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\begin{document}
\noindent Name:\\
Date:\\
Group number mod 6:\\


Homework 3 (Chapter 2 2.25-2.49)\\
Everyone must do the starred problems and you must do your own problems mod 6.

\section{Invariants}

\subsection{Euler characteristic}

\begin{thm} * Let $M^2$ be a connected, compact,
triangulated  2-manifold with triangulation $T$. Let $T'$
be a subdivision of $T$. Then
 $\chi  (M^2, T) = \chi (M^2,T')$.
%, and they are both
%orientable or both  non-orientable.
\end{thm}

\begin{thm} * Let $M_1^2$ and $M_2^2$ be connected, compact,
triangulated  2-manifolds. If
$M_1^2$ is PL-homeomorphic to $M_2^2$, then $\chi  (M_1^2) = \chi
(M_2^2)$.
%, and they are both
%orientable or both  non-orientable.
\end{thm}

\begin{thm} {\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item $\chi (\Sph^2) =2$.
\item $\chi (\T^2)=0$.
\item $\chi (\PP^2)=1$.
\item $\chi (\K^2)=0$.
\ee
\end{thm}

\begin{thm} 
* Let $M_1^2$ and $M_2^2$ be two connected, compact,
triangulated  2-manifolds. Then
$\chi (M_1^2 \num M_2^2) = \chi (M_1^2) + \chi  (M_2^2) -2$.
\end{thm}

\begin{thm}
Let $\T_i^2$ be the torus for $i=1,\ldots,n$. Then
\[
\chi \biggl( \num_{i=1}^n \T_i^2\biggr) = 2- 2n\/.
\]
\end{thm}

\begin{thm} Let $\PP_i^2$ be the projective plane for $i=1,\ldots,n$.
Then
\[
\chi
\biggl( \num_{i=1}^n \PP^2\biggr) = 2-n\/.
\]
\end{thm}


\subsection{Orientability}

\begin{exercise}
Show that the induced orientation on an edge of a $2$-simplex is
well defined; in other words, that it is independent of the choice
of positive equivalence class representative.
\end{exercise}

\begin{thm} Suppose $(M^2,T)$ is a $2$-manifold with triangulation
$T$ and $T'$ is a subdivision of $T$. Then if $(M^2,T)$ is
orientable, so is $(M^2,T')$.
\end{thm}

\begin{thm} Orientability is preserved under PL homeomorphism.
\end{thm}

\begin{thm}
 $M^2$ is orientable if and only if it contains no M\"obius band.
\end{thm}


\begin{thm}
* Let $M=M_1\num \ldots \num M_n$. Then
$M$ is orientable if and only if $M_i$
is orientable for each $i\in\{1,\ldots,n\}$.
\end{thm}

\begin{thm}[Classification of compact, connected $2$-manifolds]
\index{$2$-manifold!Classification Theorem!Euler characteristic}
* If $M^2$ is a  connected, compact,
triangulated 2-manifold then:
\be
\item[(a)] if $\chi (M^2) = 2$, then $M^2 \cong \Sph^2$.
\item[(b)] if $M^2 $  is orientable and $\chi (M^2) = 2-2n$, for $n\geq 1$, then
\[ M^2 \cong \biggl(
\num_{i=1}^n T_i^2\biggr)\ .\]
\item[(c)] if $M^2$ is non-orientable  and $\chi (M^2) =2-n$, for $n\geq 1$, then
\[ M^2 \cong \biggl( \num_{i=1}^n  \PP_i^2\biggr)\ .
\]
\ee
\end{thm}


\begin{pr}
*    Identify the following $2$-manifolds as a sphere, or a connected sum
of $n$ tori (specifying $n$), or a connected sum
of $n$ projective planes (specifying $n$).
    \be
    \item[a.]   $\T\#\PP$
    \item[b.]   $\K\#\PP$
    \item[c.]   $\PP\#\T\#\K\#\PP$
    \item[d.]   $\K\#\T\#\T\#\PP\#\K\#\T$
    \ee
\end{pr}

\section{CW complexes}

\begin{thm}
Let $(M^2,T)$ be a triangulated $2$-manifold. Suppose
$\sigma=\{uvw\}$ and $\sigma'=\{uvw'\}$ are two distinct
$2$-simplexes in $T$ that share the edge $e = \{uv\}$.  Then we can create a new structure for $M^2$ alternative to $T$, namely, $P$ where
$P$ = $T\cup\{\tau\}-\{\sigma,\sigma', $e$\}$, where $\tau=\sigma\cup\sigma'$
is the polygon formed by the union
of the two $2$-simplices along their shared edge. If $v'$,
$e'$, $f'$ are the numbers of vertices, edges, and polygons in $P$, then the Euler Characteristic
$\chi(M^2,T)=v'-e'+f'$.
\end{thm}

\vspace*{1.5in}

\begin{thm} Let $(M^2,T)$ be compact, triangulated $2$-manifold with Euler characteristic $\chi(M^2,T)$.
Suppose we create a polygonal structure $P$ on $M^2$ inductively as follows. Let $P_0$ = $T$. Suppose we have created $P_i$.  Suppose two $2$-dimensional objects $\sigma$ and $\sigma'$ in $P_i$ share a connected path of edges in the boundary of each
from vertex $u$ to $w$ ($v\neq w$). We create $P_{i+1}$ by removing $\sigma$ and $\sigma'$ from $P_i$, removing all the edges in the path from vertex $u$ to $w$, removing all vertices of the edges in that path except for $u$ and $w$, and putting in the single two dimensional object  $\sigma\cup \sigma'$. 
Then if $v$, $e$, $f$ are the
numbers of vertices, edges, and $2$-dimensional objects in $P_{i+1}$, then
$\chi(M^2,T)=v-e+f$.
\end{thm}


\vspace*{1.5in}


\begin{thm} * Let $(M^2,T)$ be compact, triangulated $2$-manifold with
a polygonal structure $P$ as defined inductively in the previous theorem. 
Suppose we substitute $P$ with a new structure obtained inductively as follows. Let $P=P_0$. If $P_i$ has an edge $e$ with a free vertex $v$, that is, $v$ is not the boundary of any other edge in $P_i$, then remove $v$ and $e$ from $P_i$ to create $P_{i+1}$.   If $P_i$ has a vertex $v$ that is one end of an edge $e$ in $P_i$ and one end of an edge $f$ in $P_i$ and $v$ is not on the end of any other edge, then remove $v$, $e$, and $f$ from $P_i$ and put in the new $1$-dimensional object $e\cup\ f$ to create $P_{i+1}$. 
Then if $v'$, $e'$, $f'$ are the
numbers of vertices, $1$-dimensional objects, and $2$-dimensional objects in an inductively defined $P$, then
$\chi(M^2,T)=v'-e'+f'$.
\end{thm}

\vspace*{1.5in}

\begin{exercise} Start with a triangulation of $\Sph^2$ and carry out
the preceding process as far as
possible. What ''structure'' do you get? Confirm that you get the
right Euler Characteristic.
\end{exercise}

\begin{exercise} * Start with a triangulation of $\T^2$ and carry out
the preceding process as far as
possible. What ``structure'' do you get? Confirm that you get the
right Euler characteristic.
\end{exercise} 

\begin{thm}
Let $(M^2, T)$ be a compact, triangulated $2$-manifold
with triangulation $T$. Then $M^2$ equals the disjoint union of
the $\interior \sigma_i$ where $\sigma_i\in T$.
\end{thm}

\begin{thm}
*Let $S$ be a cellular decomposition of a compact, triangulated
$2$-manifold $(M^2,T)$.  If $v$, $e$, and $f$ are the number of $0$, $1$ and $2$
cells in $S$, then the Euler Characteristic $\chi(M^2,T)=v-e+f$.
\end{thm}

\begin{pr}
    Identify the following surfaces:
    \be
    \item[a.] The surface obtained by identifying the edges of
the octagon as indicated:

    \vspace*{1.5in}

    \item[b.] The surface obtained by identifying the edges of
the decagon as indicated:

    \vspace*{1.5in}

    \ee
\end{pr}


\section{2-manifolds with boundary}


\begin{exercise}
What should be the definition of a connected, compact, triangulated $2$-manifold-with-boundary?
\end{exercise}

\begin{exercise}
Formulate the necessary definitions and theorem statements that classify
compact, connected, triangulated $2$-manifolds-with-boundary. Prove
your theorems.
\end{exercise}

\begin{pr}
    Identify the following surfaces made by two disks joined by
bands as indicated:

    \vspace*{1in}
    \be
    \item[a.]  \hfill $n$ twisted bands \hspace*{\fill}
    \vspace*{1in}

    \item[b.] \hfill $1$ untwisted band and $n-1$ twisted bands \hspace*{\fill}

    \ee
\end{pr}

\begin{exercise}
 *   Fill out the following table, using the connected sum decomposition. The
number of boundary components is  denoted by $|\partial|$.

\vspace*{.3in}

    \begin{tabular}{||c|l|l|l|l|l|l|l|l||}
        \hline \hline
        $\ \ |\partial|$ & \multicolumn{2}{c|}{$0$} &
                \multicolumn{2}{c|}{$1$} &
\multicolumn{2}{c|}{$2$} & \multicolumn{2}{c||}{$3$} \\
        \cline{2-9}

        \rule{0cm}{.7cm}$\chi\ \ \ $ & orient.      &
non-or.    &  orient.      &  non-or.  &
orient.      &  non-or.
                & orient.      &  non-or.   \\
        \hline \hline
            $2$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline
            $1$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline
        $0$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline
        $-1$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline
        $-2$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline
        $-3$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline
        $-4$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline
        $-5$ \rule{0cm}{.7cm}& & &  & & & & & \\
        \hline \hline
        \end{tabular}

\end{exercise}

\end{document}