% vim: tw=50
% 27/01/2023 11AM

\subsubsection*{More Examples}

\begin{enumerate}[(1)]
    \item Octagon $P$:
        \begin{center}
            \includegraphics[width=0.3\linewidth]
            {images/645f661a9e3311ed.png}
        \end{center}
        Cut in half:
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/b05329809e3311ed.png}
        \end{center}
        So $P / \sim$ looks like:
        \begin{center}
            \includegraphics[width=0.3\linewidth]
            {images/ea6fb1569e3311ed.png}
        \end{center}
    \item Now projective plane example:
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/452a98f49e3411ed.png}
        \end{center}
        $\mathbb{RP}^2 = S^2 / \pm 1 =
        (\text{closed upper hemisphere}) / (\theta
        \sim -\theta)$.
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/7cb5cd5c9e3411ed.png}
        \end{center}
\end{enumerate}

\subsection{Triangulations and Euler Characteristic}

\begin{flashcard}[polygon-subdivision]
\begin{definition*}
    A subdivision of compact topological surface
    $\Sigma$ comprises
    \begin{enumerate}[(i)]
        \item \cloze{a finite set $V$ of
            \emph{vertices}.}
        \item \cloze{a finite collection of edges $E =
            \{e_i \colon [0, 1] \to \Sigma\}$ such
            that:
            \begin{itemize}
                \item For all $i$, $e_i$ is a
                    continuous injection on its
                    interior and $e_i^{-1} V =
                    \{0, 1\}$
                \item $e_i$ and $e_j$ have
                    disjoint image except perhaps
                    at their endpoints in $V$.
            \end{itemize}}
        \item \cloze{We \emph{require} that each
            connected component of $\Sigma
            \setminus (\bigcup_i e_i [0, 1] \cup
            V)$ is homeomorphic to an open disc
            called a \emph{face}. (so the closure
            of a face has boundary $\ol{F}
            \setminus F$ lying in $\bigcup_i e_i
            [0, 1] \cup V$).}
    \end{enumerate}
    \fcscrap{A subdivision is a
    \emph{triangulation} if every closed
    face (closure of a face) contains
    exactly 3 edges and two closed faces
    are distinct, meet in exactly one edge
    or just one vertex.}
\end{definition*}
\end{flashcard}

\subsubsection*{Examples}

A subdivision of $S^2$:
\begin{center}
    \includegraphics[width=0.2\linewidth]
    {images/44ba30fe9e3511ed.png}
\end{center}
A triangulation of $S^2$:
\begin{center}
    \includegraphics[width=0.2\linewidth]
    {images/5ad90b589e3511ed.png}
\end{center}
Subdivision of $T^2$ (1 vertex, 2 edges and 1
face):
\begin{center}
    \includegraphics[width=0.2\linewidth]
    {images/857d9ef09e3511ed.png}
\end{center}
Here the left drawing is not a triangulation of
$T^2$ but the right one is:
\begin{center}
    \includegraphics[width=0.4\linewidth]
    {images/c71aadda9e3511ed.png}
\end{center}
A very degenerate subdivision of $S^2$ ( 1 vertex,
0 edges, 1 face):
\begin{center}
    \includegraphics[width=0.2\linewidth]
    {images/ee30d0e89e3511ed.png}
\end{center}

\begin{definition*}
    The Euler characteristic of a subdivision is
    \[ \#V - \#E + \#F \]
\end{definition*}

\begin{theorem}
    \begin{enumerate}[(i)]
        \item Every compact topological surface
            admits subdivisions and triangulation.
        \item The Euler characteristic, denote by
            $\chi(\Sigma)$, does \emph{not} depend
            on the choice of subdivision and
            defines a topological invariant of the
            surface (depends on on the
            homeomorphism type of $\Sigma$).
    \end{enumerate}
\end{theorem}

\begin{remark*}
    Hard to prove particularly (i). For (ii) there
    are cleaner approaches (Algebraic Topology
    part II).
\end{remark*}

\subsubsection*{Examples}

\begin{enumerate}[(1)]
    \item $\chi(S^2) = 2$.
    \item $\chi(T^2) = 0$.
    \item $\Sigma_1$, $\Sigma_2$ compact
        topological surfaces, and we form
        $\Sigma_1 \# \Sigma_2$.  We remove open
        discs $D_i \subset \Sigma_i$ which is a
        face of a triangulation in each surface.
        \begin{center}
            \includegraphics[width=0.2\linewidth]
            {images/504da1ec9e3711ed.png}
        \end{center}
        \[ \implies \chi(\Sigma_1 \# \Sigma_2) =
        \chi(\Sigma_1) + \chi(\Sigma_2) - 2 \]
        In particular if $\Sigma_g$ is a surface
        with $g$ holes ($\Sigma = \#_{c = 1}^g
        T^2$, then $\chi(\Sigma_g) = 2 - 2g$, $g$
        is called the \emph{genus}.
\end{enumerate}

\subsection{Abstract Smooth Surfaces}

$\Sigma$ topological surface.

\begin{definition*}
    Let $(U, \varphi)$ be a pair where $U \subset \Sigma$
    is open and $\varphi \colon U \to V \subset
    \RR^2$ is a homeomorphism (with $V$ open).
    Then this pair is called a \emph{chart}. THe
    inverse $\sigma = \varphi^{-1} \colon V \to U
    \subset \Sigma$ is called a \emph{local
    parametrisation} for $\Sigma$.
\end{definition*}

\begin{hiddenflashcard}[chart]
    What is a chart? \\
    \cloze{
    $(U, \varphi)$, $U \subset \Sigma$
    \fcemph{open} and $\varphi : U \to V
    \subset \RR^2$ \fcemph{homeomorphism}
    (with $V$ open).
    }
\end{hiddenflashcard}

\begin{hiddenflashcard}[local-parametrisation] 
    What is a local parametrisation? \\
    \cloze{
    $\sigma = \varphi^{-1} : V \to U \subset
    \Sigma$, where $(U, \varphi)$ is a chart.
    }
\end{hiddenflashcard}

\begin{definition*}
    A collection of charts
    \[ \{(U_i, \varphi_i)_{i \in I}\} \]
    such that $\bigcup_{i \in I} U_i = \Sigma$ is
    called an \emph{atlas} for $\Sigma$.
\end{definition*}

\begin{hiddenflashcard}[atlas]
    What is an atlas? \\
    \cloze{
    A collection of charts
    \[ \{(U_i, \varphi_i)_{i \in I}\} \]
    such that $\bigcup_{i \in I} U_i = \Sigma$.
    }
\end{hiddenflashcard}

\subsubsection*{Examples}

\begin{enumerate}[(1)]
    \item If $z \subset \RR^2$ closed then $\RR^2
        \setminus z$ is a topological surface with
        an atlas with \emph{one} chart: $(\RR^2
        \setminus z, \varphi = \id)$.
    \item For $S^2$, we have an atlas with 2
        charts: the 2 stereographic projections.
\end{enumerate}

\begin{definition*}
    Let $(U_i, \varphi_i)$, $i = 1, 2$ be two
    charts containing $p \in \Sigma$. The map
    $\varphi_2 \circ
    \varphi_1^{-1}|_{\varphi_1(U_1 \cap U_2)}$ is
    called a \emph{transition map} between charts.
\end{definition*}

\begin{hiddenflashcard}[transition-map]
    What is a transition map (between charts)? \\
    \cloze{
    $\varphi_2 \circ \varphi_1^{-1}|_{\varphi_1(U_1
    \cap U_2)}$.
    }
\end{hiddenflashcard}

\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/c55e2c449e3811ed.png}
\end{center}
$\varphi_1(U_1 \cap U_2)
\stackrel{\varphi_2 \circ \varphi_1^{-1}}
{\longrightarrow} \varphi_2(U_1 \cap U_2)$ is a
homeomorphism.

\myskip
Recall: If $V \subset \RR^n$ and $V' \subset
\RR^m$ are open, a map $f \colon V \to V'$ is
called \emph{smooth} if it is infinitely
differentiable, i.e. it has continuous partial
derivatives of all orders. A homeomorphism $f
\colon V \to V'$ is called a \emph{diffeomorphism}
if it is smooth and it has a smooth inverse.

\begin{definition*}
    An abstract smooth surface $\Sigma$ is a
    topological surface with an atlas of charts
    $\{(U_i, \varphi_i)\}_{i \in I}$ such that all
    transition maps are \emph{diffeomorphisms}.
\end{definition*}

\begin{hiddenflashcard}[abstract-smooth-surface]
    What is an abstract smooth surface? \\
    \cloze{
    A topological surface with an atlas of charts
    such that all transition maps are
    \fcemph{diffeomorphisms}.
    }
\end{hiddenflashcard}