\classheader{2012-10-29}

\subsection*{CW-Complexes and Cellular Homology}

\begin{definition}
CW-complexes are built up in an inductive process. The base of the inductive process is $X^{(0)}$, the 0-skeleton, consisting of discrete points. The inductive step is as follows: assuming we have constructed $X^{(n-1)}$, we are given
\begin{itemize}
\item a collection of $n$-balls $\{\D_\alpha^n\mid \alpha\in I\}$,
\item maps $\{\phi_\alpha:\partial \D_\alpha^n\to X^{(n-1)}\}$ attaching the boundaries of the balls to the $(n-1)$-skeleton.
\end{itemize}
Then we define
\[X^{(n)}=\frac{X^{(n-1)}\sqcup\coprod_{\alpha\in I}\D_\alpha^n}{x\sim\phi_\alpha(x)\text{ for all }\alpha\in I, x\in\partial \D_\alpha^n}.\]
The space $X=\bigcup_{n\geq 0}X^{(n)}$ built in this manner is called a CW-complex. When $X\cong X^{(N)}$, we say that $X$ is $N$-dimensional.

Note that we give $X$ the weak topology, i.e. we declare $A\subseteq X$ to be open precisely when $\phi_\alpha^{-1}(A)\subseteq \D_\alpha^n$ is open for all cells $\phi_\alpha:\D_\alpha^n\to X$.
\end{definition}

\begin{examples}
$\text{}$

\begin{itemize}
\item A 1-dimensional CW-complex is equivalent to a graph. Here is an example:



\begin{center}
\begin{tikzpicture}[>=stealth,scale=1.5]
\begin{scope}[every node/.style={inner sep=0pt,outer sep=0pt,minimum size=5pt,fill,circle}]
\node (a) at (-4,0)  {};
\node (b) at (-3.5,0.86)  {};
\node (c) at (-3,0)  {};
\end{scope}
\node at (-2.8,0.86) {$X_0$};
\begin{scope}[every node/.style={inner sep=0pt,outer sep=0pt,minimum size=5pt,fill,circle}]
\node (a) at (0,0)  {};
\node (b) at (0.5,0.86)  {};
\node (c) at (1,0)  {};
\end{scope}

\begin{scope}[thick,green!60!black]
\begin{scope}[every node/.style={inner sep=0pt,outer sep=0pt,minimum size=5pt,fill,circle}]
\node (d) at (-0.8,0.4)  {};
\node (e) at (-0.3,1.26) {};
\end{scope}
\path (d.center) edge (e.center);
\node at (-0.85,1) {$\D_\alpha^1$};
\path[dashed,thick,->,shorten >=4pt,shorten <=4pt] (e) edge (b);
\path[dashed,thick,->,shorten >=4pt,shorten <=4pt] (d) edge (a);
\path[thick,->,white] (-0.4,0.8) edge node[auto,pos=0.3,green!60!black] {$\phi_\alpha$} (0.16,0.54);
\end{scope}

\begin{scope}[thick,red!60!black]
\begin{scope}[every node/.style={inner sep=0pt,outer sep=0pt,minimum size=5pt,fill,circle}]
\node (f) at (0,2) {};
\node (g) at (1,2) {};
\end{scope}
\path (f.center) edge (g.center);
\node at (0.5,2.3) {$\D_\beta^1$};
\path[dashed,thick,->,shorten >=4pt,shorten <=4pt] (f) edge[out=270,in=120] (b);
\path[dashed,thick,->,shorten >=4pt,shorten <=4pt] (g) edge[out=270,in=60] (b);
\path[thick,white,->] (0.5,1.9) edge node[auto,pos=0.5,red!60!black] {$\phi_\beta$} (0.5,1.5);
\end{scope}

\begin{scope}[thick,blue!60!black]
\begin{scope}[every node/.style={inner sep=0pt,outer sep=0pt,minimum size=5pt,fill,circle}]
\node (h) at (0,-0.7) {};
\node (i) at (1,-0.7) {};
\end{scope}
\path (h.center) edge (i.center);
\path[dashed,thick,->,shorten >=4pt,shorten <=4pt] (h) edge[out=90,in=270] (a);
\path[dashed,thick,->,shorten >=4pt,shorten <=4pt] (i) edge[out=90,in=270] (c);
\node at (0.5,-1) {$\D_\gamma^1$};
\path[thick,white,->] (0.5,-0.6) edge node[auto,swap,blue!60!black,pos=0.5] {$\phi_\gamma$} (0.5,-0.1);
\end{scope}


\begin{scope}[every node/.style={inner sep=0pt,outer sep=0pt,minimum size=5pt,fill,circle}]
\node (aa) at (4,0)  {};
\node (ba) at (4.5,0.86)  {};
\node (ca) at (5,0)  {};
\end{scope}
\begin{scope}[thick]
\path (ba.center) edge[out=45,in=135,loop,distance=1cm] (ba.center);
\path (aa.center) edge (ba.center);
\path (aa.center) edge (ca.center);
\end{scope}
\node at (5.2,0.86) {$X_1$};
\end{tikzpicture}
\end{center}


\item The $n$-sphere $\S^n$ can be given a CW-complex structure with one 0-cell $p$ and one $n$-cell $\phi$, by letting $\phi:\D^n\to\{p\}$ be the constant map.

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[fill,circle,inner sep=0pt,outer sep=0pt,minimum size=5pt] [label=285:$p$]  at (-4,1) {};

\colorlet{blah}{blue!90!white}
\colorlet{myblue}{blah!80}

\node[fill,circle,inner sep=0pt,outer sep=0pt,minimum size=5pt] [label=285:$p$]  at (0,1) {};
\begin{scope}
\fill[color=myblue] (0,-0.8) ellipse (1.3 and 0.6);
\clip (0,-0.8) ellipse (1.3 and 0.6);
\end{scope}


\draw[color=myblue!50!black,ultra thick] (0,-0.8) ellipse (1.3 and 0.6);

\fill[ball color=blue!60] (4,0) circle (1.3); % 3D lighting effect
\begin{scope}[every label/.style={white}]
\node[fill,circle,inner sep=0pt,outer sep=0pt,minimum size=5pt,white] [label=285:$p$]  at (4,1) {};
\end{scope}
\begin{scope}[thick,myblue!50!black,>=stealth]
\path[dashed,thick,->,shorten >=6pt,shorten <=12pt] (-1.3,-0.8) edge[out=90,in=180] (0,1);
\path[dashed,thick,->,shorten >=6pt,shorten <=12pt] (1.3,-0.8) edge[out=90,in=0] (0,1);
\end{scope}
\end{tikzpicture}
\end{center}

\item The surface $\Sigma_g$ of genus $g\geq 1$ can be given a CW-complex structure with one 0-cell $v$, $2g$ 1-cells $a_1,b_1,\ldots,a_g,b_g$, and one 2-cell $\phi:\partial\D^2\to X^{(1)}$ defined by the word $a_1b_1a_1^{-1}b_1^{-1}\cdots a_gb_ga_g^{-1}b_g^{-1}$. The 1-skeleton just looks like
\begin{center}
\tikz[>=stealth,scale=0.5]{
\begin{scope}[every node/.style={inner sep=0pt,outer sep=2pt,minimum size=5pt,fill,circle}]
\node (a) at (0,0)  {};
\end{scope}
\path[thick,->] (a) edge[out=0,in=20,loop,distance=3cm] node[above right] {$a_1$} (a);
\path[thick,->] (a) edge[out=40,in=60,loop,distance=3cm] node[above right] {$b_1$} (a);
\path[thick,->] (a) edge[out=80,in=100,loop,distance=3cm] node[above] {$a_2$} (a);
\path[thick,->] (a) edge[out=120,in=140,loop,distance=3cm] node[above left] {$b_2$} (a);
\path[thick,->] (a) edge[out=280,in=300,loop,distance=3cm] node[below right] {$a_g$} (a);
\path[thick,->] (a) edge[out=320,in=340,loop,distance=3cm] node[below right] {$b_g$} (a);
\node[rotate=340] (b) at (210:0.5)  {$\ddots$};
}
\end{center}

We attach $\partial\D^2$ to the 1-skeleton $X^{(1)}$ as follows. We break up $\partial\D^2$ into $4g$ arcs of angle $\frac{2\pi}{4g}$,
\begin{center}
\begin{tikzpicture}[scale=2]
\fill[gray!30] (0,0) circle (1);
\node at (0,0) {$\D^2$};
\draw[thick] (0,0) circle (1);
\draw[thick] (0:0.95) -- (0:1.05);
\draw[thick] (25:0.95) -- (25:1.05);
\draw[thick] (50:0.95) -- (50:1.05);
\draw[thick] (75:0.95) -- (75:1.05);
\draw[thick] (100:0.95) -- (100:1.05);
\node at (12.5:1.2) {$a_1$};
\node at (37.5:1.2) {$b_1$};
\node at (62.5:1.2) {$a_1$};
\node at (87.5:1.2) {$b_1$};
\node[rotate=35] at (120:1.2) {$\cdot$};
\node[rotate=35] at (127:1.2) {$\cdot$};
\node[rotate=35] at (134:1.2) {$\cdot$};
\begin{scope}[outer sep=2pt,decoration={markings,mark=at position 0.6 with {\arrow[scale=1.7]{stealth}}}]
\draw[postaction={decorate}]
(1,0) arc (1:25:1) ;
\draw[postaction={decorate}]
(25:1) arc (25:50:1) ;
\draw[postaction={decorate}]
(75:1) arc (75:50:1) ;
\draw[postaction={decorate}]
(100:1) arc (100:75:1) ;
\end{scope}
\end{tikzpicture}
\end{center}
and define $\phi$ by mapping an arc to the indicated edge in the 1-skeleton, with degree $\pm 1$ depending on the sign determined by the word (this is reflected in the orientations of the arcs in the picture).


\item Real projective space $\R P^n$ is defined to be
\[\{\text{lines through origin in }\R^{n+1}\}=\frac{\R^{n+1}\setminus\{0\}}{\genfrac{}{}{0pt}{}{u\sim v\text{ iff }u=rv}{\text{for some }r\in\R^\times}}=\frac{\{v\in\R^{n+1}\setminus\{0\}: \|v\|=1\}}{v\sim -v}=\frac{\S^n}{v\sim-v}.\]
Denoting the upper hemisphere of $\S^n$ by $\D_+^n$,
\begin{center}
\begin{tikzpicture}[scale=2,every node/.style={minimum size=1cm},>=latex]
\fill[ball color=white!60] (0,0) circle (1.3);
\begin{scope}
\path[clip] (-1.3,0) arc (180:360:1.3 and 0.6) arc (0:180:1.3);
\fill[ball color=blue!60] (0,0) circle (1.3);
\end{scope}
\fill[color=gray!80,opacity=0.4] (0,0) ellipse (1.3 and 0.6);
\draw[thick,dashed] (1.3,0) arc (0:180:1.3 and 0.6);
\draw[thick] (-1.3,0) arc (180:360:1.3 and 0.6);
\node[myblue!50!black,inner sep=0pt,outer sep=0pt] (x) at (1.2,1) {$\D_+^n$};
\end{tikzpicture}
\end{center}
then we can also see that
\[\RP^n=\frac{\D_+^n}{v\sim -v\text{ for all }v\in\partial\D_+^n}.\]
Note that $\partial\D_+^n=\S^{n-1}$. We have the quotient map $\phi:\S^{n-1}\to\RP^{n-1}$ identifying $v$ and $-v$ for all $v\in\S^{n-1}$; this is just the attaching map for the $n$-cell $\D_+^n$ when we give $\RP^n$ a CW-complex structure.

In general, $\RP^n$ is a CW-complex with 1 $i$-cell for $i=0,1,\ldots,n$. Thus, $\RP^0$ is just a point, $\RP^1$ is $\S^1$, and
\[\RP^n=\frac{\RP^{n-1}\sqcup \D_+^n}{v\sim\phi(v)\text{ for all }v\in\partial\D^n}.\]
\end{itemize}
\end{examples}

\subsection*{How to Compute CW Homology}


Let $X$ be a CW-complex. We define a chain complex $\cal{C}^\CW{}(X)=\{C_n^{\CW}(X),d_n\}$ by letting $C_n^{\CW}(X)$ be the free abelian group on the $n$-cells, and the boundary maps $d_n:C_n^{\CW}(X)\to C_{n-1}^{\CW}(X)$ satisfy $d_{n-1}\circ d_n=0$, but we'll skip their definition to get to some calculations.

Of course, lastly we define $H_i^{\CW}(X)=H_i(\cal{C}^{\CW}(X))$.

\begin{examples}
$\text{}$

\begin{itemize}
\item Let's compute the CW homology of $\S^n$, for $n\geq 2$. The CW chain complex is just
\begin{center}
\begin{tikzcd}[row sep=0.1in,outer sep=5pt]
C_{n+1}^{\CW}(\S^n) \ar{r} & C_n^{\CW}(\S^n) \ar{r} & C_{n-1}^{\CW}(\S^n) \ar{r} & \cdots \ar{r} & C_0^{\CW}(\S^n) \ar{r} & 0\\
0 \ar{r} & \Z \ar{r} & 0 \ar{r} & \cdots \ar{r} & \Z \ar{r} & 0
\end{tikzcd}
\end{center}
and therefore
\[H_i^{\CW}(\S^n)\cong C_i^{\CW}(\S^n)=\begin{cases}
\Z & \text{ if }i=0\text{ or }n,\\
0 & \text{ otherwise.}
\end{cases}\]
This agrees with singular, and hence also simplicial, homology.
\item The CW chain complex for $\Sigma_g$ is
\begin{center}
\begin{tikzcd}[row sep=0.01in,column sep=0.8in,outer sep=3pt]
\cdots \ar{r} & 0 \ar{r} & \Z \ar{r}{d_2} & \Z^{2g} \ar{r}{d_1} & \Z \ar{r} & 0\\
 & &\overset{\text{\rotatebox{90}{$\;=$}}}{\mathclap{\langle f\rangle}} & \overset{\text{\rotatebox{90}{$\;=$}}}{\mathclap{\langle a_1,b_1,\ldots,a_g,b_g\rangle}} & \overset{\text{\rotatebox{90}{$\;=$}}}{\mathclap{\langle v\rangle}}& 
\end{tikzcd}
\end{center}
Every $a_i$ and $b_i$ is sent to $v-v=0$, so that $d_1=0$. Now note that $d_2$ sends $f$ to
\[a_1+b_1-a_1-b_1+a_2+\cdots+a_g+b_g-a_g-b_g=0\]
and therefore $d_2=0$. Thus,
\[H_i^{\CW}(\S^n)\cong C_i^{\CW}(\S^n)=\begin{cases}
\Z & \text{ if }i=0\text{ or }2,\\
\Z^{2g} & \text{ if }i=1,\\
0 & \text{ otherwise.}
\end{cases}\]
Morally, the boundary map in CW homology is measuring degree of the attaching maps of the cells.
\end{itemize}
\end{examples}