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\begin{document}
 \parindent=0cm
  \section*{$\pi_1$-action on $p^{-1}(x)$}



Let G be a group acting on a set X on the left, i.e.,
\\$\exists\,\,\alp : G \times X \rightarrow 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.\\

{\bf Note.} $\alp$ induces a homomorphism :  $G\rightarrow
Perm(X)$.

Orbit of $x=G(x)=G\cdot x=\{g\cdot x\,\,|\,\,g \in G\}$ ,\\
Isotropy subgroup at $x=G_x=\{g\in G\,\,|\,\,gx=x\}$ �� ����
������ �����Ѵ�.

 \textit{$\exists$ a bijection $\phi:G/G_x\rightarrow G\cdot x  $} :

(����)Define $\psi:G\rightarrow G\cdot x $ by $g\mapsto gx$. ��
��, $\psi^{-1}(g\cdot x)=gG_x$���� ������.

���� $h\cdot x=g\cdot x$���,

$\Leftrightarrow (g^{-1}h)x=x$

$\Leftrightarrow g^{-1}h\in G_x$

$\Leftrightarrow h\in gG_x$ .

���� $\psi$�� bijection $\phi:G/G_x \rightarrow G\cdot x$ ��
�ش�.


If the action is \textit{transitive}, i.e., $G\cdot x=X$ for some
$x\in X$,
then $G/G_x \simeq X$.\\(�� �� X�� homogeneous�ϴ� ��� �Ѵ�.)\\

{\bf Note.} $G_{gx}=gG_xg^{-1}$.\\

 $p^{-1}(x_0)$�� ������ ���� $\pi_1(X,x_0)$ action�� right action����  ��������.

\textit{$\,\,\,\,\,\,\,\,\,\,\widetilde{x_0}\cdot
\{\alp\}:=\widetilde{\alp}_{\widetilde{x_0}}(1)$}.

��, $p^{-1}(x_0)$ ���� �� ���� ���� $\alp$��
$\widetilde{x_0}$���� �����ϴ�  lifting
$\widetilde{\alp}_{\widetilde{x_0}}$ �� �������� ���� �ָ�, �̴�
action�� ������ �����Ѵ�. �̸� ���̱� ���� ���� �ΰ����� ������.

(1)
$\widetilde{x_0}(\{\alp\}\{\beta\})=(\widetilde{x_0}\{\alp\})\{\beta\}$.

(2) $\widetilde{x_0}\cdot 1= \widetilde{x_0}$.

(����)(1)�� �翬�ϰ�, (2)�� constant loop�� lifting��Ű�� ����
constant loop�� ���� ���� ������ �����Ѵ�. ���� right action��
�ǰ�, �� �� ���� �ΰ����� �����Ѵ�.

{\bf 1 . $\pi_1$-action is transitive.}:


$X=p^{-1}(x_0)$�� ���� $\pi_1$-action �� transitive���� ������
�ϹǷ�, ������ $\widetilde{x_0}'\in p^{-1}(x_0)$ �� ����
$\widetilde{x_0}\cdot \alp=\widetilde{x_0}'$�� �����ϴ� $\alp$��
������ ���̸� �ȴ�.

$\widetilde{X}$�� path connected�̹Ƿ�  $\widetilde{x}_0$���� $\widetilde{x}_0'$���� ���� path $\gamma$�� ��� $X$�� ���� ���� $\alp$
�� ����. �̸� �ٽ� lifting��Ű��, �̴� $\widetilde{x}_0$���� $\widetilde{x}_0'$���� ���� path $\widetilde{\alp}=\gamma$�� �ȴ�.\\


{\bf 2 . Isotropy subgroup at
$\widetilde{x_0}=p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$}:

(����)($\subseteq$)\textit{Isotropy subgroup at
$\widetilde{x_0}$}��
$\,\widetilde{x_0}\{\alp\}=\widetilde{x_0}\,$�� �����ϴ�
$\,\{\alp\}$���̹Ƿ�
$\,\widetilde{\alp}_{\widetilde{x_0}}(1)=\widetilde{x_0}\,$��
�����ϰ� ���� $\,\widetilde{\alp}$�� loop�� �ȴ�.
$\,\{\widetilde{\alp}\}\in \pi_1(\widetilde{X},\widetilde{x_0})\,$
���� �纯�� $p_{\sharp}$ �� ���ϸ�
$\{\alp\}=p_{\sharp}\{\widetilde{\alp}\}\in
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$ �� �ȴ�.

($\supseteq$) ������ $\{\beta\}\in
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$�� ����. �׷���
� $\{\beta '\}\in \pi_1(\widetilde{X},\widetilde{x_0})$�� ����
$p_{\sharp}(\{\beta '\})=\{\beta\}$ �̴�. �� ��, $\beta$��
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�.\\

���� 1,2 �� ���� ���� ���������� �����Ѵ�.\\

\begin{cor}
$p^{-1}(x_0)$��
$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})\setminus
\pi_1(X,x_0)$ �� 1-1 correspondence �� ������.

\end{cor}

\begin{cor}
The cardinality of $p^{-1}$(= $|p^{-1}(x)|$) is constant,
$\forall\,\, x\in X$.
\end{cor}
\begin{proof}
�� ���� �ٸ� $x_0,x_1 \in X$�� ���� $p^{-1}$�� cardinality��
������ ������. ���� �� ����  lifting
$\widetilde{x_0},\widetilde{x_1}$�� ���� �� ������ �մ� path
$\rho$ �� �����ϰ�, $\rho$�� ���� ���� $p\circ \rho$ �� ����.
�׷��� ���� diagram �� commute�ϰ�

$\hspace{9.3em}\phi_{{\rho}}$

$\hspace{3em}\pi_1(\widetilde{X},\widetilde{x_0})\hspace{2em}\rightarrow\hspace{2em}\pi_1(\widetilde{X},\widetilde{x_1})$

$\hspace{4em}p_{\sharp}\downarrow\hspace{8.5em}\downarrow
p_{\sharp}\hspace{5em}$  diagram ($*$)'

$\hspace{3em}\pi_1(X,x_0)\hspace{2em}\rightarrow\hspace{2em}\pi_1(X,x_1)$

$\hspace{9.3em}\phi_{p\circ \rho}$

���� $p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})\setminus
\pi_1(X,x_0)$ ��
$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_1})\setminus
\pi_1(X,x_1)$�� isomorphism $\phi_{p\circ \rho}$ �� ���� 1-1
correspondence �� �����µ�,  �������� 1�� ���� �º���
$p^{-1}(x_0)$ �� cardinality �� ���� �캯 ���� ����������
$p^{-1}(x_1)$ �� cardinality �� ����. ����
$|p^{-1}(x_0)|=|p^{-1}(x_1)|$ �� �����Ѵ�.


\end{proof}\\

 Ư�� $|p^{-1}(x)|=n $ �� ���, $p:\widetilde{X}\rightarrow X$ ��
 n-sheeted(n-fold) covering �̶�� �θ���.\\

\begin{defn}\textit{
(1) A path connected space X is simply connected if $\pi_1(X)=1$.}

\textit{$\hspace{3.2em}$ (2)$\,p:\widetilde{X}\rightarrow X$ is
called a universal covering if $\widetilde{X}$ is simply
connected.}
\end{defn}

�� ���ǿ� �������� 1 �κ��� ���� ������ �� �� �ִ�.

\begin{cor}
If $\,\,p:\widetilde{X}\rightarrow X$  is a universal covering,
then $|p^{-1}(x)|=|\pi_1(X,x)|$
\end{cor}

\begin{cor}
If $\,\,X$ is simply connected , then $p:\widetilde{X}\rightarrow
X$ is a homeomorphism.
\end{cor}

\begin{proof}
$\pi_1(X,{x})\,$��  trivial�̹Ƿ�  ����  subgroup ��
$\,p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})\,$ ����
 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}




  \end{document}
