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\begin{document}

\section*{\bf Compact Surfaces with Boundary}

 Let $M$ be a compact surface with boundary, then a closed surface $M^*$ is obtained from $M$ by {\it capping off} the boundaries of $M$, i.e. gluing each boundary component of $M$ with a boundary of a disk.

$$ M^* = M \bigcup_{\partial} \big( \coprod_i D_i \big) $$
\begin{center}
\includegraphics{capping}
\end{center}
In general, given $Y$ and $A \subset X$, let $ f : A \to Y $ be an imbedding of A. Then,
 $$ X \bigcup_f Y := X \bigcup Y / \sim $$
 where the relation is to identify $a \sim y=f(a) $.
\begin{center}
\includegraphics{generalcapping}
\end{center}
Let $\phi : \amalg_i S^1 _i \to \partial M $ be a homeomorphism, then
$$ M^* = M \bigcup _\phi \big( \coprod D_i \big) $$ 
 
\begin{thm}
Let $M$, $N$ be compact surfaces with boundary. Then $M \cong N$ if and only if $M^* \cong N^*$ and the number of boundary components are same.
\end{thm}

\begin{proof}

 ( $\Rightarrow$ ) The theorem of the invariance of domain implies that an interior point is mapped to an interior point by the homeomorphism between $M$ and $N$. Also it maps a boundary point to a boundary point. It follows that the number of boundary components are the same.

 ( $\Leftarrow$ )
 (1) Let $x$ and $y$ be different points in $M$, a compact surface. Then

$$ M - B^\circ _\epsilon ( x) \cong M - B^\circ _\epsilon ( y ) $$

 (2) Let $p_i \in M^* $, $i = 1,2,\ldots ,b$ where $b$ is the number of the boundary components of $M$. From (1),
  $$ M \cong M^* - \coprod _{i=1}^b B_\epsilon ( p_i ) $$

\end{proof}

\begin{cor}
The topological type of compact surface with boundary is determined by the orientability, the Euler Characteristic, and the number of boudary components.
\end{cor}

\begin{note}
$ \chi(M) = \chi(M^* ) - b$
\end{note}

\begin{figure}[h]
\begin{center}
\includegraphics{eg_sf_bd}
\end{center}
\caption{ An example of compact surface with boundary. $ \chi = -2 $ and $b=2$, and hence  $ \chi^* = 0 $ , and non-orientable. Therefore it is a Klein bottle minus two discs.}
\end{figure}

{\bf Exercises} Determine the topological type of the following surface.
\begin{center}
\includegraphics{hw_sf_bd}
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\end{document}
