\classheader{2012-10-03}

There are some corrections for the homework: on problem 2a, it should read
\[H_0(\cal{C})=\widetilde{H}_0(\cal{C})\oplus\Z\]
and on problem 3, you should show that the following diagram commutes:
\[\xymatrix{\Delta^n \ar[d]_{[v_0\cdots v_n]} \ar[r] & \Delta^m\ar[r]^{\tau} & Y\\ X\ar[rru]_{f} & & }\]

\subsection*{Simplices and $\Delta$-complexes}
\begin{definition}
The standard (ordered) $n$-simplex $[v_0\cdots v_n]$ is the convex hull of the set $\{e_1,\ldots,e_{n+1}\}$ where $e_i=(0,\ldots,1,\ldots,0)\in\R^{n+1}$. For example, when $n=2$,\vspace{0.3in}

\begin{center}
\begin{tikzpicture}[scale=2.4]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (a) at (0,-0.1) [label={135:$e_0$}] {};
\node (b) at (1.1,0) [label={45:$e_1$}] {};
\node (c) at (.5,.86) [label={165:$e_2$}] {};
\end{scope}
\node (d) at (.5,0.2) {};
\fill[gray,path fading=south] (a.center) -- (b.center) -- (c.center) -- cycle;
\draw[thick] (a.center) -- (b.center) -- (c.center) -- cycle;
\draw[shorten >=-1cm] (d.center) --  (a.center);
\draw[shorten >=-1cm] (d.center) --  (b.center);
\draw[shorten >=-1cm] (d.center) --  (c.center);
\end{tikzpicture}\vspace{0.3in}
\end{center}

 Equivalently, it is the set
\[\left\{\textstyle\sum_{i=1}^na_ie_i\in\R^{n+1}\mid a_i\geq 0,\sum_{i=1}^na_i=1\right\}.\]
We will often want to specify using coordinates. Let $v_0=0$, and $v_i=e_i$ for $i=1,\ldots,n+1$. Then
\[\Delta^n=\text{ convex hull }(\{v_1-v_0,\ldots,v_{n+1}-v_0\}).\]
\end{definition}
For example,\vspace{-0.1in}
\begin{center}
\begin{tabular}{ccc}
$\Delta^0\quad=\quad$ \raisebox{-0.13in}{\begin{tikzpicture}[scale=2.4]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (a) at (0,0) [label={330:$v_0$}] {};
\end{scope}
\end{tikzpicture}} & $\qquad$ & $\Delta^1\quad=\quad$ \raisebox{-0.13in}{\begin{tikzpicture}[scale=2.4]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (a) at (0,0) [label={210:$v_0$}] {};
\node (b) at (1,0) [label={330:$v_1$}] {};
\end{scope}
\begin{scope}[thick,outer sep=2pt,decoration={markings,mark=at position 0.57 with {\arrow[thin]{>}}}]
\draw[postaction={decorate}] (a.center) -- node[auto,swap] {} (b.center);
\end{scope}
\end{tikzpicture}}\\
 & & \\
$\Delta^2\quad=\quad$ \raisebox{-0.6in}{\begin{tikzpicture}[scale=2.4]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (a) at (0,0) [label={210:$v_0$}] {};
\node (b) at (1,0) [label={330:$v_1$}] {};
\node (c) at (.5,.86) [label={90:$v_2$}] {};
\end{scope}
\begin{scope}[thick,outer sep=2pt,decoration={markings,mark=at position 0.57 with {\arrow[thin]{>}}}]
\draw[postaction={decorate}] (a.center) -- node[auto,swap] {} (b.center);
\draw[postaction={decorate}] (b.center) -- node[auto,swap,outer sep=1pt] {} (c.center);
\draw[postaction={decorate}] (a.center) -- node[auto] {} (c.center);
\end{scope}
\end{tikzpicture}} & $\qquad$ &  $\Delta^3\quad=\quad$ \raisebox{-0.8in}{\begin{tikzpicture}[scale=2.4]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (v0) at (0,0) [label={210:$v_0$}] {};
\node (v1) at (0.65,-0.4) [label={210:$v_1$}] {};
\node (v2) at (1,0.2) [label={330:$v_2$}] {};
\node (v3) at (.5,0.8) [label={90:$v_3$}] {};
\end{scope}
\begin{scope}[outer sep=2pt,decoration={markings,mark=at position 0.57 with {\arrow[thin]{>}}}]
\draw[thick,postaction={decorate}] (v0.center) -- node[auto,swap] {} (v1.center);
\draw[dashed,postaction={decorate}] (v0.center) -- node[auto,swap] {} (v2.center);
\draw[thick,postaction={decorate}] (v0.center) -- node[auto,swap] {} (v3.center);
\draw[thick,postaction={decorate}] (v1.center) -- node[auto,swap] {} (v2.center);
\draw[thick,postaction={decorate}] (v1.center) -- node[auto,swap] {} (v3.center);
\draw[thick,postaction={decorate}] (v2.center) -- node[auto,swap] {} (v3.center);
\end{scope}
\end{tikzpicture}}
\end{tabular}
\end{center}

Note that an ordered $n$-simplex has $n+1$ ``face maps''; for each $i=0,\ldots,n$,
\[[v_0\cdots\widehat{v_i}\cdots v_n]=\text{ an ordered }(n-1)\text{-simplex (``subsimplex of $[v_0 \cdots v_n]$'')}.\]
For example, for $n=2$, we get canonical linear maps
\begin{center}
\begin{tikzpicture}[>=latex,xscale=2]
\node (a) at (0,0) {$[v_0v_1v_2]$};
\node (b1) at (2,2) {$[v_1v_2]$};
\node (b2) at (2,0) {$[v_0v_2]$};
\node (b3) at (2,-2) {$[v_0v_1]$};
\node (c1) at (4,2.5) {$[v_2]$};
\node (c2) at (4,1.5) {$[v_1]$};
\node (c3) at (4,0.5) {$[v_2]$};
\node (c4) at (4,-0.5) {$[v_0]$};
\node (c5) at (4,-1.5) {$[v_1]$};
\node (c6) at (4,-2.5) {$[v_0]$};
\draw[thick,->] (a) -- (b1);
\draw[thick,->] (a) -- (b2);
\draw[thick,->] (a) -- (b3);
\draw[thick,->] (b1) -- (c1);
\draw[thick,->] (b1) -- (c2);
\draw[thick,->] (b2) -- (c3);
\draw[thick,->] (b2) -- (c4);
\draw[thick,->] (b3) -- (c5);
\draw[thick,->] (b3) -- (c6);
\end{tikzpicture}
\end{center}



\begin{definition}
Let $X$ be a topological space. A $\Delta$-complex structure on $X$ is a decomposition of $X$ into simplices. Specifically, it is a finite collection $S=\{\Delta_i\}$ of simplices with continuous maps that are injective on their interiors, that also satisfies
\newcommand{\interior}{\operatorname{Int}}
\begin{itemize}
\item $\bigcup_{i}\sigma_i(\Delta_i)=X$,
\item For all $x\in X$, there is a unique $i$ such that $x\in\interior(\sigma_i(\Delta_i))$.
\item If $\sigma:\Delta\to X$ is an element of $S$, then all subsimplices $\tau$ of $\sigma$ are also elements of $S$. In other words, $S$ is closed under taking faces.
\end{itemize}
\end{definition}
\begin{example}
The torus $\T^2$ can be given a $\Delta$-complex structure as follows:
\begin{center}
\begin{tikzpicture}[scale=3]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (ll) at (0,0) [label={225:$v$}] {};
\node (lr) at (1,0) [label={315:$v$}] {};
\node (ul) at (0,1) [label={135:$v$}] {};
\node (ur) at (1,1) [label={45:$v$}] {};
\end{scope}
\begin{scope}[thick,outer sep=2pt,decoration={markings,mark=at position 0.54 with {\arrow[thin]{>}}}]
\draw[postaction={decorate}]
(ll.center) -- node[auto,swap] {$a$} (lr.center); % Bottom
\draw[postaction={decorate}]
(ul.center) -- node[auto] {$a$} (ur.center); % Top
\draw[postaction={decorate}]
(ll.center) -- node[auto] {$b$} (ul.center); % Left
\draw[postaction={decorate}]
(lr.center) -- node[auto,swap] {$b$} (ur.center); % Right
\draw[postaction={decorate}]
(lr.center) -- node[auto,swap,outer sep=0pt] {$c$} (ul.center); % Diagonal
\end{scope}
\end{tikzpicture}
\end{center}
\end{example}
\begin{example}
The Klein bottle can be given a $\Delta$-complex structure as follows:
\begin{center}
\begin{tikzpicture}[scale=3]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (ll) at (0,0) [label={225:$v$}] {};
\node (lr) at (1,0) [label={315:$v$}] {};
\node (ul) at (0,1) [label={135:$v$}] {};
\node (ur) at (1,1) [label={45:$v$}] {};
\end{scope}
\begin{scope}[thick,outer sep=2pt,decoration={markings,mark=at position 0.54 with {\arrow[thin]{>}}}]
\draw[postaction={decorate}]
(ll.center) -- node[auto,swap] {$a$} (lr.center); % Bottom
\draw[postaction={decorate}]
(ul.center) -- node[auto] {$a$} (ur.center); % Top
\draw[postaction={decorate}]
(ll.center) -- node[auto] {$b$} (ul.center); % Left
\draw[postaction={decorate}]
(ur.center) -- node[auto] {$b$} (lr.center); % Right
\draw[postaction={decorate}]
(lr.center) -- node[auto,swap,outer sep=0pt] {$c$} (ul.center); % Diagonal
\end{scope}
\end{tikzpicture}
\end{center}
\end{example}
\begin{example}
Let $X$ be a topological space, and let $\{U_i\}$ be an open cover of $X$ (assume indexed by an ordered set, induces order for simplices). The nerve of the cover is the $\Delta$-complex specified by
\begin{itemize}
\item A vertex $v_i$ for each $U_i$
\item If $U_i\cap U_j\neq\varnothing$, the 1-simplex $[v_iv_j]$
\begin{center}
\begin{tikzpicture}[scale=1.5]
\coordinate[label=below:$v_i$] (vi) at (-0.7cm,0);
\coordinate[label=below:$v_j$] (vj) at (0.7cm,0);
\draw[thick] (vi) circle (1cm);
\draw[thick] (vj) circle (1cm);
\draw[thick] (vi) -- (vj);
\node at (1.4cm,1cm) {$U_j$};
\node at (-1.4cm,1cm) {$U_i$};
\node at (vi) {$\bullet$};
\node at (vj) {$\bullet$};
\end{tikzpicture}
\end{center}
\item If $U_i\cap U_j\cap U_k\neq\varnothing$, the 2-simplex $[v_iv_jv_k]$


\begin{center}
\begin{tikzpicture}[scale=1.5]
\coordinate[label=left:$v_i$] (vi) at (-0.7cm,0);
\coordinate[label=right:$v_j$] (vj) at (0.7cm,0);
\coordinate[label=below:$v_k$] (vk) at (0cm,-1.2cm);
\begin{scope}
\clip (vi) circle (1cm);
\clip (vj) circle (1cm);
\fill[gray!40] (vk) circle (1cm);
\end{scope}
\draw[thick] (vi) circle (1cm);
\draw[thick] (vj) circle (1cm);
\draw[thick] (vk) circle (1cm);
\draw[thick] (vi) -- (vj);
\draw[thick] (vi) -- (vk);
\draw[thick] (vj) -- (vk);
\node at (1.4cm,1cm) {$U_j$};
\node at (-1.4cm,1cm) {$U_i$};
\node at (0cm,-2.4cm) {$U_k$};
\node at (vi) {$\bullet$};
\node at (vj) {$\bullet$};
\node at (vk) {$\bullet$};
\end{tikzpicture}
\end{center}
\end{itemize}
\end{example}

\subsection*{Simplicial Homology}

We want, for any $\Delta$-complex $X$, and $i\geq 0$, an abelian group $H_i(X)$. We want this to be
\begin{enumerate}
\item \textbf{Computable}
\item Topologically invariant: if $X$, $Y$ are $\Delta$-complexes and $X\isom Y$, then $H_i(X)\cong H_i(Y)$.
\item Non-triviality: $H_n(\S^n)\neq 0$.
\end{enumerate}

Step 1 (topology): We input a $\Delta$-complex structure on $X$. We output, for each $n\geq 0$,
\[C_n(X)=\text{free abelian group on }\{\text{(ordered) $n$-simplices in $X$}\}.\]
\begin{example}
Let $X$ be a 2-simplex,
\begin{center}
\begin{tikzpicture}[scale=2.4]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (a) at (0,0) [label={210:$v_0$}] {};
\node (b) at (1,0) [label={330:$v_1$}] {};
\node (c) at (.5,.86) [label={90:$v_2$}] {};
\end{scope}
\begin{scope}[thick]
\draw[postaction={decorate}] (a.center) --  (b.center);
\draw[postaction={decorate}] (b.center) --  (c.center);
\draw[postaction={decorate}] (c.center) --  (a.center);
\end{scope}
\end{tikzpicture}
\end{center}
Then we have
\begin{itemize}
\item $C_0(X)=\Z^3=\langle [v_0],[v_1],[v_2]\rangle$
\item $C_1(X)=\Z^3=\langle [v_0v_1],[v_0v_2],[v_1v_2]\rangle$
\item $C_2(X)=\Z=\langle [v_0v_1v_2]\rangle$
\item $C_n(X)=0$ for $n\geq 3$
\end{itemize}
\end{example}
\begin{example}
Let $X$ be the torus,
\begin{center}
\begin{tikzpicture}[scale=3]
\begin{scope}[every node/.style={fill,circle,inner sep=0pt,outer sep=1pt,minimum size=0.15cm}]
\node (ll) at (0,0) [label={225:$v$}] {};
\node (lr) at (1,0) [label={315:$v$}] {};
\node (ul) at (0,1) [label={135:$v$}] {};
\node (ur) at (1,1) [label={45:$v$}] {};
\end{scope}
\begin{scope}[thick,outer sep=2pt,decoration={markings,mark=at position 0.54 with {\arrow[thin]{>}}}]
\draw[postaction={decorate}]
(lr.center) -- node[auto] {$a$} (ll.center); % Bottom
\draw[postaction={decorate}]
(ur.center) -- node[auto,swap] {$a$} (ul.center); % Top
\draw[postaction={decorate}]
(ll.center) -- node[auto] {$b$} (ul.center); % Left
\draw[postaction={decorate}]
(lr.center) -- node[auto,swap] {$b$} (ur.center); % Right
\draw[postaction={decorate}]
(lr.center) -- node[auto,swap,outer sep=0pt] {$c$} (ul.center); % Diagonal
\end{scope}
\node at (barycentric cs:ll=1,ul=0.5,lr=0.5) {$T_1$};
\node at (barycentric cs:ur=1,ul=0.5,lr=0.5) {$T_2$};
\end{tikzpicture}
\end{center}
Then we have
\begin{itemize}
\item $C_0(X)=\Z=\langle [v]\rangle$
\item $C_1(X)=\Z^3=\langle a,b,c\rangle$
\item $C_2(X)=\Z^2=\langle T_1,T_2\rangle$
\item $C_n(X)=0$ for $n\geq 3$
\end{itemize}
\end{example}
Note that we are abusing notation and identifying simplices with their maps to the space.

Step 2 (algebra): Because $C_n(X)$ is the free abelian group on the simplices, a $\tau\in C_n(X)$ is uniquely expressed as
\[\tau=\sum_{i=1}^ra_i\sigma_i,\;a_i\in\Z.\]
For all $n\geq 1$, we define homomorphisms $\delta_n:C_n(X)\to C_{n-1}(X)$ by specifying them on the generators of $C_n(X)$: for any $n$-simplex $\sigma:\Delta^n\to X$, we let
\[\delta_n\sigma=\sum_{i=0}^n(-1)^i\sigma|_{[v_0\cdots\widehat{v_i}\cdots v_n]}.\]
The key property of these homomorphisms is that $\delta_n\circ\delta_{n-1}=0$. Note that we also needed the ordering to define this.

We will compute some homology next class.