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\begin{document}
 \parindent=0cm
  \section*{Lifting Property  }

\begin{thm}\textit{\textbf{(Uniqueness of lifting)}}\\
\textcolor{blue}{�־��� covering
p:$(\widetilde{X},\widetilde{x_0})\rightarrow(X,x_0 )$�� ���� Y��
connected�̰� $f:(Y,y_0 )\rightarrow(X,x_0 )$�� lifting
$\widetilde{f} : (Y,y_0 ) \rightarrow
(\widetilde{X},\widetilde{x_0 })$�� ������, �̴�
unique�ϴ�.}\end{thm}

\framebox{\hspace*{1em}\parbox[b]{14cm}{
$\hspace*{6em}(\widetilde{X},\widetilde{x_0})$\\
$\hspace*{2.3em}\exists
\widetilde{f}\nearrow\hspace{3em}\downarrow p$ : covering
$\hspace{1em}\Rightarrow\hspace{1em}\widetilde{f}$,
a lifitng of $f$, is unique if it exists.\\
$(Y,y_0)\hspace{1em}\overset{f}{\rightarrow}\hspace{1em}(X,x_0)$
}}\\

\begin{proof}\\
$\widetilde{f} , \widetilde{f}'$��  f�� lifting�̶� ����.\\

A := $\{y \in Y| \widetilde{f}(y) = \widetilde{f} '(y)\}\ni y_0$\\

A is open : \\
$y\in A$�� ���� f($y$)�� evenly covered�� open neighborhood U��
ã��.\\
�׷��� $p^{-1} (U) = \coprod_{\alp} \widetilde{U_{\alp}}$�̰�,
$\exists \widetilde{U_{\alp}} \ni \widetilde{f} (y) =
\widetilde{f} '(y)$�̴�.\\
�׷��� $y\in \widetilde{f}^{-1} (\widetilde{U_{\alp}}) \cap
\widetilde{f}'^{-1}
(\widetilde{U_{\alp}}) \subset A$�̹Ƿ� A�� open�̴�. \\

���������� $A^c = \{y \in Y|\widetilde{f}(y)\neq\widetilde{f}
'(y)\} $�� open�̴�.\\

�׷��� Y�� connected�̹Ƿ� $A^c = \phi$�̰� �̷μ� uniqueness of
lifting�� �����Ǿ���.
\end{proof}

\begin{lem}\textit{\textbf{(Lebesgue Covering lemma)}}\\
\textcolor{blue}{In a compact metric space, every open cover
$\mathcal{ U}$ has a Lebesgue number, i.e. , $\exists \epsilon
>0$(depending on $\mathcal {U})$ s.t. $\forall$ A $\subset X$ with
diam(A)$<\epsilon$, $\exists$ U $\in \mathcal{U}$ s.t. A$\subset
U$.}\end{lem}
\begin{proof}\\
D�� $\{$diam(A) $\in \mathbb{R} |$ A is not contained in any
$U\in\mathcal{U}\}$��� ����. �̷� A�� "big set"�̶� �θ���.

�׷��� $\epsilon $=inf D $>$0���� ���̸� ������ ������.

���� inf D�� 0 �̶�� $\forall n, \exists A_n :$ big set s.t.
diam($A_n) < \frac{1}{n}$ �̰�, $x_n \in A_n$ �� $\{x_n\}$�� ã��
�� �ִ�.

�׷��� X �� compact�ϹǷ� $x_n \rightarrow x$(by passing to a
subsequence)�� $x$�� �����Ѵ�.

�׷��� $x$�� �����ϴ� U$\subset \mathcal{U}$�� ���ԵǴ� $A_n$(n��
����� Ŭ ��)���� ã�� �� �ְ� �̴� big set�� ���ǿ� ����ȴ�.\end{proof}\\

\begin{thm}\textit{\textbf{
(Unique path lifting property)}}\\
\textcolor{blue}{Let $p: \widetilde{X} \rightarrow X$ be a
covering map and let $\alp : I \rightarrow X$ be a path with
$\alp(0)=x_0 \in X $ and $ p(\widetilde{x_0})=x_0$. Then $\alp$
has a unique path lifting $\widetilde{\alp}:I\rightarrow X$ with
$\widetilde{\alp}(0)=\widetilde{x_0}$ i.e., $p\circ
\widetilde{\alp}(t)=\alp (t).\,\,\forall \,\,t \in I.$}\end{thm}
\begin{proof}\\\textit{\textbf{(Existence)}}\\ For each t, $\alp(t)\in X$ has an
open neighborhood $U_t$ which is evenly covered by
$\displaystyle{\coprod_{a \in A}}V_{t,a}$ .  Since I=[0,1] is
compact, we can choose a Lebesgue number $\epsilon > 0$ for a
cover $\{\alp^{-1}(U_t)| t
\in I \}$ of I. Choose a partition of $\,\,$ I,\\
$\,\,0=t_0<t_1<...<t_{n+1}=1\,\,$ so that
$t_{i+1}-t_{i}<\epsilon,\,\,\,\,i=1,...n$.

 Then note that $\alp[t_i,t_{i+1}] \subset U_t$ for some t and we
 lift $\alp|_{[t_i,t_{i+1}]}$ inductively :

 Suppose $\alp|_{[t_0,t_{i}]}$ is already lifted (note that the
 initial point $x_0$ is lifted to $\widetilde{x_0}$). Then $\alp[t_i,t_{i+1}]\subset
 U_t$ for some t and $p^{-1}(U_t)=\displaystyle{\coprod_{a \in
 A}}V_{t,a}$ and there exists a unique $a \in A$ such that
 $\tilde{\alp}(t_i)\in V_{t,a}\,.$\\And since $p|_{V_{t,a}}:V_{t,a}\rightarrow
 U_t\,\,$is homeomorphism we can lift $\alp|_{[t_i,t_{i+1}]}$ using $(p|_{V_{t,a}})^{-1}$ and the proof is
 completed.\\

\textit{\textbf{(Uniqueness)}}\\ The uniqueness follows from the
uniqueness property of general lifting.
\end{proof}\\

\begin{thm}
\textit{\textbf{(Covering homotopy
property)}}\\\textcolor{blue}{Let
$p:(\widetilde{X},\widetilde{x_0})\rightarrow (X,x_0)$ be a
covering.\\
Let $f :(Y,y_0)\rightarrow (X,x_0)$ and
$\widetilde{f}:(Y,y_0)\rightarrow
(\widetilde{X},\widetilde{x_0})\,\,\,$ be a lifting of f. \\
Then for a homotopy F : $Y\times I \rightarrow X$ s.t.
$F|_{Y\times \{0\}} =f $, there exists a unique lifting
$\widetilde{F}$ of F s.t. $\widetilde{F} |_{Y\times \{0\}} =
\widetilde{f}$.}
\end{thm}

\framebox{\hspace*{1em}\parbox[b]{14cm}{
$\hspace*{6em}(\widetilde{X},\widetilde{x_0})\hspace{13em}\widetilde{X}$\\
$\hspace*{2.3em}\exists
\widetilde{f}\nearrow\hspace{3em}\downarrow
p\hspace{4em}\Rightarrow\hspace{3em}\exists ! \widetilde{F}$
\textcolor{blue}{$\nearrow$}$\hspace{2em}\downarrow p\hspace{1em}$
s.t. $\widetilde{F}
|_{Y\times \{0\}} = \widetilde{f}$\\
$(Y,y_0)\hspace{1em}\overset{f}{\rightarrow}\hspace{1em}(X,x_0)\hspace{7em}Y\times
I \hspace{1em}\overset{F}{\rightarrow}\hspace{1em} X $}}\\
\newpage
\begin{proof}\\Step 1 . Lift F on $U_y \times I $ for each $y\in
Y$.\\
���⼭ $y$�� neighborhood $U_y$�� I�� compactness�� �̿��Ͽ�
construct�Ѵ�.

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1)$f(y\times I)$ ������ evenly cover �Ǵ� covering $\{W_t\}$�� ��´�.\\
2) �� �� $\{f^{-1}(W_t )\}$�� $y\times I$�� covering�� �ǰ� �� ��
(y,t)���� $f^{-1}(W_t)$�� ���ԵǴ� product neighborhood $U_t
\times V_t $�� ������ I��
compactness�� ���� �̵� �� ���Ѱ��� $y\times I$�� cover�� �� �ִ�. \\
3) �� finite�� product neighborhood�鿡�� $U_t$���� intersection��
$U_y$�� �д�.\\
4) ������ $U_y \times I$�� lift�ϱ� ���Ͽ� I�� ������ partition��
�־� �� $U_y \times [t_i , t_{i+1}]$�� 2)���� ���� finite product
neighborhood�� ���Եǵ��� �Ѵ�.\\
5) path lifting ���� ���������� inductive�ϰ� lifting�Ѵ�.
\\

Step 2. Piece together $\widetilde{F}|_{U_y \times I}$
to get $\widetilde{F}$ on $Y\times I$.\\
Step1���� �� $y\in Y$�� ���� $U_y \times I$�󿡼�
$\widetilde{F}$�� �����ߴµ� �̵��� ��ġ��
�κп��� ��ġ���� ���̸� $Y\times I$���� $\widetilde{F}$�� �� ���ǵ��� �� �� �ִ�.\\
��, $z \in U_y \cap U_y'$�� ���� $\widetilde{F}|_{U_y \times I}
(z,t) =
\widetilde{F}|_{U_y' \times I} (z,t)$���� ������ �Ѵ�.\\
�̰��� $U_y$���� ���� $U_y'$ ���� ���� $ y\times I$�� lifting�̰�
���� initial point�� ������ �����Ƿ� uniqueness of lifting�� ����
��ġ���� �� �� �ִ� .\end{proof}
\\
\begin{cor}\textit{\textcolor{blue}{$\alp \sim \beta \Rightarrow \widetilde{\alp}\sim
\widetilde{\beta}$.}}\\\textcolor{blue}{���⼭ $\alp,\beta$�� X��
path���̰� $\widetilde{\alp}$,$\widetilde{\beta}$�� ���� initial
point�� ������ lifting ���̴�. ���� $\widetilde{\alp}$��
$\widetilde{\bet}$�� ���� terminal point�� ���´�.}
\end{cor}
\newpage

\textbf{ Some Consquences}\\

\textbf{1.} \textit{\textcolor{blue}{$\pi_1(S^1)\cong \textbf{Z}$.}}\\

\begin{proof}
Let $p : \textbf{R}\rightarrow S^1$ be a covering map given by
$p(x)=e^{2\pi ix}$. \\If $[\alp]\in \pi(S^1,1)$, then $\alp$ can
be lifted uniquely to $\widetilde{\alp}:I\rightarrow \textbf{R}$
with $\widetilde{\alp}(0)=0$.\\Define
$\phi:\pi(S^1,1)\rightarrow\textbf{Z}\,\,$ by $\,\,[\alp] \mapsto
\widetilde{\alp}(1)$. $\phi $ is well-defined by �������� 5.\\

�� $\phi$�� isomorphism���� ���̱� ���� ����  homomorphism����
������.

$\phi([\alp][\beta])=\phi([\alp*\beta])=\widetilde{\alp*\beta}(1)$�̰�
�̰��� $\widetilde{\alp}(1)+\widetilde{\beta}(1)$���� ���̱� ����
$\tau$�� ������ ���� ��������. \\$\tau : \textbf{R}\rightarrow
\textbf{R}$ be a translation given by
$\tau(x)=x+\widetilde{\alp}(1).$

�� ��,  $\widetilde{\alp*\beta}=\widetilde{\alp}*(\tau\circ
\widetilde{\beta})$�� �����Ѵ�. ���� $\widetilde{\alp}*(\tau\circ
\widetilde{\beta})$�� �� ���ǵǾ����� ��
$\widetilde{\alp}(1)=(\tau\circ \widetilde{\beta})(0)$ ����
���캸��. $\widetilde{\alp}$�� $\widetilde{\bet}$�� �� �� 0����
����ϴ� path�̰�  $\tau$�� $\widetilde{\alp(1)}$��ŭ translation
�����ִ� map�̹Ƿ�,  $\widetilde{\beta}(0)$�� $\tau$�� ����
$\widetilde{\alp}(1)$���� �Ű����� $\widetilde{\alp}*(\tau\circ
\widetilde{\beta} )$�� �� ���ǵȴ�.\\��������
$\widetilde{\alp}*(\tau\circ \widetilde{\beta})$ �� $\alp*\beta$��
lifting ���� ������.

$p\circ (\widetilde{\alp}*(\tau \circ \widetilde{\beta}))=(p\circ
\widetilde{\alp} )*(p\circ (\tau \circ \widetilde{\beta}))=(p\circ
\widetilde{\alp} )*(p\circ \widetilde{\beta})=\alp*\beta$

���� $\widetilde{\alp*\beta}=\widetilde{\alp}*(\tau\circ
\widetilde{\beta})$�� �����ϰ� ������ �����Ѵ�.

$\widetilde{\alp*\beta}(1)=\widetilde{\alp}*(\tau\circ
\widetilde{\beta})(1)=(\tau\circ
\widetilde{\beta})(1)=\tau(\widetilde{\beta}(1))=\widetilde{\alp}(1)+\widetilde{\beta}(1)$.\\

�������� $\phi$ �� 1-1 ���� ������.\\ ���� $\phi([\alp])=0$ �̶��
$\widetilde{\alp}(1)=0$ �� �Ǿ� $\widetilde{\alp}$ �� 0�� base
point�� �ϴ� loop�� �ȴ�. �� $[\widetilde{\alp}]\in
\pi_1(\textbf{R},0)=0$�̹Ƿ�\\
$0=p_{\sharp}[\widetilde{\alp}]=[p\circ
\widetilde{\alp}]=[\alp]\in \pi(S^1,1)$ .  ��
$[\alp]=0$�̴�.\\

���������� $\phi$�� onto���� ���̱� ���� $\forall\,\,n \in
\textbf{Z}$�� ���� $ \widetilde{\alp}(t)=nt\,\,$,$\,\,\alp=p\circ
\widetilde{\alp}$ �� ������,
$\,\,\phi([\alp])=\widetilde{\alp}(1)=n$ �� �ȴ�. ���� $\phi$��
onto�̴�.

\end{proof}\\

\textbf{2.} \textcolor{blue}{Let $\widetilde{X}$ be a path-
connected space and $p: (\widetilde{X},\widetilde{x_0})
\rightarrow (X,x_0 )$ be a covering. Then $p_{\sharp} : \pi_1
(\widetilde{X},\widetilde{x_0})\rightarrow\pi_1(X,x_0)$ is
injective.}\\

\begin{proof} Clear from Covering homotopy property(�������� 5)
\end{proof}\\
\newpage
\textbf{3.}\textcolor{blue}{Let $\widetilde{X}$ be a path-
connected space and $p: (\widetilde{X},\widetilde{x_0})
\rightarrow (X,x_0 )$ be a covering. Then $p_{\sharp}\pi_1
(\widetilde{X},\widetilde{x_0})$ and $p_{\sharp}\pi_1
(\widetilde{X},\widetilde{x_0}')$ are conjugate in $\pi_1 (X,x_0 )
$.}\\

\begin{proof}$\hspace*{1em} \widetilde{x_0}$�� $\widetilde{x_0}'$�� �����ϴ� path $\tau$�� ������ ���� diagram
�� commute�Ѵ�.\\

 $\hspace*{5em}\pi_1
(\widetilde{X},\widetilde{x_o})\hspace{3em}\overset{\phi
_\tau}{\longrightarrow}\hspace{3em} \pi_1
(\widetilde{X},\widetilde{x_o}')$\\
$\hspace*{7em}\downarrow p_{\sharp} \hspace{9em}\downarrow
p_{\sharp}$\\
$\hspace*{5em}\pi_1
(X,x_o)\hspace{2em}\overset{conjugate\hspace{1mm} by [p\cdot
\tau]^{-1}}{\longrightarrow}\hspace{1em} \pi_1 (X,x_o)$\end{proof}\\

\begin{thm}
\textit{\textbf{(General lifting theorem)}} \\
\textcolor{blue}{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}
\framebox{\hspace*{1em}\parbox[b]{14cm}{
$\hspace{6em}(\widetilde{X},\widetilde{x_0})$

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

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

$\hspace{4em}f$}}\\

\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� ���� 1�� ����
�����ȴ�.
\end{proof}\\

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

������ \textit{Unique path lifting property}�� $Y$�� $I$��
����̰�, $Y\times I$�� ��찡  \textit{Covering homotopy property}�̴�.\\
\newpage
\textbf{ Review of locally (path-) connected space.}\\

1. $\forall x \in X$ has a (path-) connected neighborhood.
$\Rightarrow$(path-) component is open ( and hence closed).

\begin{proof}$\hspace{1em}$clear\end{proof}\\

2. X : path-connected $\Leftrightarrow$ X : connected and $\forall
x \in X$ has a path-connected neighborhood.

\begin{proof}$\hspace{1em}(\Rightarrow$)clear\\
$\hspace*{3em}(\Leftarrow)$1.���� path component�� open and closed�̹Ƿ� �������� �ȴ�.\end{proof}\\

3. X : locally path connected $\Rightarrow$ path-component =
component

\begin{proof}$\hspace{1em}$�Ϲ������δ� $\subset$�� �����ϴµ� component�� connected�̹Ƿ� 2�� ���� =�� �����Ѵ�.\end{proof}\\

\begin{thm}\textcolor{blue}{X is locally (path-) connected and p:$\widetilde{X} \rightarrow X$
is a covering.\\If $\widetilde{C}$ is a (path-) component of
$\widetilde{X}$, then\\ $(i)$ p($\widetilde{C}$) is a (path-)
component of X.\\ $(ii)$ p$|_{\widetilde{C}} : \widetilde{C}
\rightarrow p(\widetilde{C})$ is a covering.}
\end{thm}

\begin{proof}$\hspace{1em}$ HW $\sharp$ 5
\end{proof}
\\

���� covering space theory���� path-connected�� ����
�����Ͽ��� �������. ���� fundamental group�� ���ÿ� �����ϱ�
���ؼ��� path-connected�� �ʿ䰡 �ִ�.\\
������ general lifting therorem�� covering space theory���� ����
�߿��� �����̰� �̸� �����Ӱ� ����ϱ� ���ؼ� covering space
theory���� ������ �����鿡 ���� ���� path-connectedness, locally
path-connectedness�� �����Ѵ�.
\end{document}
