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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}

%따라서 임의의 $x\in X$에 대해 $p^{-1}(x)$ 는 discrete하다.\\

{\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 $\,\,\,\,\,\,p(z) = z^n$.

이와 같이 $p^{-1}$의 image가 $n$개인 경우를 $n$-sheeted covering
혹은 $n$-fold covering 이라고 부른다.\\


%   figure  %
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\caption{4-fold covering of $S^1$}
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5. $p : S^n\hspace{1em}\rightarrow \hspace{1em} \mathbb{R}P^n =
S^n/ \sim$   where $x\sim -x$.

이 경우 quotient map $p$는 covering map이 되고 특히 two fold
covering(double covering) 이 된다.\\

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\\
아래 그림과 같이 figure eight의 copy 3개를 잘라서 표시한대로 다시
붙이면 된다.\\

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

마찬가지 방법으으로 torus의 경우에도 단면을 잘라서 붙이면 되는데,
그 결과 torus의 3-fold covering은 torus이며, genus가 2인 surface의
3-fold covering은 genus가 4인 surface가 됨을 알 수 있다. \\

{\bf Note.}

1. $M$ is a ($\cc^\infty$-)manifold $\Rightarrow$ $\widetilde{M}$
is also a ($\cc^\infty$-)manifold.\\
(증명) 각 $x\in \widetilde{M}$에 대해 $p(x)\in M$ 가  coordinate
chart ($U$,$\varphi$) 를 가지고, 또한 $p(x)$ 는 evenly cover 되는
neighborhood V 를 가진다. 이 때 $U\cap V$ 에 대해 $p^{-1}(U \cap
V)$를 생각해 보면, 이 중 $x$를 포함하는 $U\cap V$의 copy가 있고
이 copy와 $\varphi\circ p$가 $x$의 coordinate chart 를 준다.\\
{\bf Exercise.} $\cc^\infty$- case에는 이렇게 정의된 coordinate
chart들이 서로 $\cc^\infty$-relate되어 있음을 보이라. \\

2. $M$ is orientable $\Rightarrow$  $\widetilde{M}$ is
also orientable.\\
(증명) 먼저 $\widetilde{M}$의 각 점 $x$에 orientation을 주자.
$\widetilde{M}$의  orientation은 local homeomorphism $p$ 를
이용하여 $p(x)$의  orientaion을 그대로 가져다 쓴다. 그러면 각
$x\in \widetilde{M}$에 대해 orientation이 locally constant가 되는
$p(x)$의 근방 $U$를 잡을 수 있고, 또한 $p(x)$에서 evenly cover
되는 $V$를 잡을 수 있다. 이 때 $p^{-1}(U\cap
V)=\displaystyle{\coprod_{a \in A}} W_{a}$ 중 $x$를
포함하는 $W_{a}$에서 orientation은 $p$에 의해 $U\cap V$의 orientation과 같으므로 locally constant이다.\\

3. $M$ is a compact manifold, $p$ is a finite sheeted covering
$\Rightarrow$ $\widetilde{M}$ is compact.\\
(증명) $M$이 compact하므로 evenly cover되는 coordinate
neighborhood 유한개로 덮을 수 있고,
compact set의 finite union은 compact이므로 자명하다. \\

4. Every non-orientable manifold has an
orientable double covering manifold.\\
(증명) non-orientable manifold의 경우 각 점마다 2개의
orientation이 존재하는데, 이를 double covering의 각 sheet에
분리하여 할당한 후 orientation이 일치하도록 붙여나가면 double
covering manifold가 orientable하게 만들 수 있다. \\

(예) M\"{o}bius band의 double covering은 두번 꼬인 band가 되는데
이는 annulus와 같고 3-fold covering은 다시 M\"{o}bius band가 된다.
같은 방법으로 Klien bottle의 double covering은 torus가 된다.\\

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

%   figure : Klien Bottle의 double covering %
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\caption{Klien Bottle의 double covering}
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5. $M$이 orientable이고 genus가 $g$인 경우, 이에 따른
$\widetilde{M}$를 살펴보자. \\
위의 내용에 따라 $\widetilde{M}$는 역시 orientable이므로 $\chi$만
알면 $\widetilde{M}$를 결정할 수 있다.($\chi=2-2g$) 그런데,
$\widetilde{M}$가 $n$-fold라면 $M$의 triangulation을
$\widetilde{M}$ 위로 올리면 $V,E,F$ 모두 n배가 되므로 $\chi$ 역시
$n$배가 된다. ($\chi=V-E+F$) 비슷한 방법으로 $M$이
non-orientable일 때도 $n$-fold covering $\widetilde{M}$의 $\chi$를
알 수 있고,
orientability가 결정되면 $\widetilde{M}$를 결정할 수 있다.\\

{\bf 숙제4.} 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}
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위 그림에서 끝점의 $p$-image 점 근방에서는 evenly cover되는 근방을 잡을 수 없으므로 covering map이 될 수 없다.\\
그러나, 이 때 $p$는 {\it local homeomorphism}이 되는데, 어떤 map
$f: X \to Y$가 {\it local homeomorphism}이라는 것은 임의의 $x\in
X$에 대하여 $x$의 neighborhood $U$와 $f(x)$의 neighborhood $V$가
존재하여 $f|_U$가 $U$와 $V$사이의 homeomorphism이 된다는 것이다.\\


{\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}