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\begin{document}
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  \section*{General Lifting Theorem}

  \begin{thm}
(General lifting theorem) \\ Let
$p:(\widetilde{X},\widetilde{x_0})\rightarrow (X,x_0)$ be a
covering and  $Y$ be a path-connected and locally path-connected
space.
Let $f :(Y,y_0)\rightarrow (X,x_0)$. Then\\
$\exists \,\, \widetilde{f}:(Y,y_0)\rightarrow
(\widetilde{X},\widetilde{x_0})\,\,\,$ : a lifting of f
$\,\,\,\Leftrightarrow\,\,\,$ $f_{\sharp}\pi_1(Y,y_0)\subset
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$.\\In this case
 $\widetilde{f}$is unique.
\end{thm}

$\hspace{6em}(\widetilde{X},\widetilde{x_0})$

$\hspace{2.3em}\exists \widetilde{f}\nearrow\hspace{3em}\downarrow
p\hspace{4em}\Leftrightarrow\hspace{4em}$
$f_{\sharp}\pi_1(Y,y_0)\subseteq
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$

$(Y,y_0)\hspace{1em}\rightarrow\hspace{1em}(X,x_0)$

$\hspace{4em}f$\\

\begin{proof}
($\Rightarrow$)

$f=p\circ \widetilde{f}$ �̹Ƿ� $f_{\sharp}=p_{\sharp}\circ
\widetilde{f}_{\sharp}$ �̰�
$f_{\sharp}\pi_1(Y,y_0)=p_{\sharp}(\widetilde{f_{\sharp}}\pi_1(Y,y_0))\subseteq
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0}) $.

($\Leftarrow$)

������ $y\in Y$�� ���� $\widetilde{f}(y)$�� ������ ���� ��������.
$y_0$�� $y$ ������ path $\rho$ �� ���� ��  $f\circ \rho$��
$\widetilde{x_0}$���� �����ϴ� lifting $\widetilde{f\circ \rho}$
�� ���� $\widetilde{f}(y)=\widetilde{f\circ \rho}(1)$ �� �����ϸ�
�̰��� �ٷ� ���ϴ� $\widetilde{f}$ �� �ȴ�.

���� $\widetilde{f}$ �� �� ���ǵǾ������� ������. ��, $y_0$��
$y$������ path�� ���ÿ�  �������� ���̱� ����, $\sigma$�� �Ǵٸ�
path�� ����. �׷���, $f\circ \rho * \overline{f\circ \sigma}$��
loop�� �ǰ� ������ ����  $\{f\circ \rho * \overline{f\circ
\sigma}\}=f_{\sharp}\{\rho * \overline{\sigma}\}\in
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$ �̴�.

���� $f\circ \rho * \overline{f\circ \sigma}$�� lifting��
loop�̰�, $\widetilde{f\circ \sigma}(1)=\widetilde{f\circ
\rho}(1)$
�� �ǹǷ� $\widetilde{f}$�� �� ���ǵǾ����� �� �� �ִ�.\\

�������� $\widetilde{f}$�� �������� ������. $\widetilde{f}$��
�������� ���̱� ���� $\forall y\in Y$ �� ���� � �ٹ� $W_y$��
�־ $\widetilde{f}|_{W_y}$�� �����̶�� ���� ���̸� ����ϴ�.
$x=f(y)$�� ���� evenly cover�Ǵ� �ٹ��� $U_x$�� �θ� $f$��
�����̰� $Y$�� locally path connected�̹Ƿ� $f(W_y)\subset U_x$��
�����ϴ� path connected $W_y$�� �����Ѵ�. �׷��� $\forall z \in
W_y$ ��  $y$�κ��� $W_y$�ȿ��� path $\tau_z$��  ������ �� �ְ�,
$\rho*\tau_z$�� $y_0$�κ��� $z$������ path�� �ش�. ����
$\widetilde{f}(z)=\widetilde{f\circ (\rho
*\tau_z)}(1)=\widetilde{f\circ \rho}*\widetilde{f\circ
\tau_z}(1)=\widetilde{f\circ \tau_z}(1)$�� �ǰ� �� ��
$\widetilde{f\circ \tau_z}$�� $\widetilde{f}(y)$ ���� �����ϴ�
lifting�̴�.  �� $W_y$���� ������ �� $z$�� $y_0$���� path��
�������� ���� �� path���� $y$�� $z$�� �մ� �κ��� $f$�� ���� ����
$U_x$�� ���� �̿��ؼ� $U_x$������ $V_a$���� homeomorphsim
$p^{-1}$�� ����.

 $p^{-1}(U_x)=\displaystyle{\coprod_{a\in A}}V_a$ and
$\widetilde{f}(y)\in V_a$ �̶�� �θ�   $V_{a}$���� $p^{-1}$��
homeomorphism�̹Ƿ� $\tau_z$�� �̿��ؼ�
$\,\,\widetilde{f}|_{W_y}=p|_{V_a}^{-1}\circ f|_{W_y}$ ���� �� ��
�ְ�, �� $\,\,\,\widetilde{f}$�� path�� ���ÿ� �����ϴٴ�
��Ƿκ��� �׷��� �� �� �ۿ� ����. ���� $\widetilde{f}|_{W_y}$
�� �����̴�.

���������� uniqueness �� Y�� connected�̹Ƿ� ������ ������ ����.

\end{proof}

{\bf Remark.}  This is a generalization of \textit{Unique path
lifting property} and \textit{Lifting of path homotopy theorem}
and \textit{Covering homotopy property}.\\

������ \textit{Unique path lifting property}�� $Y$�� $I$��
����̰�, \textit{Lifting of path homotopy theorem}�� $Y$��
$I\times I$ �� ����̴�. �װ��� �� �Ϲ�ȭ�ؼ�  $I\times I$��
$Y\times I$�� ��
��찡  \textit{Covering homotopy property}�̴�.\\

\textit{Covering homotopy property :}

covering $p:\widetilde{X}\rightarrow X$�� ���� $f:Y\rightarrow
X$�� lifting $\widetilde{f}:Y\rightarrow \widetilde{X}$��
�־����ٰ� ����. $f$��  $g:Y\rightarrow X$ �� homotopy $F:Y\times
I\rightarrow X(F|_{Y\times\{0\}}=f)$ �� ���� homotopic�ϴٰ� ��
�� �� homotopy $F$��  ���� $\widetilde{F}$ �� lifting ��ų ��
�ִ�.

$\widetilde{F}:Y\times I\rightarrow \widetilde{X}\,\,$,
$\,\,\,\,\widetilde{F}|_{Y\times \{0\}}=\widetilde{f}$.\\

{\bf ���� 8. } \textit{$X=M^2$ with triangulation
T$\,\,\Rightarrow$ T
induces a triangulation $\widetilde{T}$ on $\widetilde{X}$.}\\



 {\bf ���� 9.} �Ϲ������� closed surface ���� n-fold covering��
 classify�϶�.



\end{document}
