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\begin{document}
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\section*{III.3 Applications}
1. $X=$ \resizebox{2cm}{0.4cm}{%
\begin{pspicture}(2,.4)%
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 \hspace{3ex}
 $\Rightarrow$\hspace{3ex}$\pi_1(X)=\zb \underset{\{e\}}{*} \zb=\zb*\zb=F_2$\\
\hspace*{16em}(Free group with 2 generators)\\
�� circle�� �����ϴ� open set�� $U$, $V$��� �ϸ� (��, $U=$\resizebox{1cm}{0.3cm}{%
\begin{pspicture}(1.1,.4)%
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\end{pspicture}},$V=$\resizebox{1cm}{0.3cm}{%
\begin{pspicture}(.9,0)(2,.4)%
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\rput{180}(1,0.2){\pscurve(0,0)(-0.5,-.2)(-1,0)(-0.5,0.2)(0,0)}
\endpsclip
\end{pspicture}}), $\pi_1(U)=\pi_1(V)=\zb$�̰� $\pi_1(U\bigcap V)=\{e\}$�̹Ƿ�
amalgamated product�� �����ϸ� $\pi_1(X)=F_2$���� �� �� �ִ�.\\
���� ������� $\pi_1( \raisebox{-2.5ex}{
\resizebox{6ex}{6ex}{%
\begin{pspicture}(2,2)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
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\rput{120}(1,1){\pscurve(0,0)(-0.5,-.2)(-1,0)(-0.5,0.2)(0,0)}
\end{pspicture}}})=F_3$(Free group with 3 generators)�̴�.\\

2.
\begin{floatingfigure}[l]{6cm}
\begin{pspicture}(3,2)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
% S^2%
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\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-1,0)(1,0)(1,0.5)(-1,0.5)(-1,0)
\psellipse[linestyle=dashed](0,0)(.7,0.2) \pscircle(0,0){.7}
}%
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\end{floatingfigure}

$S^2$�� �����ϴ� open set�� circle�� �����ϴ� open set�� �����ϸ�,
$\pi_1(S^2)=1$, $\pi_1(S^1)=\zb$�̰� �������� fundamental group��
trivial group�̹Ƿ�, �׸��� ���� surface�� fundamental group��
\begin{displaymath}
1*\zb =\zb
\end{displaymath}
�̴�.\\

3. $X=M^2$, closed surface.\\
(1) orientable case \\
���� Torus�� ��츦 �����ϸ�, �������� ���캻 �ٿ� ����
Fundamental group�� $\zb\bigoplus\zb$�̴�. �̸� Van-Kampen
Theorem�� �̿��Ͽ� ���غ���.
\begin{center}
\begin{pspicture}(0,0.3)(4,1.7)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
% Torus%
\rput(1,1){\psellipse(0,0)(1,.7) }
% genus
\rput(1,1){\psarc(0,0.9){1}{240}{300}\psarc(0,-0.95){1}{70}{110} }
 \rput(2.4,1){=}%
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    \rput(.5,-.1){$a$}
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    \rput(.5,1.1){$a$}

    % ����
    \rput(.8,.8){$U$}
    \rput(1.2,.1){$V$}
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\end{center}
Torus�� �簢���� �� ���� ���� Identify�� �ذͰ� ����. ��������
�׸������� ���� Torus���� ���� ���� open set�� $U$, ������
�����ϴ� open neighborhood�� $V$��� �ϸ�, $U\bigcap V$��
annulus�� �����Ƿ� $\pi_1(U\bigcap V)=\zb$�̰�, $V$�� disk�̹Ƿ�
$\pi_1(V)=\{e\}$�̴�. ���� ���� 1���� $\pi_1(U)=\pi_1($figure
eight$)$���� �˾ƺ� �� �����Ƿ�, (������ �׸����� figure eight
$a\cup b$�� $U$�� strong deformation retract�� ���� ���� �� ��
�ִ�.) $\pi_1(U)=\zb*\zb=F_2$�̴�.\\
group presentation�� �̿��Ͽ� amalgamated product�� ���ϱ� ���Ͽ�
�� group�� generator�� �˾ƺ���, $\pi_1(U)$�� generator�� ��
�׸����� loop $a$, $b$�� equivalence class���̰�, $\pi_1(U\bigcap
V)$�� generator�� �������� �ѹ��� ���� loop\\
($c$��� ����)�� equivalence class�̴�. $\pi_1(V)$�� generator��
�����Ƿ� Amalgamated product�� $a,b$�� generate�Ǵ� free group��
relation�� �ָ� �Ǵµ�, loop $c$�� $U$�� loop�� ����
$aba^{-1}b^{-1}$�̰� $V$�� loop�� ���� constant loop�̹Ƿ�
relation�� $aba^{-1}b^{-1}=1$���� �� �� �ִ�.\\
����, $\pi_1(Torus)=<a,b~|~aba^{-1}b^{-1}>=\zb\bigoplus\zb$�̴�. \\

���������� genus�� 2�� surface�� ���ؼ� �����غ���, genus�� 2��
surface�� Torus�� 2�� connected sum���� ���̹Ƿ� ���� �׸��� ����
8������ �̿��Ͽ� ǥ���� �� �ִ�.
\begin{center}
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% Torus%
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% genus
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\rput(1,.8){\psarc(0,0.9){1}{240}{300} \psarc(0,-0.95){1}{70}{110}}%
\rput(2.3,1){=}%
\SpecialCoor  %����ǥ��
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    \rput(1.2;8){$a^{-1}$}
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}
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\end{center}
������ �׸����� ������ ���� �ڸ��� ������ Torus�� disk�� ���
�Ͱ� ������ �� �� �ִ�. (�����κ��� disk�� �ȴ�.) ���� �� �׸���
2���� Torus���� disk�� ����� disk���� ���� ���̹Ƿ�(�ٽ� ����
connected sum)
genus�� 2�� surface�� ��Ÿ����.\\
Torus�� ���� ���������� �����غ���, �� surface�� fundamental
group��
\begin{displaymath}
<a,b,c,d~|~aba^{-1}b^{-1}cdc^{-1}d^{-1}>
\end{displaymath}
�̴�. �Ϲ������� genus�� $g$�� surface�� $4g$�������� ǥ���� ��
�ִ�. ���� ���� ������� $U$, $V$�� ��� amalgamated product�ϸ�
\begin{displaymath}
\pi_1(\Sigma_g)=<a_1,b_1,\cdots,a_g,b_g~|~\prod^g_{i=1}[a_i,b_i]>
\end{displaymath}
���� �� �� �ִ�. (���⼭ $[a_i,b_i]=a_i b_ia_i^{-1}b_i^{-1}$ :
commutator of $a$ and $b$)\\

\newpage
(2) Non-orientable case\\
 Non-orientable surface $M$��
$M=N_k=\rb P^2\sharp\rb P^2\sharp\cdots\sharp\rb P^2$�� ��Ÿ�� ��
�ִ�.\\
����, $M$�� ���� �׸��� ���� �ٰ������� ǥ���� �� �ִ�.
\begin{center}
\begin{pspicture}(2,2)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\SpecialCoor  %����ǥ��
\degrees[6]
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\multido{\ni=0.5+1.0,\nj=1.5+1.0,\i=1+1}{6} {
    \psline(1;\ni)(1;\nj)
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    \rput(1.05;1){$a_1$}
    \rput(1.05;2){$a_3$}
    \rput(1.05;3){$a_3$}
    \rput(1.05;4){$a_2$}
    \rput(1.05;5){$a_2$}
    \rput(1.05;6){$a_1$}
    \psline[linestyle=dashed](1;1.5)(1;5.5)
    \psline[linestyle=dashed](1;1.5)(1;3.5)
    \psline[linestyle=dashed](1;3.5)(1;5.5)
}
\end{pspicture}
\end{center}
�� �׸��� $M=N_3$�� ����ε� ������ ���� �ڸ��� ��� �ﰢ����
$S^2$���� disk�� 3�� ��� �Ͱ� ���� ������ �ﰢ������ $\rb
P^2$���� disk�� ��� �Ͱ� ����.(���������� �����κ��� disk��
�ȴ�.) ���� �� ������ $\rb P^2$ 3���� connected sum�� �Ͱ� ����.\\
orientable case�� ���������� open set $U$,$V$�� ��� Fundamental
group�� ����ϸ�,
\begin{displaymath}
\pi_1(N_k)=<a_1,\cdots,a_k ~|~a_1^2a_2^2\cdots a_k^2 >
\end{displaymath}
���� �� �� �ִ�. Ư�� $k=1$�̸� $\pi_1(\rb
P^2)=<a~|~a^2>=\zb/2$�̴�.\\

{\bf ���� 10.} $K$ : Klein Bottle \\
Since $K=\rb P^2\sharp\rb P^2=N_2$, We know $\pi_1(K)=<a,b~|~a^2
b^2>$.\\
But from $K=$%
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    \psline{-<}(1,0)(1,0.55)
\rput(-.15,.5){$d$} \rput(1.15,.5){$d$} \rput(.5,1.15){$c$}
\rput(.5,-.15){$c$}
\end{pspicture}
}%
, we get $\pi_1(K)=<c,d~|~ cdc^{-1}d>$.\\
(1) Compare two presentations.\\
(2) Find $\pi_1(T^2)$ in $\pi_1(K)$ as a subgroup of index 2. \\
(3) Can you show $\pi_1(K)$ is non-abelian?\\

\newpage
4. $X=S^n$, $n\geq 2$ $\Longrightarrow \pi_1(S^n)=0$ \\
�Ϲݱ��� �����ϴ� open ball�� $U$, ���ݱ��� �����ϴ� open ball��
$V$�� �ϸ� $U$, $V$�� ��� disk�̹Ƿ� $\pi_1(U)=\pi_1(V)=0$�̴�.
���� �̵��� amalgamated product�ϸ� $\pi_1(S^n)=0$�̴�.(���⼭ $n\geq 2$�̾�� $U\bigcap V$�� path connected�̴�.)\\
���� $S^n$�� $\rb P^n$�� double covering space��� ��Ƿκ���
$\pi_1(\rb P^n)=\zb/2$���� �� �� �ִ�.\\

�̸� �̿��ϸ� ���� ������ ��´�.
\begin{thm}[Borsuk-Ulam]
\hfill\\
\textcolor{blue}{(1) There is no anti-pode preserving map $f:S^n \to S^1$. \\
(2) Let $f:S^n \to \rb^2$ $\Rightarrow$ $\exists x\in S^n$ such
that $f(x)=f(-x)$.}
\end{thm}
\begin{proof}
(1) anti-pode preserving map $f$�� �����Ѵٸ� ���� diagram��
commute�Ѵ�.\\
$\hspace*{5em}S^n\hspace{1em}\overset{f}{\longrightarrow}\hspace{1em}S^1$\\
$\hspace*{4.3em}q\downarrow\hspace{5em}\downarrow r$\\
$\hspace*{4.5em}\rb P^n\hspace{.7em}\overset{\bar{f}}{\longrightarrow}\hspace{1em}\rb P^1 =S^1$\\
�� ��, $q$�� $r$�� quotient map�̸� $f$�� anti-pode
perserving�̹Ƿ�, $\bar{f}$�� �� ���ǵȴ�. \\
$\bar{f}_\sharp : \pi_1(\rb P^n) \to  \pi_1(\rb P^1)$�� �����ϸ�,
$\pi_1(\rb P^n)=\zb/2$�̰� $\pi_1(\rb P^1)=\zb$�̹Ƿ�
$\bar{f}_\sharp=0$�̴�. ����, $\rb P^n$�� loop��
$\bar{f}$-image�� $\rb P^1$���� constant loop�� homotopic�� loop��
�ǰ� �� loop�� $S^1$���� �ø��� constant loop�� homotopic�� loop��
�Ǿ�� �Ѵ�. \\
�׷���, $\pi_1(\rb P^n)=\zb/2$�̹Ƿ� $\rb P^n$���� constant loop��
homotopic���� ���� loop�� �����ϰ�, $S^n$���� $x_0$�� $-x_0$��
�����ϴ� path�� project��Ų ���� �̷��� ������ ������ �ִ�. ��
path�� $f$-image�� anti-pode preserving�̶�� �������� $S^1$��
loop�� ���� �����Ƿ� diagram�� commute�Ѵٴ� ��ǿ� ����̴�.
���� anti-pode preserving map $f$�� �������� �ʴ´�. \\

(2) ������ $x\in S^n$�� ���Ͽ� $f(x)\neq f(-x)$�̶�� ��������. \\
$g:S^n \to S^1$�� $g(x)=\frac{f(x)-f(-x)}{|f(x)-f(-x)|}$��
�����ϸ� $g$�� anti-pode preserving map�� �ǹǷ� ����̴�.
\end{proof}\\

5. $\pi_1(M^n\sharp N^n)=\pi_1(M^n)*\pi_1(N^n)$ if $n\geq 3$. \\
$M\setminus B^n$�� ��¦ �����ϴ� open set $U$, $N\setminus B^n$��
��¦ �����ϴ� open set $V$�� $\pi_1(U\bigcap
V)=\pi_1(S^{n-1})=0$�� �ǵ��� ���� �� �����Ƿ� �ڸ��ϴ�.




\end{document}
