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\begin{document}
 \parindent=0cm
  \section*{Deck transformation}
G�� covering $p:\widetilde{X}\rightarrow X$��  deck transformation
group �̶�� ����. �� �� ���� �� ������ �����Ѵ�.

{\bf 1. G acts on $\widetilde{X}$ on the \textit{left}}.

$\,\,g:\widetilde{X}\rightarrow \widetilde{X}$

$\hspace{2em}x\mapsto g\cdot x=g(x)$ and $(g \circ f)(x)=g(f(x))$.

{\bf 2. And the action is \textit{free}, i.e., $\forall g\neq
1\,\,$,$\,\,g(\widetilde{x})\neq \widetilde{x}\,\,$,$\,\forall
\widetilde{x} \in X$ .}

( Or equivalently, $ G_{\widetilde{x}}=\{1\}\,\,$, $\,\,\forall
\widetilde{x}\in X$.)

\begin{proof}

$\hspace{9em}g$

$\hspace{5em}(\widetilde{X},\widetilde{x})\hspace{0.5em}\rightarrow\hspace{0.5em}(\widetilde{X},\widetilde{x})$


$\hspace{6em}p \searrow \hspace{2em}\swarrow p$

$\hspace{8em}(X,x)$

�� diagram ���� $g$�� $p$�� lifting(�� morphism)���� �� �� �ְ�,
$id$ map�� $\widetilde{x}$�� $\widetilde{x}$�� ������
morphism�̹Ƿ� \textit { Uniqueness of morphism(lifting)} �� ����
$g=id$ �� �ȴ�.
\end{proof}\\

{\bf Note.}(\textit{Uniqueness}) �� ��Ƿκ��� �ΰ��� deck
transformation $g$�� $h$ �� �� point $x$���� ��ġ�ϸ�, $g$��
$h$�� map���μ� ������ ���ƾ� �Ѵٴ� ���� �� �� �ִ�.\\


G action on $\widetilde{X}$��  G action on $p^{-1}(x)$ �� �ְ�
�̰� ���� free�̴�. �׷��� �տ��� �����ߴ� right action of
$\pi_1(X,x)$ on $p^{-1}(x)$�� �ٽ� ����. �׷���

\begin{prop}
Two actions commute, i.e., $g(\widetilde{x}\cdot
\{\alp\})=g(\widetilde{x})\cdot \{\alp\}\,\,\,\,\,$ for
$\,\forall \,\,\widetilde{x}\,,\,\forall
\,\,\{\alp\}\,,\,\forall\,\, g$.
\end{prop}

\begin{proof}
*�׸�*


$p\circ (g\circ \widetilde{\alp})=p\circ \widetilde{\alp}$ �̹Ƿ�
$g\circ \widetilde{\alp}$��  $(g\circ
 \widetilde{\alp})(0)=g(\widetilde{x})$���� $(g\circ
 \widetilde{\alp})(1)=g(\widetilde{x}\cdot\{\alp\})$�� ����
$\alp$�� lifting �� �ȴ�. �׷��� $g(\widetilde{x})\cdot
\{\alp\}=(g\circ \widetilde{\alp})(1)$ �̹Ƿ�
$g(\widetilde{x}\cdot \{\alp\})=g(\widetilde{x})\cdot \{\alp\}$��
�ȴ�.
\end{proof}\\

\begin{prop}
Let $p:\widetilde{X}\rightarrow X$ be a covering and
$\widetilde{x},\widetilde{x}'\in p^{-1}(x)$. Then

 $\,\,\exists g\in G$ such that $g(\widetilde{x})=\widetilde{x}'$

$ \,\,{\Leftrightarrow}\,\,
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})=
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}')$

 $\,\,\Leftrightarrow$ $\widetilde{x}'=\widetilde{x}\cdot \{\alp\}$,
$\{\alp\}\in N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))$.

 ���⼭ $\pi_1(X,x)$�� subgroup $H$�� ����,
$N(H)$�� $H$�� normalizer�̴�.
\end{prop}

\begin{proof}

ù��° equivalence�� �տ��� ���� ���� �ְ�, �ι�° equivalence��
��������.

($\Rightarrow$) �Ϲ������� $x$�� $x'$�� �մ� path �� $\rho$�� ��
�� $\pi_1(\widetilde{X},\widetilde{x})$ �� $
\pi_1(\widetilde{X},\widetilde{x}') $ ���̿� isomorphism
$\phi_{\rho}$ �� �����Ѵ�. �� $\rho$ �� ������ $\pi_1(X,x)$�󿡼�
loop �� �ǰ� �� loop $p\circ \rho$�� $\alp$�� ����. �׷���
$\{\alp\}\in \pi_1(\widetilde{X},\widetilde{x})$ with
$\widetilde{x}'=\widetilde{x}\cdot \{\alp\}$ �� ����
$\{\alp\}p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}')\{\alp\}^{-1}=
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$  �̴�. �׷��� $
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})=
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}')$ �̹Ƿ�
$\{\alp\}p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})\{\alp\}^{-1}=
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$  �� �Ǿ�
$\{\alp\}\in N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))$ ��
�ȴ�.\\

($\Leftarrow$) $\{\alp\}\in
N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))$  �̸�

$ p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}')=
\{\alp\}^{-1}p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})\{\alp\}=
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$ �̹Ƿ�

$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})=
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}')$.

\end{proof}



�� �������� ���� $\phi$ �� ������ ����.

$\phi:N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))\rightarrow
G\,\,\,,\,\,\,\,\,\,g_{\alp}(\widetilde{x})=\widetilde{x}'=\widetilde{x}\cdot
\{\alp\}$.

$\hspace{4em}\{\alp\}\hspace{1.8em}\mapsto g_{\alp}$

{\bf 1. $\phi$ is a homomorphism. }

(����) $g_{\alp \beta}=g_{\alp}\circ g_{\beta}$ �� ������.

$\pi_1$ action�� deck transformation group $G$ action�� commute
�ϹǷ�

 $g_{\alp}(\widetilde{x}\cdot \{\beta\})=g_{\alp}(\widetilde{x})\cdot
 \{\beta\}=(\widetilde{x}\cdot \{\alp\})\cdot\{\beta\}=\widetilde{x}\cdot (\{\alp\}\{\beta\})=g_{\alp
 \beta}(\widetilde{x})$.

���� �� Note���� ����� deck transformation��  uniqueness�� ����
$g_{\alp \beta}=g_{\alp}\circ g_{\beta}$ �̴�.

{\bf 2. $\phi$ is onto.}

(����)������ �־��� $g\in G $�� ����
$\widetilde{x}'=g(\widetilde{x})$�� ����. �׸���  $\rho$��
$\widetilde{x}$���� $\widetilde{x}'$ �� ���� path�� ����. �׷���
loop $\alp=p\circ \rho$�� ���� ���� 2�� ����  $\widetilde{x}'=x\cdot \{\alp\}$
,$\{\alp\} \in
N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})) $�� �ȴ�.

 {\bf 3. $ker\,\,\phi=p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$.}

(����)���� $\phi(\{\alp\})=id$ ��� $g_{\alp}=id$ �̰�,

$\widetilde{x}\cdot \{\alp\}=g_{\alp}(\widetilde{x})
=\widetilde{x} $ �̹Ƿ�  $ \{\alp\} \in
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$�̴�. ������
$\{\alp\} \in p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$�̸�
$g_{\alp}(\widetilde{x})= \widetilde{x}\cdot \{\alp\}
=\widetilde{x}$�̰� uniqueness�� ���� $g_{\alp}=id$�̴�. \\

���� 1,2,3 �� ���� ���� ���������� �������.

\begin{cor}
$\displaystyle{G \cong
N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))\diagup
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})}$.
\end{cor}

HW. �� ${\phi}$�� ���� induce�� isomorphism ${\theta}:N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))\diagup
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})\rightarrow G$�� base point $\widetilde{x}$�� depend�Ѵ�.
base point�� �޶��� �� ��� �޶������� ���϶�.\\

 Ư�� $p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$ ��
 $\pi_1({X},{x})$ �� normal subgroup�� ���,
 ������ �����Ѵ�.

\begin{cor}Let $p:\widetilde{X}\rightarrow X$ be a regular
covering. Then $G\cong \pi_1(X,x)\diagup
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$.

�׸��� �̰��� $p^{-1}(x)$�� 1-1 correspondence �� ������.
\end{cor}
(����) $\pi_1$-action �� �������� 1���� 1-1 correspondence ��
������ ������.

\begin{cor}
Let $p:\widetilde{X}\rightarrow \widetilde{X}$ be a universal
covering. Then $G \cong \pi_1(X,x)$.
\end{cor}
\begin{proof}
$\widetilde{X}$��  simply connected �̹Ƿ�
$\pi_1(\widetilde{X})=1$ �� �ȴ�. ���� $
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})=1$ �� �ǰ�, �̴�
�翬�� $\pi_1(X,x)$�� normal subgroup�� �ǹǷ�

$\displaystyle{G \cong
N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))\diagup
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})}$

$\hspace{1em}= \pi_1(X,x)\diagup
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$

$\hspace{1em}= \pi_1(X,x)$.

\end{proof}
\\
Q. $g_{\alp}$�� ���� ��� map�ΰ�?\\

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

\begin{proof}
($\Rightarrow$)� $\widetilde{x}$�� ���� $G\cdot \widetilde{x}=p^{-1}(x)$ ��
�������� ���̸� �ȴ�.

$p$�� regular covering�̸�
$N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))=\pi_1(X,x)$�̹Ƿ�
  ������ $\widetilde{x}'\in p^{-1}(x)$�� ���� $\widetilde{x}$����
  $\widetilde{x}'$�� ���� path $\rho$�� ��� $\alp=p\circ \rho$ ��
  �θ� ���� 2�� �ι�° equivalence�� ����
$g\cdot \widetilde{x}=\widetilde{x}'$ �� �����ϴ� $g\in G$ ��
�����Ѵ�.

($\Leftarrow$) G action on $p^{-1}(x)$ is transitive�ϹǷ� ������
$\widetilde{x}$�� ���� $G\cdot \widetilde{x}=p^{-1}(x)$ ��
�����Ѵ�. ������ $\{\alp\}\in \pi_1(X,x)$�� ����
$\widetilde{x}'=\widetilde{x}\cdot \{\alp\}$�� �θ�
transitivity�� ����  $g(\widetilde{x})=\widetilde{x}'$�� �����ϴ�
$g$�� �����Ѵ�. ���� ���� 2�� �ι�° equivalence�� ����
$\{\alp\}\in N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))$
�̴�. �ٽ� ����
$\pi_1(X,x)=N(p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}))$�̰�
$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x})$�� normal subgroup
�̴�.

\end{proof}
\\
HW. "Figure eight"�� regular covering�� non-regular covering�� construct�϶�.

\end{document}
