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\begin{document}
 \parindent=0cm
\section*{II. Review of covering space}
\section*{Definitions and Examples}
\begin{defn}\textit{
Let p : $\widetilde{X} \rightarrow X$.  $\widetilde{X}$ is a
covering space of X with a covering map p if \\(1) p is onto and
\\(2) each x $\in$X has a neighborhood U which is evenly covered,
i.e.,
\\$p^{-1}(U)=\displaystyle {\coprod_{a \in A} V_{a}}$ is a disjoint union of open sets $V_{a}$ of
$\widetilde{X} $ such that
\\$ \,\,p|_{V_a} : V_a \rightarrow U$ is a homeomorphism, $\forall \,\, a \in
A$.}\\
\end{defn}

%���� ������ $x\in X$�� ���� $p^{-1}(x)$ �� discrete�ϴ�.\\

{\bf Examples.}

1. $ id : X \hspace{1em} \rightarrow \hspace{1em}X$.

2. $ p : \mathbb{R} \hspace{1em}\rightarrow \hspace{1em}S^1
(\subset \mathbb{C})\,\,\,\,\,\,$ given by    $p(x) = e^{2 \pi
ix}$.

3. $ p : \mathbb{C} \hspace{1em}\rightarrow
\hspace{1em}\mathbb{C}^* =\mathbb{C}-\{0\} \,\,\,\,\,\,$ given by
$p(z) = e^z$.

4. $ p : S^1 \hspace{1em}\rightarrow \hspace{1em}S^1\,\,\,\,\,\,$
 given by $\,\,\,\,\,\,p(z) = z^n$.

�̿� ���� $p^{-1}$�� image�� $n$���� ��츦 $n$-sheeted covering
Ȥ�� $n$-fold covering �̶�� �θ���.\\


%   figure  %
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\caption{4-fold covering of $S^1$}
\end{figure}





5. $p : S^n\hspace{1em}\rightarrow \hspace{1em} \mathbb{R}P^n =
S^n/ \sim$   where $x\sim -x$.

�� ��� quotient map $p$�� covering map�� �ǰ� Ư�� two fold
covering(double covering) �� �ȴ�.\\

6. $ p : \mathbb{R}^2\hspace{1em} \rightarrow \hspace{1em}T^2=S^1
\times S^1\,\,\,\,\,\,$ given by $\,\,\,\,\,\,(x,y) \mapsto (e^{2
\pi
ix},e^{2 \pi iy})$.\\

{\bf Excercise.} If $p:\widetilde{X}\rightarrow X$  , $q :
\widetilde{Y} \rightarrow Y$  are  covering  maps, then \\$p
\times q : \widetilde{X}\times \widetilde{Y} \rightarrow X \times
Y$ is
also a covering map. \\

7. 3-fold covering of figure eight\\
�Ʒ� �׸��� ���� figure eight�� copy 3���� �߶� ǥ���Ѵ�� �ٽ�
���̸� �ȴ�.\\

%   figure : 3-fold covering of figure eight %
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\caption{3-fold covering of figure eight}
\end{figure}

�������� ��������� torus�� ��쿡�� �ܸ��� �߶� ���̸� �Ǵµ�,
�� ��� torus�� 3-fold covering�� torus�̸�, genus�� 2�� surface��
3-fold covering�� genus�� 4�� surface�� ���� �� �� �ִ�. \\

{\bf Note.}

1. $M$ is a ($\cc^\infty$-)manifold $\Rightarrow$ $\widetilde{M}$
is also a ($\cc^\infty$-)manifold.\\
(����) �� $x\in \widetilde{M}$�� ���� $p(x)\in M$ ��  coordinate
chart ($U$,$\varphi$) �� ������, ���� $p(x)$ �� evenly cover �Ǵ�
neighborhood V �� ������. �� �� $U\cap V$ �� ���� $p^{-1}(U \cap
V)$�� ������ ����, �� �� $x$�� �����ϴ� $U\cap V$�� copy�� �ְ�
�� copy�� $\varphi\circ p$�� $x$�� coordinate chart �� �ش�.\\
{\bf Exercise.} $\cc^\infty$- case���� �̷��� ���ǵ� coordinate
chart���� ���� $\cc^\infty$-relate�Ǿ� ������ ���̶�. \\

2. $M$ is orientable $\Rightarrow$  $\widetilde{M}$ is
also orientable.\\
(����) ���� $\widetilde{M}$�� �� �� $x$�� orientation�� ����.
$\widetilde{M}$��  orientation�� local homeomorphism $p$ ��
�̿��Ͽ� $p(x)$��  orientaion�� �״�� ������ ����. �׷��� ��
$x\in \widetilde{M}$�� ���� orientation�� locally constant�� �Ǵ�
$p(x)$�� �ٹ� $U$�� ���� �� �ְ�, ���� $p(x)$���� evenly cover
�Ǵ� $V$�� ���� �� �ִ�. �� �� $p^{-1}(U\cap
V)=\displaystyle{\coprod_{a \in A}} W_{a}$ �� $x$��
�����ϴ� $W_{a}$���� orientation�� $p$�� ���� $U\cap V$�� orientation�� �����Ƿ� locally constant�̴�.\\

3. $M$ is a compact manifold, $p$ is a finite sheeted covering
$\Rightarrow$ $\widetilde{M}$ is compact.\\
(����) $M$�� compact�ϹǷ� evenly cover�Ǵ� coordinate
neighborhood ���Ѱ��� ���� �� �ְ�,
compact set�� finite union�� compact�̹Ƿ� �ڸ��ϴ�. \\

4. Every non-orientable manifold has an
orientable double covering manifold.\\
(����) non-orientable manifold�� ��� �� ������ 2����
orientation�� �����ϴµ�, �̸� double covering�� �� sheet��
�и��Ͽ� �Ҵ��� �� orientation�� ��ġ�ϵ��� �ٿ������� double
covering manifold�� orientable�ϰ� ���� �� �ִ�. \\

(��) M\"{o}bius band�� double covering�� �ι� ���� band�� �Ǵµ�
�̴� annulus�� ���� 3-fold covering�� �ٽ� M\"{o}bius band�� �ȴ�.
���� ������� Klien bottle�� double covering�� torus�� �ȴ�.\\

%   figure : Mobius band�� double covering %
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\caption{M\"{o}bius band�� double covering}
\end{figure}

%   figure : Klien Bottle�� double covering %
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\caption{Klien Bottle�� double covering}
\end{figure}


5. $M$�� orientable�̰� genus�� $g$�� ���, �̿� ����
$\widetilde{M}$�� ���캸��. \\
���� ���뿡 ���� $\widetilde{M}$�� ���� orientable�̹Ƿ� $\chi$��
�˸� $\widetilde{M}$�� ������ �� �ִ�.($\chi=2-2g$) �׷���,
$\widetilde{M}$�� $n$-fold��� $M$�� triangulation��
$\widetilde{M}$ ���� �ø��� $V,E,F$ ��� n�谡 �ǹǷ� $\chi$ ����
$n$�谡 �ȴ�. ($\chi=V-E+F$) ����� ������� $M$��
non-orientable�� ���� $n$-fold covering $\widetilde{M}$�� $\chi$��
�� �� �ְ�,
orientability�� �����Ǹ� $\widetilde{M}$�� ������ �� �ִ�.\\

{\bf ����4.} Let $M, \widetilde{M}$ be a closed surface. Suppose
$\chi(\widetilde{M})=n\chi(M)$. Is there a $n$-fold covering
$\widetilde{M} \to M$?\\

{\bf non-covering}\\

%   figure  %

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\caption{non-covering}
\end{figure}

�� �׸����� ������ $p$-image �� �ٹ濡���� evenly cover�Ǵ� �ٹ��� ���� �� �����Ƿ� covering map�� �� �� ����.\\
�׷���, �� �� $p$�� {\it local homeomorphism}�� �Ǵµ�, � map
$f: X \to Y$�� {\it local homeomorphism}�̶�� ���� ������ $x\in
X$�� ���Ͽ� $x$�� neighborhood $U$�� $f(x)$�� neighborhood $V$��
�����Ͽ� $f|_U$�� $U$�� $V$������ homeomorphism�� �ȴٴ� ���̴�.\\


{\bf Remark}\\
(1) $p^{-1}(x)$ is discrete. \\
(2) A covering $p:\widetilde{X} \to X$ is a local homeomorphism,
and hence, an open map.\\
(3) If $p:\widetilde{X} \to X$ is a covering and $A\subset X$,
then $p^{-1}(A) \overset{p}{\to}A$ is a covering. \\

\end{document}
