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\begin{document}
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  \section*{The Induced Covering Over a Subspace}

�� note ��ü�� ����, $p:\widetilde{X} \longrightarrow X$�� covering map�̶� �ϰ�, $A$�� $X$�� subspace�� �����Ѵ�. �� ��, $\wtA$�� $p^{-1}(A)$�� �� path component�� ��Ÿ���ٰ� ����. ����, $i: A \longrightarrow X$�� inclusion map���� �����Ѵ�.

\begin{prop}
Let $A$ be a path-connected and locally path-connected subspace of $X$, and let $\wtA$ be a path-component of $p^{-1}(A)$. Then $p|_{\wtA} : \wtA \longrightarrow A$ is a covering map, and $p_{\sharp} \pi_1(\wtA, \tilde{a}) = i^{-1}_{\sharp}(p_{\sharp} \pi_1(\wtX, \tilde{a}))$.
\end{prop}

\begin{proof}
\begin{enumerate}
\item {\bf ����} $p|_{\wtA} : \wtA \longrightarrow A$�� covering map�� ���� �����Ͻÿ�.

\item �켱, $p_{\sharp} \pi_1(\wtA, \tilde{a}) \subset i^{-1}_{\sharp}(p_{\sharp} \pi_1(\wtX, \tilde{a}))$���� ������. ���� ���̾�׷�
$$\begin{CD}
(\wtA, \tda)  @> i >>  (\wtX, \tda) \\
@V{p|}VV                    @VV{p}V \\
(A,a)  @>> i >                (X,a)
\end{CD} $$
�� commute�ϹǷ�, functor�� ��� �ۿ��Ͽ� ����, $i_{\sharp} p_{\sharp} \pi_1 (\wtA, \tda) = p_{\sharp} i_{\sharp} \pi_1(\wtA, \tda)$���� �� �� �ִ�. �׷���, $i_{\sharp} \pi_1(\wtA, \tda) \subset \pi_1(\wtX, \tda)$�̹Ƿ�,  $i_{\sharp} p_{\sharp} \pi_1 (\wtA, \tda) \subset p_{\sharp} \pi_1(\wtX, \tda)$�̴�. �׷��Ƿ�, $p_{\sharp} \pi_1(\wtA, \tilde{a}) \subset i^{-1}_{\sharp}(p_{\sharp} \pi_1(\wtX, \tilde{a}))$�� ��´�.

\item �̹����� �ݴ� ����, ��, $p_{\sharp} \pi_1(\wtA, \tilde{a}) \supset i^{-1}_{\sharp}(p_{\sharp} \pi_1(\wtX, \tilde{a}))$���� ������.
$\alpcl \in i^{-1}_{\sharp}(p_{\sharp} \pi_1(\wtX, \tilde{a}))$�� ����. �׷��� $i_{\sharp}\alpcl \in p_{\sharp}\pi_1(\wtX, \tda)$�̰�, ����, $i_{\sharp}\alpcl = p_{\sharp}\{ \tilde{\bet} \}$ for some $\{ \tilde{\bet} \} \in \pi_1(\wtX, \tda)$. �� ��, $i_{\sharp}\alpcl = \{ i \circ \alp \} = \{ \alp \}$ in $\pi_1(X,a)$�̰�, $p_{\sharp}\{ \tilde{\bet} \} = \{ p \circ \tilde{\bet} \} =: \{ \bet \}$�̴�. �׷��Ƿ� "$\alp \sim \bet$ " �̰�, ���� $\tilde{\alp}(1) = \tilde{\bet}(1) = \tilde{a}$�� �Ǿ�, $\{\tilde{\alp}\} \in \pi_1(\wtA, \tda)$���� �� �� �ִ�.
���� �츮�� ���ϴ� ���, $\{\alp\} = \{p\circ\tilde{\alp}\} = p_{\sharp}\{\tilde{\alp}\} \in p_{\sharp}\pi_1(\wtA, \tda)$�� ��´�.
\end{enumerate}
\end{proof}

\begin{cor}
Let $p: \wtX \longrightarrow X$ be a universal covering, and let $A$ be a subspace of $X$. (As usual, $A$ is assumed to be path-connected and locally path-connected.) Let $\wtA$ be a path component of $p^{-1}(A)$. Then, $p_{\sharp} \pi_1(\wtA, \tda) = \ker i_{\sharp}$, and hence $p: \wtA \longrightarrow A$ is a universal covering if and only if $i_{\sharp}$ is injective.
\end{cor}

\begin{prop}
$p^{-1}(A) = \wtA$ if and only if $i_{\sharp} \pi_1(A,a)$ meets every coset of the subgroup $p_{\sharp}\pi_1(\wtX, \tda)$ in $\pi_1(X,a)$.
\end{prop}

\begin{proof}
�츮�� ù ��° ������, $p^{-1}(A) = \wtA$�� �ʿ���������� $p^{-1}(a) = p|^{-1}_{\wtA}(a)$��� ���̴�. �켱 $\Longrightarrow$ ������ �ڸ��ϴ�. �ݴ� ������ ���̱� ���Ͽ�, $x \in p^{-1}(A)$�� ����. �׷��� $p(x) \in A$�̰�, $p(x)$���� $a$������ �մ� $A$-���� path $\rho$�� �ϳ� �����Ѵ�. �̸� lifting�Ͽ�, $\tilde{\rho}(0) = x$�� path $\tilde{\rho}$�� ������, �� ���� $\tilde{\rho}(1)$�� $p^{-1}(a)$�� ����, $p^{-1}(a) = p|^{-1}_{\wtA}(a)$�̹Ƿ�, �ᱹ $\wtA$�� ���� �ȴ�. ����, $x$�� ���� path component�� �־�� �ϹǷ�, $x \in \wtA$�̴�.

����, �츮�� $i: p|^{-1}_{\wtA}(a) \longrightarrow p^{-1}(a)$�� onto map���� ���̸� �ȴ�.

�켱, �Ϲ������� $(G,X)$�� group action�̶� ����. �� $x \in X$�� ���Ͽ�, $ev_x : G \longrightarrow X$�� $ev_x(g)=g \cdot x$�� ��������. �׷��� ���� ���̾�׷�

$ \hspace{11em} ev_x$

$ \hspace{10em} G \longrightarrow G \cdot x $

$ \hspace{8em} can. \downarrow \hspace{1em} \nearrow \cong $

$ \hspace{10em} G/G_x$

�� commute�ϹǷ�, Ư���� ���� �� diagram�� ��� commute�Ѵ�.

$ \hspace{10em} ev                                                      \hspace{15em} ev    $

$ \hspace{6em} \pi_1(A,a) \longrightarrow p|_{\wtA}^{-1}(a)             \hspace{7em} \pi_1(X,a) \longrightarrow p^{-1}(a) $

$ \hspace{5em} can. \downarrow \hspace{3em} \nearrow \cong              \hspace{8em} can. \downarrow \hspace{3em} \nearrow \cong $

$ \hspace{6em} \pi_1(A)/p_{\sharp}\pi_1(\wtA)                           \hspace{8em} \pi_1(X)/p_{\sharp}\pi_1(\wtX)$

����, ���� ���̾�׷�
$$\begin{CD}
\pi_1(\wtX,\tda)        @> p_{\sharp} >>   \pi_1(X,a)          \\
@A i_{\sharp} AA                  @A i_{\sharp} AA      \\
\pi_1(\wtA,\tda)        @> p_{\sharp} >>    \pi_1(A,a)
\end{CD}$$
�� commute�ϹǷ�, $i_{\sharp} : \pi_1(A,a) \longrightarrow \pi_1(X,a)$�� induce�ϴ� map $\bar{\i}_{\sharp} : \pi_1(A)/p_{\sharp}\pi_1(\wtA) \longrightarrow \pi_1(X)/p_{\sharp}\pi_1(\wtX)$�� ���Ͽ�, ���� ���̾�׷��� commute�Ѵ�.
$$\begin{CD}
\pi_1(X,a)        @>can.>>   \pi_1(X)/p_{\sharp}\pi_1(\wtX)       \\
@A i_{\sharp} AA                  @A \bar{\i}_{\sharp} AA      \\
\pi_1(A,a)        @>can.>>   \pi_1(A)/p_{\sharp}\pi_1(\wtA)
\end{CD}$$

����������, ���̾�׷�
$$\begin{CD}
\pi_1(X,a)        @>ev>>             p^{-1}(a)       \\
@A i_{\sharp} AA                      @A i AA      \\
\pi_1(A,a)        @>ev>>             p|^{-1}_{\wtA}(a)
\end{CD}$$
�� commute�ϴ� ���� �ڸ��ϴ�.

�̻��� �����ϸ�,
$$\begin{CD}
\pi_1(X)/p_{\sharp}\pi_1(\wtX)        @>\cong>>             p^{-1}(a)       \\
@A \bar{\i}_{\sharp} AA                                   @A i AA       \\
\pi_1(A)/p_{\sharp}\pi_1(\wtA)        @>\cong>>             p|^{-1}_{\wtA}(a)
\end{CD}$$
�� commute���� �� �� �ְ�, ���� $i$�� onto�� �Ͱ� $\bar{\i}_{\sharp}$�� onto�� ���� ��ġ�� �ȴ�. �̰��� �� $i_{\sharp} \pi_1(A,a)$�� $\pi_1(X,a)$���� ��� $p_{\sharp}\pi_1(\wtX, \tda)$�� coset�� �����ٴ� �Ͱ� ��ġ�̴�.
\end{proof}

\begin{cor}
If $p: \wtX \longrightarrow X$ is a universal covering, then
\begin{enumerate}
\item $p^{-1}(A)$ is path-connected if and only if $i_{\sharp}:\pi_1(A,a) \longrightarrow \pi_1(X,a)$ is onto.
\item $p : p^{-1}(A) \longrightarrow A$ is a universal covering if and only if $i_{\sharp}$ is an isomorphism. 
\end{enumerate}
\end{cor}


\end{document}