---- > [!definition] > A **cell complex**, or **CW complex**,[^1] is any [[topological space]] built inductively out of the following procedure: > 1. Start with a [[discrete topology|discrete space]] $X^{0}$. The set of points in $X^{0}$ is denoted $I_{0}$. > 2. If $X^{n-1}$ has been constructed, then we may choose a family of [[continuous|maps]] $\{ \varphi_{\alpha}: \mathbb{S}^{n-1} \to X^{n-1} \}_{\alpha \in I_{n}}$, and set $X^{n}= \left(X^{n-1} \amalg (\coprod \mathbb{D}_{\alpha}^{n})\right) / \{ x \in \partial \mathbb{D}^{n}_{\alpha} \sim \varphi_{\alpha}(x) \in X^{n-1} \}.$ > We call $X^{n}$ the **$n$-skeleton** of $X$. Note $X^{0} \subset X^{1} \subset X^{2} \subset \cdots$. > 3. Finally, we define $X=\bigcup_{n \geq 0}X^{n}$ > with the weak topology, namely that $A \subset X$ is open if $A \cap X^{n}$ is open in $X^{n}$ for all $n$. > > We write $\Phi_{\alpha}:\mathbb{D}^{n}_{\alpha} \to X^{n} \to X$ for the obvious [[inclusion map|inclusion]], called the **characteristic map** for $\alpha$. $\Phi_{\alpha}(\mathbb{D}^{n}_{\alpha}-\partial \mathbb{D}^{n}_{\alpha}) \subset X^{n} \subset X$ is called the **open cell** $e_{\alpha}$. > > > If $X=X^{n}$ for some $n$, we say $X$ is **finite-dimensional**. If moreover the $I_n$ are all finite, we say $X$ is **finite**. > > A **subcellcomplex** $A$ of $X$ is a cell complex obtained by using a subset $I_{n}' \subset I_{n}$. The pair $(A,X)$ is always [[good pair|good]].[^2] [[relative homology for a good pair is reduced homology of the quotient|Thus]], $H_{n}(X,A) \cong \widetilde{H}_{n}(X / A)$. > ![[CleanShot 2024-10-29 at [email protected]]] - [ ] characterize regular cell complexes > [!basicexample] Example. (Pedantically building $\mathbb{S}^{1}$) > We can make a circle by starting with a singleton and attaching a 1-cell. Explicitly, WLOG start with the origin $(0,0)$ (a $0$-cell) and the [[closed interval]] $[0,1]$ (a $1$-cell). The map $f: \{ 0,1 \} \to (0,0)$ defined (obviously) by $f(x)=(0,0)$ induces the [[equivalence relation]] declaring $x \sim (0,0)$ if $x=1$ and $x \sim x$ for all other $x \in [0,1]$. The [[disjoint union topology|space]] $(0,0) \sqcup [0,1]$ is [[homeomorphism]] to $[0,1]$ itself, and homeomorphisms pass to quotients, thus the space $(0,0) \cup_{f} [0,1]$ obtained by attaching a 1-cell to a singleton is [[homeomorphism]] to the circle $[0,1] / {\sim}$. In practice one is rarely as pedantically precise as in this example. > > > ```handdrawn-ink > { > "versionAtEmbed": "0.1.19", > "filepath": "Ink/Drawing/2024.4.27 - 20.58pm.drawing" > } > ``` > > ---- #### [^1]: These terms are not always treated as synonyms, but they will be here. [^2]: See Hatcher 0.16, or [[subcell complexes form good pairs|here]]. # Legacy (but still useful) > [!definition] Definition. ([[cell complex]]) > For a [[topological space|space]] $X$ and [[continuous]] map $f: \mathbb{S}^{n-1} \to X$, the space obtained by **[[adjunction space|attaching]] an $n$-[[cell]] to $X$ along $f$** is the [[quotient space]]$X \cup_{f} \overline{\mathbb{B}^{n}} := (X \sqcup \overline{\mathbb{B}^{n}}) / \sim $ > where $\sim$ is the [[equivalence relation]] generated by $x \sim f(x)$ for $x \in \mathbb{S}^{n-1} \subset \overline{\mathbb{B}^{n}}$ and $\sqcup$ yields the [[disjoint union topology]]. > \ > A (finite) **cell complex** is the space $X$ obtained by > 1. starting with a finite set $X^{0}$ with the [[discrete topology]], called the $0$**-skeletion**; > 2. having defined the $(n-1)$**-skeleton** $X^{n-1}$, form the $n$**-skeleton** $X^{n}$ by attaching a collection of $n$-cells along finitely many maps $\{ \varphi_{\alpha}: \mathbb{S}^{n-1} \to X^{n-1} \}_{\alpha \in I_{n}}$, i.e., $X^{n}=(X^{n-1} \sqcup \bigsqcup_{\alpha \in I_{n}} \mathbb{D}_{\alpha}^{n}) / {\sim},$ > where $\sim$ corresponds to the identification of each boundary point $x \in \partial{\mathbb{D}_{\alpha}^{n}} = \mathbb{S}^{n-1}_{\alpha}$ with an element $\varphi_{\alpha}(x)$ of $X^{n-1}$. > 3. stop at some finite stage $k$, so $X=X^{k}$; this $k$ is the **dimension** of $X$. > > For the $n$-cell corresponding to index $\alpha \in I_{n}$ we may define its **characteristic map** via the composition of [[inclusion map|inclusions]] $\Phi_{\alpha}:\mathbb{D}_{\alpha}^{n} \xhookrightarrow{} \overbrace{X^{n}}^{(X^{n-1} \sqcup \bigsqcup_{I_{n}}\mathbb{D}_{\alpha}^{n}) / {\sim}} \xhookrightarrow{}X$ Applying $\Phi_{\alpha}$ to the [[topological interior|interior]]: $\Phi(\mathbb{D}_{\alpha}^{n} - \partial{\mathbb{D}}_{\alpha}^{n})=e_{\alpha} \subset X;$ where $e_{\alpha}$ is an **open cell**. As a set (but not as a [[topological space]], $X$ is a [[disjoint union]] of $X^{0}$ and open cells). > A **subcomplex** is a [[subspace topology|subspace]] $A \subset X$ such that $A \cap e_{\alpha} \neq \emptyset \implies \overline{e_{\alpha}} \subset A$ (i.e., if you want to include part of a cell you've got to include the whole cell). Ends up meaning that $A$ is also a cell complex with $A^{0} \subset X^{0}$ and in a given dimension $n$ we attach as subset of $I_{n}$. ^definition ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch03