\classheader{2012-11-05}


Today we'll discuss Mayer-Vietoris, which is one of the most powerful computational tools we have, and we'll be moving to cohomology soon. 

Let $X$ be a space and consider injective maps $i:A\hookrightarrow X$ and $j:B\hookrightarrow X$ (we may as well consider $A$ and $B$ as subsets of $X$) with the property that $A\cup B=X$, and $(X,A)$ and $(X,B)$ are good pairs.

\begin{theorem}[Mayer-Vietoris]
There is a long exact sequence
\begin{center}
\begin{tikzcd}
\cdots\ar{r} & H_n(A\cap B) \ar{r} & H_n(A)\oplus H_n(B) \ar{r} & H_n(X) \ar{r}{\partial} & H_{n-1}(A\cap B) \ar{r} & \cdots
\end{tikzcd}
\end{center}
\end{theorem}
\begin{proof}
There is a short exact sequence of chain complexes
\begin{center}
\begin{tikzcd}[row sep=0.02in,column sep=0.55in]
0 \ar{r} & C_n(A\cap B) \ar{r}{i_\#\oplus j_\#} & C_n(A)\oplus C_n(B) \ar{r}{\phi} & C_n(A+B) \ar{r} & 0\\
 & z \ar[|-to]{r} & (i(z),j(z)) & & \\
& & (x,y) \ar[|-to]{r} & x-y & 
\end{tikzcd}
\end{center}
where $C_n(A+B)$ is the subgroup of $C_n(X)$ consisting of chains of the form $\sum a_i\sigma_i+\sum c_i\tau_i$ where $\sigma_i\in C_n(A)$ and $\tau_i\in C_n(B)$.

The fundamental theorem of homological algebra implies that we get a long exact sequence
\begin{center}
\begin{tikzcd}
\cdots \ar{r} & H_n(A\cap B) \ar{r} & H_n(A)\oplus H_n(B) \ar{r} & H_n(A+B) \ar{r} & H_{n-1}(A\cap B) \ar{r} & \cdots
\end{tikzcd}
\end{center}
where $H_n(A+B)$ is the $n$th homology of the chain complex $\{C_n(A+B)\}$.

The key claim is that the inclusion $h:C_n(A+B)\to C_n(X)$ induces an isomorphism on homology, $h_*:H_n(A+B)\xrightarrow{\;\cong\;} H_n(X)$ for all $n\geq 0$. The idea of the proof of this key claim is as follows.

If $X$ is a $\Delta$-complex and we can triangulate $X$ such that the triangulation restricts to triangulations on $A$, $B$, and $A\cap B$, then the key claim is easy since $C_n^\Delta(A+B)\xrightarrow{\;\cong\;} C_n^\Delta(X)$. In general, we need to subdivide $X$ (use simplicial approximation), because for example, we'd run into trouble with a simplex like this:\vspace{-0.2in}
\begin{center}
\begin{tikzpicture}
\draw[thick] (0,0) circle (1);
\draw[thick] (1.25,0) circle (1);
\node at (0,1.3) {$A$};
\node at (1.25,1.3) {$B$};
\begin{scope}[every node/.style={fill,circle,outer sep=0pt,inner sep=0pt,minimum size=5pt}]
\node (a) at (-0.1,0) {};
\node (b) at (1.25,0.58) {};
\node (c) at (1.25,-0.58) {};
\end{scope}
\draw[thick] (a) -- (b) -- (c) -- (a);
\end{tikzpicture}\vspace{-0.2in}
\end{center}\qedhere
\end{proof}

\begin{applications}
$\text{}$

\begin{itemize}
\item Let $X=\S^n$, $A=\D^n_+$, $B=\D^n_{-}$, so that $A\cap B=\S^{n-1}$ (really, we mean a little regular neighborhood around these subsets).
\begin{center}
\begin{tikzpicture}[scale=0.7,every node/.style={minimum size=1cm},>=latex]
\fill[ball color=white!60] (0,0) circle (1.3);
\begin{scope}
\path[clip] (-1.32,-0.2) to (-1.28,-0.2) arc (180:360:1.28 and 0.6) to (1.3,1.4) to (-1.3,1.4) to (-1.32,-0.2) to (-1.28,-0.2);
\fill[ball color=blue!60] (0,0) circle (1.3);
\end{scope}
\fill[color=gray!80,opacity=0.4] (0,-0.2) ellipse (1.28 and 0.6);
\draw[dashed] (1.3,0) arc (0:180:1.3 and 0.6);
\draw[] (-1.3,0) arc (180:360:1.3 and 0.6);
\node[myblue!50!black,inner sep=0pt,outer sep=0pt] (x) at (1.3,1) {$A$};
\end{tikzpicture} 
\raisebox{-0.1in}{\begin{tikzpicture}[xscale=0.7,yscale=-0.7,every node/.style={minimum size=1cm},>=latex]
\fill[ball color=white!60] (0,0) circle (1.3);
\begin{scope}
\path[clip] (-1.32,-0.2) to (-1.28,-0.2) arc (180:360:1.28 and 0.6) to (1.3,1.4) to (-1.3,1.4) to (-1.32,-0.2) to (-1.28,-0.2);
\fill[ball color=blue!60] (0,0) circle (1.3);
\end{scope}
\fill[color=gray!80,opacity=0.4] (0,-0.2) ellipse (1.28 and 0.6);
\draw[] (1.3,0) arc (0:180:1.3 and 0.6);
\draw[dashed] (-1.3,0) arc (180:360:1.3 and 0.6);
\node[myblue!50!black,inner sep=0pt,outer sep=0pt] (x) at (-1.3,1) {$B$};
\end{tikzpicture}}
\end{center}
Then we get in the Mayer-Vietoris sequence
\begin{center}
\begin{tikzcd}[row sep=0.02in]
H_n(A\cap B)\ar{r} & H_n(A)\oplus H_n(B) \ar{r} & H_n(X)\ar{r} & H_{n-1}(A\cap B)\\
& =0\oplus 0 & &
\end{tikzcd}
\end{center}
So, there are exact sequences for all $n\geq 2$
\begin{center}
\begin{tikzcd}[row sep=0.02in]
0 \ar{r} & H_n(X)\ar{r} & H_{n-1}(A\cap B) \ar{r} & 0
\end{tikzcd}
\end{center}
so we get that $H_n(\S^n)\cong H_{n-1}(\S^{n-1})$ for all $n\geq 2$.
\item Given $n$-manifolds $M$ and $N$ for $n\geq 2$, the connect sum $M\mathbin{\#} N$ is the $n$-manifold obtained via picking $\D_1^n\subset M$ and $\D_2^n\subset N$, and
\[M\mathbin{\#} N=\frac{(M-\operatorname{int}(\D_1^n))\sqcup (N-\operatorname{int}(\D_2^N))}{\partial\D_1^n\sim \partial\D_2^n}\]
We can apply Mayer-Vietoris to $M\mathbin{\#} N$ by letting $X=M\mathbin{\#} N$, $A=M-\operatorname{int}(\D_1^n)$, $B=N-\operatorname{int}(\D_2^n)$, so that $A\cap B=\partial \D_1^n=\partial\D_2^n\cong\S^{n-1}$. We get\vspace{-0.1in}
\begin{center}
\begin{tikzcd}[row sep=0.02in]
H_n(\S^{n-1})\ar{r} & H_n(M-\operatorname{int}(\D_1^n))\oplus H_n(N-\operatorname{int}(\D_2^n)) \ar{r} & H_n(X)\ar{r} & H_{n-1}(M\# N)
\end{tikzcd}
\end{center}
and we can compute e.g. $H_n(M-\operatorname{int}(\D_1^n))$ using Mayer-Vietoris in terms of homology of $M$. 

We can use this to get another computation of the homology of $\Sigma_g$, because $\Sigma_g\cong \Sigma_{g-1}\mathbin{\#}\T^2$.
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\newcommand{\mynewskip}{0.2}

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\path[thick,<->,>=stealth,red,dashed] (al) edge[out=315,in=235] (ar);
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\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[xscale=1]
\node at (-1.3,0) {$\cong$};
\begin{scope}
\clip (-0.6,-1.1) rectangle (3.5,1.1);
\draw [thick,fill=gray!40] plot [smooth,tension=0.8] coordinates { (4.5,1)  (3.5,0.5) (2.5,1) (1.5,0.5) (0.4,1) (-0.5,0) (0.4,-1) (1.5,-0.5) (2.5,-1) (3.5,-0.5) (4.5,-1)};
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\draw[thick] (2.5,0) ellipse (0.4 and 0.2);
\end{scope}
\end{scope}
\node at (4.03,0) {$\cdots$};
\end{tikzpicture}
\begin{tikzpicture}[xscale=-1]
\begin{scope}
\clip (-0.6,-1.1) rectangle (3.5,1.1);
\draw [thick,fill=gray!40] plot [smooth,tension=0.8] coordinates { (4.5,1)  (3.5,0.5) (2.5,1) (1.5,0.5) (0.4,1) (-0.5,0) (0.4,-1) (1.5,-0.5) (2.5,-1) (3.5,-0.5) (4.5,-1)};
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\draw[thick,fill=white] (2.5,-0.1) ellipse (0.35 and 0.25);
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\begin{scope}
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\draw[thick] (2.5,0) ellipse (0.4 and 0.2);
\end{scope}
\end{scope}
\end{tikzpicture}
\end{center}
\end{itemize}
\end{applications}

Note that $M\mathbin{\#}\S^n\cong M$ for any $n$-manifold $M$.

\begin{definition}
A manifold $M$ is prime when $M\cong A\mathbin{\#} B$ for $n$-manifolds $A$, $B$, we must have $A\cong\S^n$ or $B\cong\S^n$.
\end{definition}

\begin{theorem}[Kneser, 1920's]
Every closed 3-manifold $M$ can be expressed as a connect sum of prime closed 3-manifolds,
\[M\cong M_1\mathbin{\#}\cdots\mathbin{\#} M_r,\]
and this is unique up to ordering, i.e. if $M\cong N_1\mathbin{\#}\cdots\mathbin{\#} N_s$, then $r=s$ and we can permute the indices so that $M_i\cong N_i$.
\end{theorem}

\begin{theorem}[Schoenflies]
$\S^2-\S^1$ has 2 components, each homeomorphic to $\operatorname{int}(\D^2)$.
\end{theorem}