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\begin{document}
 \parindent=0cm


\section*{II.3 Category of covering space and deck transformation group}
{\bf Note.} Assume all the spaces are path connected and locally
path connected from now on.
\begin{defn}
Given $\x$, consider a {\bf category} of covering spaces $\tx$ and morphisms between covering spaces. \\
A morphism of $\widetilde{X}_1$ to $\widetilde{X}_2$ is a
  map $\phi:\widetilde{X}_1\rightarrow \widetilde{X}_2$ such that
  the diagram

  \hspace{2em}$\txx\hspace{1em}\overset{\phi}{\rightarrow}
  \hspace{1em}\txxx$

$  \hspace{3em}p_1\searrow \hspace{2em} \swarrow p_2 \hspace{5em}$
commutes.

  \hspace{4.5em}$\x$

\end{defn}

��, �� diagram �� commute�Ѵٴ� ���� fiber�� fiber�� �����ٴ� ����
�ȴ�.
\begin{thm}
Let $p_i:\widetilde{X}_i\rightarrow X, i=1,2 $ be  covering
  maps. Then a morphism $\phi:(\widetilde{X}_1,\widetilde{x}_1)\rightarrow
(\widetilde{X}_2,\widetilde{x}_2)$ exists if and only of
$p_{1\sharp}\pi_1(\widetilde{X}_1,\widetilde{x}_1)\subset
p_{2\sharp}\pi_1(\widetilde{X}_2,\widetilde{x}_2)$. \\In this
case, $\phi$ is unique.
\end{thm}
\begin{proof}
morphism�� ���������� lifting�̹Ƿ�, General Lifting Property��
���� �ڸ��ϴ�. uniqueness ���� lifting�� uniqueness�� ���Ͽ�
�ڸ��ϴ�.
\end{proof}

\begin{thm}
A morphism $\phi$ is a covering map.
\end{thm}

\begin{proof}
Exercise.
\end{proof}\\

{\bf Note.} �Ϲ�������, commute�ϴ� diagram
\begin{center}
$X \ \overset{p}{\rightarrow}\ Y$\\
$ q\searrow \ \ \swarrow r$\\
$ Z$
\end{center}
���� 2���� covering�̸� �������� covering���� ���� �� �ִ�. ($Z$��
universal covering�� ������)



\begin{thm}
Let $p_i:\widetilde{X}_i\rightarrow X, i=1,2 $ be  covering
  maps.
 Then there exists an isomorphism
$\phi:\txx\rightarrow\txxx$ if and only if $
p_{1{\sharp}}\pi_1\txx=p_{2{\sharp}}\pi_1\txxx$
\end{thm}

\begin{proof}
($\Rightarrow$) ���� 1�� �������� �����ϸ� �ڸ��ϴ�.\\
($\Leftarrow$) ���� 1�� ���Ͽ� morphism
$\phi:\txx\rightarrow\txxx$, $\psi:\txxx\rightarrow\txx$��
�����ϰ�, $\psi\circ\phi$�� $\widetilde{x}_1$��
$\widetilde{x}_1$���� ������ lifting�̹Ƿ�, uniqueness�� ���Ͽ�
identity map�� �ȴ�. ���������� $\phi\circ\psi$�� identity
map�̹Ƿ�, $\phi$�� isomorphism�� �ȴ�.
\end{proof}\\


�� �������� base point�� �����ϰ� ���� ������ �������� �� �� �ִ�.

\begin{cor}
Let $p_i:\widetilde{X}_i\rightarrow X, i=1,2 $ be  covering
  maps.
Then there exists an isomorphism
$\phi:\widetilde{X}_1\rightarrow\widetilde{X}_2$ if and only if $
p_{1{\sharp}}\pi_1\txx$ and $p_{2{\sharp}}\pi_1\txxx$ are
conjugate in $\pi_1(X,x)$ for some
$x=p_1(\widetilde{x}_1)=p_2(\widetilde{x}_2)$
\end{cor}

\begin{proof}
�������� �Ϲ�������
$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}_0)$��
$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}_0')$�� conjugate����
�˰� �����Ƿ� ���� 3���κ��� ���� �� �ִ�.
\end{proof}

\begin{defn}
An isomorphism $g:\widetilde{X} \to \widetilde{X}$ is called a
{\it deck transformation} (or a {\it covering transformation}).
The group of deck transformations is called a deck transformation
group and will be denoted by $G$ in this section.
\end{defn}

{\bf Note.}\\
(1) $g, h \in G$, $h(\xt)=g(\xt)$ for some $\xt$ $\Rightarrow
g=h$. Hence $G$-action on $\widetilde{X}$ is free.\\
�̰��� morphism�� lifting�̶�� ��Ƿκ��� uniqueness�� ���Ͽ� �翬�ϴ�. \\
(2) $G$ acts on $p^{-1}(x)$ (as a permutation group.) \\
��, $\xt\in p^{-1}(x)$�� ���Ͽ� $p\circ g(\xt)=p(\xt)=x$�̹Ƿ�,
$g(\xt) \in p^{-1}(x)$�̴�.\\
(3) $g\in G$ and $g(\xt_0 )=\xt_1$, then what is
g($\yt$) for $\yt\in\Xt$?\\
general lifting property���� ����� argument�� ���� $\xt_0$����
$\yt$�� ���� path�� ��� �� path�� $X$�� ���� �� $\xt_1$����
����ϴ� path�� �ٽ� �ø��� �� ������ �ٷ�
$g(\yt)$�̴�. \\

{\bf (����) Group Action}\\
Let G be a group. G acts on a set X on the left if\\
$\exists\,\,\alp : G \times X \rightarrow X$, denoted by $g\cdot x
:= \alp (g,x)$ \\
such that (1)
$(g\cdot h)\cdot x=g\cdot(h\cdot x)$, (2) $ e\cdot x= x$, where $e$ is an identity in G.\\

$\bullet$ Orbit of $x=G\cdot x=\{g\cdot x\,\,|\,\,g \in G\}$ ,\\
$\bullet$ Isotropy subgroup at $x=G_x=\{g\in G\,\,|\,\,gx=x\}$\\
$\bullet$ The action is said to be {\it free} if $g\cdot x=x$ for
some $x$
implies $g=e$.\\
$\bullet$ The action is said to be \textit{transitive} if $G\cdot
x=X$ for some $x\in X$.

$\bullet$ �Ϲ������� $G/G_x\cong G\cdot x  $�� �����Ѵ�.\\
(����)$\phi:G/G_x\rightarrow G\cdot x$�� $\phi(gG_x)=g\cdot x$��
�����ϸ� well-defined�̰� bijection���� ���� ���� �� �ִ�.

$\bullet$ $G_{g \cdot \, x} = g \, G_x \, g^{-1}$ \\
(����) $\subset$ �� �Һ�,\,\,\,   $\supset$ : $ G_x = G_{g^{-1}(gx)}
\supset g^{-1}\, G_{gx} \, g$ ���� ���´�.


\newpage
\section*{Action of paths on fibers}

%   figure : action of paths on fibers %
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\begin{center}
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\rput(1.4,2.25){$\widetilde{\sig}$}%
\rput(0.9,0.05){$x_0$} \rput(2.15,-0.05){$x_1$} \rput(1.4,0.15){$\sig$}%
\rput(1,3){$p^{-1}(x_0)$} \rput(2,3){$p^{-1}(x_1)$}


\end{pspicture}
\end{center}
\caption{Action of paths on fibers}
\end{floatingfigure}

Given $\sig$ from $x_0$ to $x_1$, a lifting $\widetilde{\sig}$
defines a bijection $\theta_{\sig} : p^{-1}(x_0)\to p^{-1}(x_1)$
where $\theta_{\sig}(\widetilde{\sig}(0))=\widetilde{\sig}(1)$\\

Note. (a) $\sig \sim \tau \Rightarrow
\theta_{\sig}=\theta_{\tau}$ \\
\hspace*{33pt}(b) $
\theta_{\sig*\tau}=\theta_{\tau}\circ\theta_{\sig}$\\

�� note�� ���Ͽ�
$\theta_{\sig}\circ\theta_{\overline{\sig}}=\theta_{\sig*\overline{\sig}}
=\theta_{x_0}=id$�̹Ƿ�, $\theta_{\sig}$�� bijection���� �ȴ�.\\

\vspace{8em}
����,\\
(1) $|p^{-1}(x_0)|=|p^{-1}(x_1)|$\\
(2) Have a $\pi_1\x$ (right) action on $p^{-1}(x_0)$:\\
$\xt_0\cdot[\alp]=\theta_\alp(\xt_0)$�� �����ϸ� ���� Note (a)��
���Ͽ� �� ���ǵǰ�, Note (b)�� ���Ͽ�
$\xt_0\cdot([\alp][\bet])=\theta_{\alp*\bet}(\xt_0)=\theta_{\bet}\circ\theta_{\alp}(\xt_0)
=(\xt_0\cdot[\alp])\cdot[\bet]$�� �ȴ�. ���� constant loop��
�ø��� constant loop�� �Ǳ� ������ $\xt_0\cdot 1=\xt_0$�� �����Ͽ�
right action�� ���ǵȴ�.\\

Isotropy subgroup at
$\widetilde{x_0}=p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$:

(����)($\subseteq$)���� $[\alp]\in$ Isotropy subgroup at
$\widetilde{x_0}$���,
$\widetilde{x}_0\cdot[\alp]=\widetilde{x}_0$�̹Ƿ�
$\widetilde{\alp}$�� loop�� �ȴ�. $[\widetilde{\alp}]\in \pi_1
\tx$���� �纯�� $p_{\sharp}$ �� ���ϸ�
$[\alp]=p_{\sharp}[\widetilde{\alp}]\in
p_{\sharp}\pi_1\tx$�̴�. \\
($\supseteq$)������ $[\beta]\in p_{\sharp}\pi_1\tx$�� ����. �׷���
� $[\beta ']\in \pi_1\tx$�� ���� $p_{\sharp}([\beta
'])=[\beta]$ �̴�. �� ��, $\beta$�� lifting�� $\widetilde{\beta}$
�� �θ� $p_{\sharp}[\widetilde{\beta}]=[p\circ
\widetilde{\beta}]=[\beta]=p_{\sharp}[\beta ']$.\\
���� $\beta=p\circ \widetilde{\beta}\sim p\circ \beta'$�̹Ƿ�
$\widetilde{\beta}\sim \beta '$ �� �ǰ�, $\beta '$�� loop �̹Ƿ�
$\widetilde{\beta}$�� loop�� �ȴ�. ����
$\widetilde{\beta}_{\widetilde{x_0}}(1)=\widetilde{x_0}$ ��
�����Ѵ�.\\

(3) $\Xt$ : path connected $\Rightarrow$ $\pi_1$-action is
transitive.\\
fiber ���� ������ �� ���� �����ϴ� path�� �����Ƿ�, �� path��
$p$-image�� loop�� �ǰ� �̰Ϳ� ���� action�� �����ϸ� �ڸ��ϴ�.\\
�� ��Ƿκ��� ������ �� �� �ִ�.
\begin{displaymath}
p^{-1}(x_0)\cong \pi_1\x / p_{\sharp}\pi_1\tx
\end{displaymath}

\begin{cor}
If $X$ is simply connected, i.e., $\pi_1(X)=1$, then
$p:\widetilde{X}\rightarrow X$ is a homeomorphism.
\end{cor}

\begin{proof}
$\pi_1(X,{x})$��  trivial�̹Ƿ�  ����  subgroup��
$p_{\sharp}\pi_1\tx$ ���� trivial�ϴ�. ���� $
|p^{-1}(x)|=|\pi_1(X,{x})|=1$�� �Ǿ�  $p$�� 1-1�� �ȴ�. ����
covering map $p$ �� onto, continuous, open map�̾����Ƿ� $p$��
homeomorphism �� �ȴ�.
\end{proof}

\section*{Actions of G and $\pi_1\x$ on $p^{-1}(x)$}
{\bf Notation.} %
$G$= deck transformation group\\
$\Pi=\pi_1(X,x)$, $Y=p^{-1}(x)$,
$\Pi_y=p_{\sharp}\pi_1(\widetilde{X},y)$\\

1. Two actions commute, i.e.,
$g(\xt\cdot[\alp])=g(\xt)\cdot[\alp]$.\\

%   figure : Two actions commute. %
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\begin{center}
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\rput(1.1,-0.1){$x$} \rput(1.4,0.15){$\alp$}%
\rput(1.3,3.1){$p^{-1}(x)$}

\rput(1.1,1.4){$\xt$}%
\rput(0.9,1.8){$\xt\cdot[\alp]$}%
\rput(1.0,2.2){$g(\xt)$}%
\rput(0.35,2.6){$g(\xt)\cdot[\alp]=g(\xt\cdot[\alp])$}%


\end{pspicture}}
\end{center}
\caption{Two actions commute}
\end{floatingfigure}

\begin{proof}
$p\circ (g\circ \widetilde{\alp})=p\circ \widetilde{\alp}=\alp$
�̹Ƿ� $g\circ \widetilde{\alp}$�� $\alp$�� lifting���� �ȴ�.
�̶�, $g\circ \widetilde{\alp}$�� ��������
$g(\widetilde{x})$�̹Ƿ�, $\pi_1$-action�� ���ǿ� ����
$g(\widetilde{x})\cdot[\alp]=(g\circ \widetilde{\alp})(1)$�̴�.
�׷���, $(g\circ
\widetilde{\alp})(1)=g(\widetilde{\alp}(1))=g(\xt\cdot[\alp])$
�̹Ƿ� $g(\widetilde{x}\cdot [\alp])=g(\widetilde{x})\cdot
[\alp]$�� �ȴ�.
\end{proof}\\

\vspace{9em} 2. (a) By general lifting theorem(�Ǵ� ���� ����3��
����) , $\exists g\in G$ such that $
g(y)=y' \Leftrightarrow \Pi_y =\Pi_{y'}$.\\
\hspace*{2ex} (b) In general,
$\Pi_{y\cdot[\alp]}=[\alp]^{-1}\Pi_y[\alp]$.\\

���� (a)�� (b)�κ��� ���� ����� �� �� �ִ�.\\

\hspace*{2ex} (c) \framebox[1.05\width]{\Large $\exists g\in G$
such that $ g(y)=y\cdot [\alp] \iff [\alp]\in N(\Pi_y)$} \\

 ���⼭ $N(\Pi_y)$�� $\Pi_y$��
Normalizer�� ���Ѵ�. ��, $[\alp]\in
N(\Pi_y)$�� $[\alp]^{-1}\Pi_y[\alp]=\Pi_y$���� ���Ѵ�.\\


(d) Define a homomorphism $\theta:N(\Pi_y)\rightarrow G$ by
$\theta([\alp])(y)=y\cdot[\alp]$\\
(���⼭ $\theta([\alp])$�� deck transformation�̹Ƿ�, uniqueness��
���� �� ���� image $\theta([\alp])(y)$�� ���� ������ �����ȴ�. )\\
���� $\theta$�� homomorphism���� ������. \\
 $\theta([\alp][\bet])(y)=y\cdot([\alp][\bet])=(y\cdot[\alp])\cdot[\bet]\\
 =(\theta([\alp])(y))\cdot[\bet]=\theta([\alp])(y\cdot[\bet])$\hspace{2cm}($\because$ Two actions commute.)  \\
 $=\theta([\alp])(\theta([\bet])\cdot
 y)=(\theta([\alp])\theta([\bet]))(y)$\\

$\theta$ is onto: \\
������ �־��� $g\in G $�� ���� $y'=g(y)$�� ����. �׸���
$\widetilde{\alp}$�� $y$���� $y'$ �� ���� path�� ����. �׷��� loop
$\alp=p\circ \widetilde{\alp}$�� ���� $y'=y\cdot [\alp]$ �� �ǰ�
���� 2(c)�� ���Ͽ� $[\alp] \in N(\Pi_y)$�� �ȴ�.\\

����������, $ker\theta = \{\alp\in N(\Pi_y)|
y\cdot[\alp]=y\}=\Pi_y$���� �˼� �����Ƿ� ������ ��´�.
\begin{center}
\framebox[1.1\width]{\Large $N(\Pi_y)/\Pi_y \cong G$}
\end{center}

�̶�, $N(\Pi_y)/\Pi_y \cong G$�� isomorphism�� $\theta_y$�� �д�.
��, $\theta_y(\overline{[\alp]})=\theta([\alp])$.\\

{\bf Remark.} \\
(1) $\theta_y$ depends on a choice of $y\in p^{-1}(x)$.\\

(2) $\Xt$ is called a universal covering of $X$ if $\Xt$ is simply
connected, i.e., $\Xt$ is path-connected and $\pi_1(\Xt)=0$.
In this case, it follows that $\pi_1(X)=G$.\\

{\bf ���� 6.} Universal covering�� ���, $\theta_y$�� $y$�� ���ÿ�
���� ��� �޶����°�?\\

\newpage
\begin{defn} A covering $p:(\Xt,\xt)\to (X,x)$ is a {\it regular
(or normal) covering} if $p_\sharp \pi_1 (\Xt,\xt)$ is a normal
subgroup of $\pi_1(X,x)$.\end{defn}

{\bf Note.} The notion of regular covering is independent of
choice of base point, i.e., if $p:(\Xt,\xt_0)\to (X,x_0)$ is a
regular covering, then $p:(\Xt,\xt)\to (X,x)$ is also a regular
covering.\\
\begin{proof}
$\hspace{3em}\pi_1(\widetilde{X},\widetilde{x}_0)\hspace{2em}
\overset{\phi_\rho}{\longrightarrow}\hspace{2em}\pi_1(\widetilde{X},\widetilde{x})$\\
$\hspace*{6em}p_{\sharp}\downarrow\hspace{9.5em}\downarrow
p_{\sharp}\hspace{5em}$\\
$\hspace*{5.5em}\pi_1(X,x_0)\hspace{2em}\overset{\phi_{\overline{\rho}}}{\longrightarrow}\hspace{2em}\pi_1(X,x)$\\
$\rho$�� $\xt_0$���� $\xt$�� ���� path�� �ϰ�,
$\overline{\rho}=p\circ\rho$�� �ϸ�, $\phi_\rho,
\phi_{\overline{\rho}}$�� ��� isomorphism�̰�, �� diagram��
commute�ϹǷ�,$p_\sharp \pi_1 (\Xt,\xt_0)$�� $\pi_1(X,x_0)$��
normal subgroup�̸� $p_\sharp \pi_1 (\Xt,\xt)$�� $\pi_1(X,x)$��
normal subgroup�� �ȴ�.
\end{proof}

\begin{thm}
 $p:\widetilde{X}\rightarrow X$ is a regular covering
$\Leftrightarrow$ G action on $p^{-1}(x)$ is transitive.
\end{thm}

\begin{proof}
$p$�� regular covering�̶�� ���� � $y\in\Xt$�� ���Ͽ�
$N(\Pi_y)=\Pi$��� �Ͱ� ��ġ�̴�. ����, $N(\Pi_y)=\Pi$���, ������
$y'\in p^{-1}(x), x=p(y)$�� ���Ͽ� $y$���� $y'$���� ���� path
$\widetilde{\alp}$�� �����ϰ� �̰��� $p$-image�� $\alp$��� �ϸ�
$[\alp]\in\Pi=N(\Pi_y)$�̰� $y'=y\cdot[\alp]$�̴�. ����, 2.(c)��
���Ͽ�
$g(y)=y'$�� $g\in G$�� �����ϹǷ� $p^{-1}(x)=G\cdot y$�̴�.\\
������ $p^{-1}(x)=G\cdot y$���, ������ $[\alp]\in\Pi$�� ���Ͽ�,
$g(y)=y\cdot[\alp]$�� $g$�� �����ϹǷ� 2.(c)�� ���Ͽ� $[\alp]\in
N(\Pi_y)$�� �Ǿ� $N(\Pi_y)=\Pi$�� �ȴ�.
\end{proof}\\


{\bf Application.} If $\Pi=\pi_1(X)$ for some $X$ and $\Gamma$ is
a subgroup of $\Pi$
of index 2, then $\Gamma\lhd\Pi$.\\
\begin{proof}
���� ���� ������ ���� $p_\sharp\pi_1\tx=\Gamma$�� �Ǵ� covering
$\Xt$�� �����ϰ�, index�� 2�̹Ƿ� �� covering�� double covering��
�ȴ�. �� ��, �� fiber�� �� ���� ġȯ�ϴ� map�� deck
transformation�� �ǰ� deck transformation group $G$�� order 2��
group�� �ǹǷ� $G$-action�� transitive�ϰ� �ȴ�. ����, ����
������ ���� $\Gamma\lhd\Pi$���� �� �� �ִ�.
\end{proof}

\newpage
�Ϲ������� group $G$�� set $X$�� act�ϴ� ��쿡 orbit���� ���� $\{
G\cdot x ~|~ x\in X\}$�� orbit space��� �ϰ�, $G\setminus X$��
����. �� ��, cannonical projection $q:X\to G\setminus X$��
surjection�̹Ƿ�, $G\setminus X$�� quotient topology�� �ش�. �̶�,
$q$�� quotient map�̶�� �θ���.
\begin{thm}
If $p:\Xt\to X$ is a regular covering, then $X=G\setminus \Xt$.
\end{thm}
\begin{proof}
quotient map $q:\Xt\to G\setminus X$�� �����ϸ�, ���� 6�� ���Ͽ�
$q$�� �ٷ� covering map $p$�� ��ġ�Ѵٴ� ���� �� �� �ְ�, ����
$X=G\setminus \Xt$�̴�.
\end{proof}\\

�� ������ ���� �����ϱ� ���Ͽ� ������ �����Ѵ�.
\begin{defn}
A group action $(G,\Xt)$ is called a {\it covering action} if\\
(1) $G$-action is free \\
(2) $\forall \xt\in\Xt$, $\exists U$ a neighborhood of $\xt$ such
that $g(U)\bigcap U=\varnothing$, $\forall g\in G$.
\end{defn}
\begin{thm}
If $(G,\Xt)$ is a covering action, then the quotient map $q:\Xt\to
G\setminus \Xt$ is a regular covering.
\end{thm}
\begin{proof}
covering action�� ���� (2)�� ����, ������ $\xt\in \Xt$�� ���� ����
neighborhood $U$�� �����Ͽ� $g(U)$���� ��� disjoint�ϰ� �ȴ�.
����, $g(\xt)\in G\setminus \Xt$�� neighborhood $q(U)$�� �ٷ�
evenly covered neighborhood�� �Ǿ�, $q$�� covering map�� �ȴ�.
�̶�, $G$�� ���ҵ��� deck transformation�� �ǰ�, orbit space��
���ǿ� ����, $G$-action�� transitive���� �翬�ϴ�. ����, ����
6�� ���Ͽ� regular covering�� �ȴ�.
\end{proof}\\

{\bf ���� 7.}. Let $p:\Xt\to X$ be a covering map. Show that $\Xt$
is Hausdorff if $X$ is Hausdorff. Is the converse true?\\
(Hint. Let $f:\rb^2\setminus \{0\} \to \rb^2\setminus \{0\}$ be
given by
$f=( \begin{array}{rr}2 & 0 \\ 0 & \frac{1}{2}\end{array} )$, and\\
\hspace*{3em}consider the action $(G,\rb^2\setminus \{0\})$ where
$G=<f>\cong\mathbb{Z}$.)


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