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\begin{document}
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  \section*{Definitions and Examples}
{\bf Assumption.}  All the spaces in this chapter are path
connected and locally path connected.

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

3 . $ 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���.

4 . $p : S^n\hspace{1em}\rightarrow \hspace{1em} P^n = S^n/ \sim$.

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

5 . $ p : \textbf{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. (�� 2���� 5������ �� �� �ִ�)\\



6 . non-covering


    \begin{figure}[htb]
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$\hspace{15em}$ �׸� 1\\



$p$�� local homeomorphism �� ������ covering map�� �� �� ����.(�� �׷���?)\\

 7 . (figure eight, torus ���� n-fold covering.)\\

\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.6,clip=true]{graph2.eps}}

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$\hspace{15em}$ �׸� 2\\

 8 . M\"{o}bius band �� double covering �� annulus �� �ǰ�
3-fold covering�� �ٽ� M\"{o}bius band �� �ȴ�.  ���� �������
Klien bottle�� double covering�� torus��
�ȴ�. �Ϲ������� ������ �����϶�.\\



{\bf Note.}

\textit{1 . M is a manifold $\Rightarrow$ $\widetilde{M}$ is also
a 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 Excercise.} If M is a $C^\infty$-manifold $\Rightarrow$
$\widetilde{M}$ is also a $C^\infty$-manifold .\\

\textit{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�̴�.\\

{\bf ���� 7.}\\ \textit{Every non-orientable manifold has an
orientable double covering manifold.}\\



\textit{3 . M is a compact manifold, p is a finite covering
$\Rightarrow$ $\widetilde{M}$ is compact.}\\

\textit{4} . M�� orientable�̰� genus �� g�� ���� ���, �̿� ����
$\widetilde{M}$�� ���캸��. ���� ���뿡 ����  $\widetilde{M}$��
���� orientable�̹Ƿ� $\chi$�� �˸� $\widetilde{M}$�� ������ ��
�ִ�. �׷��� $\widetilde{M}$�� n-fold ���  M�� triangulation��
$\widetilde{M}$ ���� �ø��� V,E,F ��� n�谡 �ǹǷ� $\chi$ ����
n�谡 �ȴ�. ����� ������� $M$�� non-orientable �� ����
$\widetilde{M}$�� orientability�� ���� $\widetilde{M}$�� ������ �� �ִ�.\\

\end{document}
