\documentclass[12pt ]{article}
\setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true
cm}
\usepackage{graphicx}
\usepackage{hangul}
\usepackage{amscd,amsmath}
\usepackage{amsfonts}
\usepackage{amssymb,theorem}
\usepackage{longtable}
\newcommand{\Aff}{\mbox{\it Aff}}
\newcommand{\aff}{\mbox{\it aff}}


\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\del}{\delta}
\newcommand{\gam}{\gamma}
\newcommand{\vep}{\varepsilon}
\newcommand{\eps}{\epsilon}
\newcommand{\lam}{\lambda}
\newcommand{\kap}{\kappa}
\newcommand{\sig}{\sigma}
\newcommand{\ome}{\omega}
\newcommand{\Gam}{\Gamma}
\newcommand{\Ome}{\Omega}
\newcommand{\Sig}{\Sigma}
\newcommand{\Del}{\Delta}
\newcommand{\Lam}{\Lambda}


\newtheorem{thm}{����}
\newtheorem{cor}[thm]{��������}
\newtheorem{lem}[thm]{��������}
\newtheorem{prop}[thm]{����}
\newtheorem{cl}{Claim}

{\theorembodyfont{\rm}
\newtheorem{ex}{��}
\newtheorem{que}{����}
\newtheorem{notation}{Notation}[section]
\newtheorem{defn}{����}
\newtheorem{rem}{��}
\newtheorem{note}{Note}
}
\renewcommand{\thenote}{}
\renewcommand{\therem}{}

\newenvironment{proof}{{\bf ����}}{\hfill\framebox[2mm]{}}
\newenvironment{proof1}{{\bf ��������}}{\hfill\framebox[2mm]{}}

\begin{document}
 \parindent=0cm
  \section*{Examples}

  {\bf 1. $\pi(S^1)=\mathbb{Z}_1=F_1$}\\

  $\pi_1(figure\,\,\,\, eight)=\mathbb{Z}*\mathbb{Z}=F_2.$

  $\pi_1(S^2\vee S^1)=1*\mathbb{Z}=\mathbb{Z}=F_1.$

  $\pi_1(S^2\vee S^2)=1.$

  $\pi_1(S^1\vee S^2 \vee S^1)=\mathbb{Z}*\mathbb{Z}=F_2.$\\

  {\bf 2. $\pi_1(T^2)=\mathbb{Z}\oplus \mathbb{Z}$ ;}\\

torus�� �Ʒ� �׸��� ���� ����.\\

 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph17.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 17\\


 �׸��� ���� �κ��� $A$�� �ΰ�
������ �κ��� $B$�� �θ�, �� �� torus $K=A\cup B$ �� �ȴ�. �̸� �Ʒ��׸��� ���� ǥ���� �� �ִ�.\\

 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph18.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 18\\

$A,B,K,A\cap B,A\cup B$ �� Van Kampen Theorem �� ������ ���
�����ϹǷ�
 ${\pi_1(K,v_0)=\pi_1(A,v_0)*\pi_1(B,v_0)}/N$���� �� �� �ִ�.
 ���� $\pi_1(A,v_0)$�� $\pi_1(B,v_0)$�� �˸� �ȴ�. $
 \pi_1(A,v_0)$�� �� �׸��� A�κп��� �簢�� ������ ���� �ٱ�
 �簢���� boundary�� retract ��Ű�� �̴� $aba^{-1}b^{-1}$ �� �Ǿ�
 figure eight �� ���� ���� �ȴ�. ���� $\pi_1(A,v_0)$ �� $F_2$ ��
 �ȴ�. $\pi_1(B,v_0)$�� 1���� ���� �� �� �ְ� $A\cap B$�� circle
 �̹Ƿ� $\pi_1(A\cap B)=\mathbb{Z}$ �̴�. �׸���\\

 $\hspace{3em}F_2=<a,b\,\,|\,\,\,>\hspace{3em}1$

$\hspace{6em}i\nwarrow\hspace{3.5em}\nearrow j$

$\hspace{6em}\mathbb{Z}=<x\,\,|\,\,>$\\

�� diagram���� $\underset{\mathbb{Z}}{*}$�� ���ǿ� ���� $i(x)$ ��
$j(x)$ �� ���� ���� relation�� �ָ� �ǹǷ� generator �� $a,b$�̰�
relation �� $x\in \pi_1(A\cap B)$�� ���� $i(x)=j(x)$ �̴�. ��
$\pi_1(K,v_0)=<a,b\,\,|\,\,i(x)=j(x)>$ �̴�. �׷���
$\pi_1(B,v_0)=1$ �̹Ƿ� $j(x)=1$ �̰� $A$�� $B$�� ������ �κ���
circle�� ���ؼ��� $i(x)$�� �ٱ� �簢���� �ǹǷ�
$i(x)=aba^{-1}b^{-1}$ �� �ȴ�. ���� relation��
$[a,b]=aba^{-1}b^{-1}=1$ �̴�. �׷��Ƿ�
$\pi_1(T^2)=F_2\underset{\mathbb{Z}}{*}1=<a,b\,\,|\,\,\,[a,b]=1>=\mathbb{Z}\oplus\mathbb{Z}$�̴�.\\

{\bf 3. $\pi_1(P^2)=\mathbb{Z}/2$} ;

(��� 1.) $S^2$ �� $P^2$�� covering ���� �˰� �ִ�. �׷��� $S^2$��
simply connected�̹Ƿ� $S^2$�� $P^2$�� universal covering �̴�.
$p^{-1}(x)$�� $\pi_1(X)/p_{\sharp}\pi_1(X)$�� �ϴ��ϴ������踦
�����µ� universal covering�̶�� ��������  $p_{\sharp}\pi_1(X)$��
1�� �Ǿ�  $\pi_1(X)$�� $p^{-1}(x)$ �� fiber�� ������ ����. $P^2$��
�� ���� ���� fiber�� �� ���̹Ƿ� $\pi_1(P^2)$�� $\mathbb{Z}/2$��
�� �ۿ� ����.\\

(��� 2.) Van Kampen theorem�� �̿��ϱ� ����  $P^2$�� �Ʒ� �׸���
����
����.\\


 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph19.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 19\\



�� �� $\pi_1(A)=\mathbb{Z}=<a\,\,|\,\,\,>$, $\pi_1(B)=1$,
$\pi_1(A\cap B)=\mathbb{Z}$ ���� ���� �� �� �ְ�
$\mathbb{Z}_2\underset{\mathbb{Z}}{*}1$ �� ������ ����. $x\in
A\cap B$ �� ���� $i(x)=aa$ �� �ǰ�, $j(x)=1$ �� �Ǿ� $a^2=1$
�̶�� relation�� ��´�. ����
$\pi_1(P^2)=<a\,\,|\,\,a^2=1>=\mathbb{Z}/_2 $ �� �ȴ�.\\

���� ������ compact surface�� ���� fundamental group�� ����� ����.\\

{\bf 1. Orientable surfaces.}\\

orientable surface�� $T^2\sharp T^2\sharp \cdot\cdot\cdot \sharp
T^2$ �÷� ǥ���ǹǷ� genus g��¥�� ���� �׸��� orientable
surface�� ���캸��.\\

 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph20.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 20\\


�� �׸����� $A$��
$\overset{2g}{\overbrace{S^1\vee\cdot\cdot\cdot\vee S^1}}$ =
$\overset{2g}{\vee}S^1$ ( = bouquet of 2g circles =
2g����
circle���� one point union�� �� ) �� homotopy equivalent�ϹǷ�
 $\pi_1(A,v_0)=F_{2g}$ �� �ȴ�. ( genus �ϳ�¥����
figure eight $S^1\vee S^1$ �� �Ǿ� �̰��� $\pi_1$�� $F_2$�� �ȴ�.)
���� $\pi_1(B,v_0)=1$ �̰� $x\in A\cap B$�� ����

$a_1b_1a_1^{-1}b_1^{-1}a_2b_2a_2^{-1}b_2^{-1}\cdot\cdot\cdot
a_gb_ga_g^{-1}b_g^{-1}=i(x)=j(x)=1$ ���� �˼� �ִ�. ����
���������� �����Ѵ�.

$\pi_1(\Sigma g)=<a_1,b_1,\cdot\cdot\cdot
,a_g,b_g\,\,|\,\,[a_1,b_1]\cdot\cdot\cdot [a_g,b_g]=1>$.\\

{\bf 2. Nonorientable surfaces.}\\

nonorientable surface $P^2\sharp P^2\sharp\cdot\cdot\cdot \sharp
P^2$�� �����غ���. ���� �׸����� \\

 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph21.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 21\\




$\pi_1(A,v_0)=<a_1,a_2,\cdot\cdot\cdot ,a_k\,\,|\,\,\,>$ �� �ǰ�
$\pi_1(B,v_0)=1$ �� �ȴ�. $x\in A\cap B$�� ���� $i(x)=j(x)$ ��
ã���� $a_1a_1a_2a_2\cdot\cdot\cdot a_ka_k=1$ �� �ȴ�. ����


$\pi_1(N_k)=<a_1,a_2,\cdot\cdot\cdot,a_k\,\,|\,\,a_1^2a_2^2\cdot\cdot\cdot
a_k^2=1
>$.\\

{\bf Dunce hat.}\\

 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph22.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 22\\




�� �׸��� $\pi_1$�� ������ ����. �տ����� ����� ���������� ���ο�
���� ��Ƽ�  $A,B$ �� ������ $\pi_1(A,v_0)=<a\,\,|\,\,\,>$,
$\pi_1(B,v_0)=1$ �̰� $x\in A\cap B$�� ���� $i(x)=aaa^{-1}$ �̹Ƿ�
$\pi_1(Dunce\,\,\, hat)=<a\,\,|\,\,aaa^{-1}=1>=<a\,\,|\,\,a=1>=1$
�� �ȴ�. �� simply connected space�� �ȴ�. ����  Dunce hat ��
contractible space �� �Ǵµ� �̴�  ������ ���� ����
�� �ִ�. ���� ���� �׸��� ������ ���� ����ó�� �� ����\\

 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph23.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 23\\


 ���� ä��� �̴� Dunce hat�� homotopy equivalent �ϰ�(���
Dunce hat�� ���� ä�� ���� strong
deformation retract �̴�.) contractible ���� ���� �� �� �ִ�.\\

���� Dunce hat($aaa^{-1}$)�� �޸� $aaa$�÷� �� ���� ���캸��.\\

 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph24.eps}}

    \end{figure}
$\hspace{15em}$ �׸� 24\\


 ��� �̰��� $\pi_1$�� $\mathbb{Z}/3$ �� �ȴ�.  $P^2(��\,\,aa)$��
universal covering�� $S^2$ �Ӱ� ����ϰ� $aaa$�� universal
covering�� ���غ��ƶ�.(Exercise) �׸��� �� covering�� 3-fold
covering���� �̿��ϸ� $\pi_1=\mathbb{Z}/3$���� ���� �� �� �ִ�.












  \end{document}
