\classheader{2012-11-21}

\subsection*{Poincar\'e Duality}

\begin{definition}
Let $X$ be a space. The cap product is a pairing between certain homology groups and cohomology groups of $X$. For $k\geq \ell$, we define $\capprod:C_k(X)\times C^\ell(X)\to C_{k-\ell}(X)$ by taking $\sigma\in C_k(X)$, a singular $k$-chain $\sigma:[v_0\cdots v_k]\to X$, and $\phi\in C^\ell(X)$, a singular $\ell$-cochain, and mapping them to
\[\sigma\capprod\phi=\phi(\sigma|_{[v_0\cdots v_\ell]})\sigma|_{[v_{\ell+1}\cdots v_k]}.\]
\end{definition}
It is easy to check the following properties:
\begin{itemize}
\item $\capprod$ is bilinear
\item $\partial(\sigma\capprod\phi)=(-1)^\ell(\partial\sigma\capprod\phi - \sigma\capprod\delta\phi)$
\item $\capprod(Z_k\times Z^\ell)\subseteq Z_{k-\ell}$, i.e. $\text{cycle}\capprod\text{cycle}=\text{cycle}$
\item $\capprod(B_k\times Z^\ell)\subseteq B_{k-\ell}$, i.e. $\text{boundary}\capprod\text{cycle}=\text{boundary}$
\end{itemize}
These facts imply that the cap product descends to a bilinear map $\capprod:H_k(X)\times H^\ell(X)\to H_{k-\ell}(X)$.

\begin{theorem}[Poincar\'e Duality]
Let $M$ be a closed, oriented $n$-manifold. Then for any $0\leq i\leq n$, the map $D_M:H^i(M)\to H_{n-i}(M)$ is an isomorphism, where $D_M$ is defined by
\[D_M([\phi])=[M]\capprod[\phi].\]
\end{theorem}
\begin{corollary}
The top homology group $H_n(M)$ is isomorphic to $\Z$, and $[M]$ is a generator.
\end{corollary}
\begin{proof}[Proof idea]
We want to use a Mayer-Vietoris argument, but there's an immediate problem: the base case is false!
\[H_n(\R^n)=0\not\cong\Z=H_0(\R^n).\]
To overcome this, we define $H_c^n(M)$, cohomology with compact support, which will satisfy
\begin{align*}
H_c^n(M)&\cong H^n(M)\text{ for }M\text{ compact}\\
H_c^i(\R^n)&\cong \begin{cases}
\Z & \text{ if }i=n,\\
0 & \text{ otherwise.}
\end{cases}
\end{align*}
Then we prove, using a Mayer-Vietoris argument, for all (not-necessarily-compact) connected, oriented manifolds $Y$ without boundary that the map
\[D_Y:H_c^i(Y)\to H_{n-i}(Y)\]
is an isomorphism, and then finally, extend $\capprod$ to the non-compact case.
\end{proof}

\subsection*{Cohomology with Compact Support}

Let $X$ be a locally finite $\Delta$-complex. Define $C_c^i(X;R)$, the $i$-cochains with compact support, to be
\[C_c^i(X;R):=\{\phi\mid \phi=0\text{ outside a finite \# of simplices}\}\subseteq C^i(X;R):=\Hom(C_i(X),R).\]
Clearly, $\delta(C_c^i)\subseteq C_c^{i+1}$, so that $\zeta=\{C_c^i(X),\delta\}$ is a cochain complex. We then define the cohomology of $X$ with compact support to be $H_c^i(X):= H^i(\zeta)$.

Note that $H_c^i$ is only a (contravariant) functor when considering \textbf{proper} maps between spaces.

\begin{example}
Let's consider $X=\R$. We give it the following triangulation:
\begin{center}
\begin{tikzpicture}
\draw[thick] (-3,0) to (3,0);
\node[mypoint] at (-2.5,0) {};
\node[mypoint] at (-1.5,0) {};
\node[mypoint] at (-0.5,0) {};
\node[mypoint] at (0.5,0) {};
\node[mypoint] at (1.5,0) {};
\node[mypoint] at (2.5,0) {};
\end{tikzpicture}
\end{center}
Then $C^0(\R;\Z)$ just consists of the functions on $\R$, and given $\phi\in C^0(\R;\Z)$, we have $\delta\phi=0$ only if $\phi(v)=\phi(w)$ for all $w,v\in\R$, i.e. $\phi$ is constant. Therefore, if $\phi\in C_c^0(\R)$, we must have $\phi=0$. Thus $Z_c^0(\R)=0$, and thus $H_c^0(\R)=0$.

Now we claim that $H_c^1(\R;\Z)\cong \Z$. Let $\Sigma :C_c^1(\R)\to \Z$ be the map sending $\phi$ to $\sum_{e\in X^{(1)}}\phi(e)$. Then for any $\psi\in C_c^0(\R)$, we have
\[\delta\psi[i,i+1]=\psi(i+1)-\psi(i),\]
so that $\Sigma(\delta\psi)=0$. Therefore, $\Sigma$ induces a homomorphism $\Sigma:H_c^1(\R)\to\Z$. It is easy to check that $\Sigma$ is a bijection, and therefore $H_c^1(\R;\Z)\cong\Z$.
\end{example}
Note that $\{K\subseteq X\mid K\text{ compact}\}$ is a poset under inclusion. We obviously have that $K\subseteq L$ implies $X-K\supseteq X-L$, so for any inclusion $K\subseteq L$, we get a map $H^i(X,X-K)\to H^i(X,X-L)$, and this gives us directed system of abelian groups.
\begin{theorem}
For any space $X$,
\[H_c^i(X)\cong \lim_{\substack{\longrightarrow\\ K}} H^i(X,X-K).\]
\end{theorem}
\begin{proof}
Do on your own.
\end{proof}

It turns out that there is a relative fundamental class $[(M,M-K)]\in H_n(M,M-K)$, where $M$ is an $n$-manifold. We can extend $\capprod$ to $H_k(X,A)\times H^\ell(X,A)\to H_{k-\ell}(X,A)$, and then use the above theroem to define
\[\capprod:H_k(X)\times H_c^\ell(X)\to H_{k-\ell}(X).\]
\begin{proposition}
The cohomology with compact support of $\R^n$ is
\[H_c^i(\R^n;\Z)\cong\begin{cases}
\Z & \text{ if }i=n,\\
0 & \text{ otherwise.}
\end{cases}\]
\end{proposition}
\begin{proof}
Let $K_r=B_0(r)$. We get an increasing sequence of compact sets, $K_1\subset K_2\subset\cdots$, and hence a decreasing sequence $X-K_1\supset X-K_2\supset\cdots$. This is a cofinal sequence in the poset mentioned above. Because
\[H^i(\R^n,\R^n-B_0(r))\cong\begin{cases}
\Z & \text{ if }i=n,\\
0 & \text{ otherwise}
\end{cases}\]
and all of the maps in the directed system are the identity, we will get the same thing when we take the algebraic limit.
\end{proof}