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\begin{document}
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  \section*{Fundamental group of $S^1$}
\begin{thm}
\textit{$\pi_1(S^1)\cong \textbf{Z}$.}
\end{thm}
\begin{proof}
Let $p : \textbf{R}\rightarrow S^1$ be a covering map given by
$p(x)=e^{2\pi ix}$. \\If $\alp\in \Omega(S^1,1)$, then $\alp$ can
be lifted uniquely to $\widetilde{\alp}:I\rightarrow \textbf{R}$
with $\widetilde{\alp}(0)=0$.\\Define
$\phi:\Omega(S^1,1)\rightarrow\textbf{Z}\,\,$ by $\,\,\alp \mapsto
\widetilde{\alp}(1)$.  Now  note  that $\,\,\phi\,\,$ induces\\
"$\phi$"\,\,:\,\,$\pi_1(S^1,1)\rightarrow\textbf{Z}$ by �������� 3.

�� $\phi$�� 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�. �� $\widetilde{\alp}\in
\Omega(\textbf{R},0) \Rightarrow \{\widetilde{\alp}\}\in
\pi_1(\textbf{R},0)=0$�̹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}

\begin{prop}
Let $f:S^1\rightarrow X$ . Then the followings are equivalent.

1 . f is homotopic to a constant map.

2 . $f_{\sharp}:\pi_1(S^1)\rightarrow\pi_1(X)$ is 0.

3 . f can be extended to a map  $\,\,\overline{f} : D^2\rightarrow
X$.

(In this case f is said to be inessential.)
\end{prop}
\begin{proof}
($1\Rightarrow 3$)

$F: S^1\times I\rightarrow X$ �� $f$��  $c_{x_0}$������
homotopy���  $F$�� $S^1\times I$�� $q : S^1 \times I \rightarrow
D^2 / S^1 \times \{ 1 \}$�� quotient�� quotient space�󿡼��� map
$\overline{F}$�� induce�Ѵ�.

$\hspace{8em}\hspace{9em}F$

$\hspace{6em}\hspace{3em} \hspace{3em}S^1\times I
\hspace{2em}\rightarrow\hspace{2em} X$\\

$\hspace{12em}\,\,q\downarrow \hspace{4em}\nearrow \exists
\overline{F}$\\

$\hspace{10em}S^1\times I/S^1 \times \{1\}$\\


�� �� diagram�� �Ʒ��ʿ� �ִ� $S^1\times I/(S^1\times \{1\})$��
$D^2$ �� homeomorphism ���� ���̱� ���� ������ ����  $\pi$��
��������.

$\,\,\,\,\,\,\,\pi : S^1\times I \rightarrow D^2 $ ,
$\pi(x,t)=(1-t)x$

�׷��� ���� diagram �� �׸� �� �ִ�.


$\hspace{9em}\pi\hspace{8em}F$

$\hspace{5em}D^2\hspace{2em}\leftarrow \hspace{2em}S^1\times I
\hspace{2em}\rightarrow\hspace{2em} X$\\

$\hspace{7em}\exists \overline{\pi}\nwarrow
\hspace{2em}q\downarrow \hspace{4em}\nearrow \exists
\overline{F}$\\

$\hspace{10em}S^1\times I/S^1 \times \{1\}$\\

�� ��, $\pi$ ���� induced�� $\overline{\pi}$�� ����
$\overline{\pi}$ �� 1-1, onto�� �����Ѵ�. �׷��� $D^2$ ��
Hausdorff space�̰�, $S^1\times I/S^1 \times \{1\}$ �� compact
�̹Ƿ�, $\overline{\pi}$�� homeomorphism �� �ȴ�. ����
$g=\overline{F}\circ \overline{\pi}^{-1} : D^2 \rightarrow X$ ��
�����غ���, �̴� $g|_{S^1}=f$�� �����ϰ�, ���� $f$�� $D^2$��
extend �ȴ�.\\
($3\Rightarrow 2$)$\hspace{3em}f\hspace{18em}f_{\sharp}$

$\hspace{3em}S^1\hspace{2em} \rightarrow\hspace{2em}X\hspace{6em}
\hspace{3em}\pi_1(S^1)\hspace{2em}
\rightarrow\hspace{2em}\pi_1(X)$

$\hspace{4em}i\searrow\hspace{2.5em}\nearrow \exists
\overline{f}\hspace{5em}\Rightarrow \hspace{1em}
\hspace{4em}i_{\sharp}\searrow\hspace{2.5em}\nearrow \overline{f}_{\sharp} $

$\hspace{6em}D^2\hspace{10em} \hspace{6em}\pi_1(D^2)$\\

�� diagram ���� ���� �׸��� commute�ϰ�, funtorial property��
���� ������ �׸��� commute�Ѵ�. $\pi_1(D^2)=0$�̹Ƿ�
$f_{\sharp} = 0$�̴�.\\
($2\Rightarrow 1$)

$\pi_1(S^1)$�� �� $\{\alp\}$�� $\alp(t)=e^{2\pi it}$�� ����� ��,
2���� �־��� $f$�� ���� $f_{\sharp}=0$ �̹Ƿ� $f\circ \alp \sim
c_{x_0}
$ �� �ȴ�.\\

$\hspace{4em}$**diagram**\\

$\overline{F}|_{S^1\times \{0\}}=f$ and
$\overline{F}|_{S^1\times\{1\}}=x_0$.








\end{proof}

\end{document}
