-- +-- {: .num_example} ###### Example <div style="float:right;margin:0 10px 10px 0;"> <img src="http://ncatlab.org/nlab/files/GluingHemispheres.jpg" width="400"></div> 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 $n$-sphere: $$ \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. $$ (graphics from Ueno-Shiga-Morita 95) =-- +-- {: .num_example } ###### Example 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 $\ast$: $$ \array{ (S^{-1} = \emptyset) &\overset{\exists !}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ (D^0 = \ast) &\longrightarrow& X \sqcup \ast } \,. $$ 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. =-- +-- {: .num_defn #CellAttachments} ###### Definition **(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 $$ (\phi_i)_{i \in I} \;\colon\; \underset{i \in I}{\sqcup} S^{n_i-1} \longrightarrow X \,. $$ Then the result of attaching _all_ the corresponding $n$-cells to $X$ is the pushout of the corresponding [[disjoint union]] of boundary inclusions: $$ \array{ \underset{i \in I}{\sqcup} S^{n_i - 1} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X \cup_{(\phi_i)_{i \in I}} \left(\underset{i \in I}{\sqcup} D^{n_i}\right) } \,. $$ =-- 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: +-- {: .num_defn #RelativeCellComplexes} ###### Definition **([[relative cell complexes]]) Let $X$ be a topological space, then a _topological [[relative cell complex]]_ of countable height based on $X$ is a [[continuous function]] $$ f \colon X \longrightarrow Y $$ and a [[sequential diagram]] of [[topological space]] of the form $$ X = X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots $$ such that 1. 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 $$ \array{ \underset{i \in I}{\sqcup} S^{n_i - 1} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X_k \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_{k+1} } \,. $$ 1. $Y = \underset{k\in \mathbb{N}}{\cup} X_k$ 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]] $\underset{\longrightarrow}{\lim}_k X_k$: $$ \array{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet } $$ 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 _topological [[cell complex]]_ of countable height. Finally, a topological (relative) cell complex of countable height is called a **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: $$ \array{ \underset{i \in I}{\sqcup} S^{k} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X_k \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{k+1} &\longrightarrow& X_{k+1} } \,. $$ A _[[finite CW-complex]]_ is one which admits a presentation in which there are only finitely many attaching maps, and similarly a _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]]. =-- A **cellular map** between CW-complexes $X$ and $Y$ is a [[continuous function]] $f\colon X \to Y$ such that $f(X_n) \subset Y_n$. ## Properties ### Topological properties {#TopologicalProperties} \begin{prop}\label{CWComplexesAreLocallyContractible} Every CW-complex is a [[locally contractible topological space]]. \end{prop} (e.g. [Hatcher 2002, prop. A.4](#Hatcher02)). \begin{prop} **([[CW-complexes are paracompact Hausdorff spaces]])** \linebreak Every CW-complex is a 1. a [[normal topological space]], in particular a [[Hausdorff space]], 1. a [[paracompact topological space]]. \end{prop} \begin{prop}\label{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 $Top$ ([prop.](Top#DescriptionOfLimitsAndColimitsInTop)) 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} In fact: \begin{prop} Every CW-complex is a [[Delta-generated topological space]]. \end{prop} See there, [this Prop.](Delta-generated+topological+space#CWComplexesAreDTopologicalSpaces) ### Products of CW-complexes 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 (see [Brooke-Taylor 2017](#BrookeTaylor17) for more and more generality, and see Prop. \ref{ClosureOfCWComplexesUnderCartesianProduct} below). 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. +-- {: .num_prop #ClosureOfCWComplexesUnderCartesianProduct} ###### Proposition **(product preserves CW-complexes in [[compactly generated topological spaces]])** \linebreak For $X$ and $Y$ [[CW-complexes]] with attaching maps $\{\phi_\alpha\}$ and $\{\Psi_\beta\}$, then the [[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 $$ (X\times Y)_c \simeq X \times Y $$ and so then the [[product topological space]] $X \times Y$ itself has CW-complex structure. =-- (e.g. [Hatcher 2002, theorem A.6](#Hatcher02)) ### Up to homotopy equivalence +-- {: .num_theorem} ###### Theorem Every CW complex is homotopy equivalent to (the [realization](simplicial+complex#geometric_realisations_and_polyhedra) of) a [[simplicial complex]]. =-- See [Gray](#Gray), Corollary 16.44 (p. 149) and Corollary 21.15 (p. 206). For more see at _[[CW approximation]]_. +-- {: .num_cor} ###### Corollary Every CW complex is homotopy equivalent to a space that admits a [[good open cover]]. =-- +-- {: .num_theorem} ###### Theorem 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. =-- ([Milnor 59](#Milnor59)) ### Subcomplexes {#Subcomplexes} +-- {: .num_prop } ###### Proposition 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 _[good pair](relative+homology#GoodPair)_ in the sense of [[relative homology]]. =-- For instance ([Hatcher 2002, prop. A.5](#Hatcher02)). +-- {: .num_remark } ###### Remark For $A \hookrightarrow X$ the inclusion of a subcomplex into a CW complex, then the pair $(X,A)$ is often called a _[[CW-pair]]_. This appears notably in the [[axioms]] for [[generalized (Eilenberg-Steenrod) cohomology]]. =-- e.g. ([AGP 02, def. 5.1.11](#AGP02)) ### Cellular approximation theorem {#CellularApproximationTheorem} The _[[cellular approximation theorem]]_ states that every [[continuous map]] between [[CW complexes]] (with chosen CW presentations) is [[homotopy|homotopic]] to a [[cellular map]] (a map induced by a morphism of [[cell complexes]]). This is the analogue for [[CW-complexes]] of the [[simplicial approximation theorem]] (sometimes also called lemma): that every continuous map between the [[geometric realizations]] of [[simplicial complexes]] is [[homotopy|homotopic]] to a map induced by a map of [[simplicial complexes]] (after subdivision). For more see at _[[cellular approximation theorem]]_. ### Fibrations Fibrations between CW-complexes also behave particularly well: [[a Serre fibration between CW-complexes is a Hurewicz fibration]]. ### Singular homology {#SingularHomology} We discuss aspects of the [[singular homology]] $H_n(-) \colon $ [[Top]] $\to$ [[Ab]] of CW-complexes. See also at _[[cellular homology]] of CW-complexes_. Let $X$ be a CW-complex and write $$ X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X $$ 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 $X$. +-- {: .num_prop #SkeletalRelativeSingularHomologyOfCW} ###### Proposition The [[relative singular homology]] of the filtering degrees is $$ H_n(X_k , X_{k-1}) \simeq \left\{ \array{ \mathbb{Z}[nCells] & if\; k = n \\ 0 & otherwise } \right. \,, $$ where $\mathbb{Z}[nCells]$ denotes the [[free abelian group]] on the set of $n$-cells. =-- The proof is spelled out at _[Relative singular homology - Of CW complexes](relative+homology#RelativeHomologyOfCWComplexes)_. +-- {: .num_prop #RelativeHomologyOfFilterStep} ###### Proposition With $k,n \in \mathbb{N}$ we have $$ (k \gt n) \Rightarrow (H_k(X_n) \simeq 0) \,. $$ 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 $$ (k \gt dim X) \Rightarrow (H_k(X) \simeq 0). $$ Moreover, for $k \lt n$ the inclusion $$ H_k(X_n) \stackrel{\simeq}{\to} H_k(X) $$ is an [[isomorphism]] and for $k = n$ we have an isomorphism $$ image(H_n(X_n) \to H_n(X)) \simeq H_n(X) \,. $$ =-- This is mostly for instance in ([Hatcher 2002, lemma 2.34 b),c)](#Hatcher02)). +-- {: .proof} ###### Proof By the [[long exact sequence]] in [[relative homology]], discussed at _[Relative homology -- long exact sequences](relative+homology#LongExactSequences)_, we have an [[exact sequence]] $$ H_{k+1}(X_n , X_{n-1}) \to H_k(X_{n-1}) \to H_k(X_n) \to H_k(X_n, X_{n-1}) \,. $$ 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 $$ H_k(X_{n-1}) \stackrel{\simeq}{\to} H_k(X_n) $$ 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 $$ H_{n-1+1}(X_n , X_{n-1}) \to H_{n-1}(X_{n-1}) \to H_{n-1}(X_n) \to 0 $$ and hence that with the above the map $H_{n-1}(X_{n-1}) \to H_{n-1}(X)$ is surjective. =-- ## Examples {#Examples} \begin{example} Any undirected [[graph]] (loops and/or multiple edges allowed) has a geometric realization as a 1-dimensional CW complex. \end{example} \begin{example} The [[geometric realization]] of any [[simplicial set]] is a CW-complex ([Milnor 57](#Milnor57)). In particular, in the context of the [[homotopy hypothesis]] the [[Quillen equivalence]] between [[∞-groupoids]] and [[nice topological spaces]] maps each [[∞-groupoid]] to a CW-complex. \end{example} \begin{example} The [[n-spheres]] have a standard CW-complex structure, with exactly 2-cells in each dimension, obtained [[induction|inductively]] by [[space attachment|attaching]] two $n$-dimensional [[hemispheres]] to the $(n-1)$-sphere regarded as the [[equator]] in the $n$-sphere. \end{example} The [[infinite-dimensional sphere]] may be realized as the CW-complex which is the [[colimit]] over the resulting [[relative cell complex]]-inclusions $S^n \hookrightarrow S^{n + 1} \hookrightarrow S^{n + 2} \hookrightarrow \cdots$. \end{example} \begin{example} Every [[projective space]] over the [[real numbers]], [[complex numbers]] or [[quaternions]] has the structure of a [[CW-complex]] with a single cell i in each dimension $k$, $2k$ or $4k$, respectively. See at _[[cell structure of K-projective space]]_. \end{example} \begin{example} \label{SmoothManifoldsAdmitCWComplexStructure} **(smooth manifolds)** Every [[compact topological space|compact]] [[smooth manifold]] admits a smooth [[triangulation]] (by the [[triangulation theorem]]) and hence a CW-complex structure. In the generality of manifolds with group actions see at _[G-CW complex -- G-manifolds](G-CW+complex#GManifolds)_. Every noncompact [[smooth manifold]] of [[dimension]] $n$ is [[homotopy equivalence|homotopy equivalent]] to an $(n-1)$-dimensional CW-complex. ([Napier & Ramachandran](#NapierRamachandran)). \end{example} ## Related concepts * [[dimension of a CW-complex]] * [[cell complex]] * [[cellular map]] * [[CW approximation]] * [[quasi-finite CW-complex]] * [[cellular homology]] * [[simplicial set]], [[geometric realization]] * [[CW-spectrum]] * [[G-CW complex]] [[!include universal constructions of topological spaces -- table]] ## References The introduction of the term is contained in * [[J. H. C. Whitehead]], _Combinatorial homotopy I_ , Bull. Amer. Math. Soc, 55, (1949), 213–245. Basic textbook accounts: * Brayton Gray, _Homotopy Theory: An Introduction to Algebraic Topology_, Academic Press, New York (1975). * [[George Whitehead]], chapter II of _Elements of homotopy theory_, 1978 * {#Bredon93} [[Glen Bredon]], Section IV.8-9 of: *Topology and Geometry*, Graduate Texts in Mathematics **139**, Springer (1993) &lbrack;[doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf)&rbrack; * {#May} [[Peter May]], _[[A Concise Course in Algebraic Topology]]_, U. Chicago Press (1999) * {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press 2002 ([ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)) * {#HatcherTopologyOfCellComplexes} [[Allen Hatcher]], _Topology of cell complexes_ ([pdf](https://www.math.cornell.edu/~hatcher/AT/ATapp.pdf)) in _Algebraic Topology_ * {#HatcherK} [[Allen Hatcher]], _Vector bundles & K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html)) * {#FP} [[Rudolf Fritsch]], [[Renzo A. Piccinini]], _Cellular structures in topology_, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). ([doi:10.1017/CBO9780511983948](https://doi.org/10.1017/CBO9780511983948), [pdf](https://epub.ub.uni-muenchen.de/4493/1/4493.pdf)) * {#AGP02} [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], section 5.1 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf)) Original articles include * {#Milnor59} [[John Milnor]], _On spaces having the homotopy type of a CW-complex_, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280. * {#Milnor57} [[John Milnor]], _The geometric realization of a semi-simplicial complex_, Annals of Mathematics, 2nd Ser., __65__, n. 2. (Mar., 1957), pp. 357-362; doi:[10.2307/1969967](https://doi.org/10.2307/1969967), [Semantic scholar pdf](https://pdfs.semanticscholar.org/7cbe/0482ce422d3adcc84be80b5ab3f68520a247.pdf) On [[products]] of CW-complexes: * {#BrookeTaylor17} [[Andrew Brooke-Taylor]], *Products of CW complexes -- the full story*, 2017 ([pdf](http://www.math.helsinki.fi/logic/arctic/2017/Slides/BrookeTaylor_arctic2017.pdf), [[Brooke-Taylor_ProductsOfCWComplexes.pdf:file]]) See also: * {#NapierRamachandran} Terrance Napier, Mohan Ramachandran, _[Elementary Construction of Exhausting Subsolutions of Elliptic Operators](http://www.unige.ch/math/EnsMath/EM_en/)_ L'Enseignement Math&eacute;matique, t. 50 (2004), p. 367 - 389. An inconclusive discussion [here](http://nforum.mathforge.org/discussion/4135/simplicial-homology/?Focus=33785#Comment_33785) about what parts of the definition of 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]] [[!redirects relative CW complex]] [[!redirects relative CW complexes]]