\documentclass[12pt ]{article}
\setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true
cm}

\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*{Product and topological group }

\begin{thm}
If X and Y are path connected, then $\pi_1(X \times Y,(x_0,y_0))
\cong \pi_1(X,x_0) \times \pi_1(Y,y_0)$.
\end{thm}
\begin{proof}
Define $\phi : \pi_1(X \times Y, (x_0,y_0))\rightarrow
\pi_1(X,x_0)\times \pi_1(Y,y_0)$ ,

$\hspace{10em}\{\alpha\}\hspace{3em}\mapsto \,\,\,\,(\{p_1 \circ
\alpha\},\{p_2 \circ \alpha\})$

where $p_1,p_2$ are projections to X and Y respectively.

�Ϲ������� �� homomorphism $\phi_1 : G\rightarrow H_1 ,\,\, \phi_2
: G \rightarrow H_2$ �� ���� $\phi(g) = (\phi_1(g),\phi_2(g)) : G
\rightarrow H_1 \times H_2 $ ���� homomorphism�� �ǹǷ� ��������
$\phi$ �� homomorphism�� �ȴ�. ���� $\phi$�� ���Լ��� ã�� ����
�Ʒ��� ���� $\psi$�� ��������.

 $\,\,\,\,\,\,\,\,\,\,\,\psi : \pi_1(X,x_0)\times\pi_1(Y,y_0) \rightarrow \pi_1(X \times Y,(x_0,y_0))$

 $\hspace{5em}(\{\beta\},\{\gamma\})\,\,\hspace{2em}\mapsto \hspace{2em}\{\delta\}$

 $\hspace{2em}$,where $\delta(t)=(\beta(t),\gamma(t))$

 �� $\psi$�� $\phi$ �� ���Լ��� �ǰ� ���� $\psi$ �� ��
 ���ǵǾ������� ���캸�� �ȴ�.

  F : $\beta \sim \beta'$ , G : $\gamma \sim \gamma'$ �� ��
 $H(t,s)=( F(t,s) , G(t,s))$ �� �ָ� �̴� $\delta$ �� $\delta'$
 ������ homotopy�� �ֹǷ� $\psi$ �� �� ���ǵǾ����� �� �� �ִ�.

\end{proof}

{\bf �� 3. Topological group}

\begin{thm}\textit{
 G : a path connected topological group with identity
 e\\$ \hspace{3em}\Rightarrow \pi_1(G,e)$ is abelian.}
\end{thm}

\begin{proof}
$\{\alpha\} , \{\beta\} \in \pi_1(G , e)$ �� ����
$\{\alpha\}\{\beta\} = \{\beta\}\{\alpha\}$ ���� ���̱� ����

$\{\alpha\}\{\beta\}\{\alpha\}^{-1}\{\beta\}^{-1} = 1$ ����
������.

$\{\alpha\}\{\beta\}\{\alpha\}^{-1}\{\beta\}^{-1}=\{\alpha*
\beta* \overline{\alpha}*\overline{\beta}\} \simeq 1$ �� ���̱�
���� $\alpha* \beta* \overline{\alpha}*\overline{\beta}$ ��
������ ���� ����.

**�׸� 1,2**


 �׸� 2���� ���� ����,

F : $I^2 \rightarrow G$ given by F(t,s)=$\alpha(t)\beta(s)$ ��
group G �ȿ� ���� ���ǵǹǷ� $I^2$�󿡼� �����Լ��� �� ���ǵǰ�
$I^2$�� boundary�� ��ǻ� $\alpha* \beta*
\overline{\alpha}*\overline{\beta}$�� �ش�.

�� �׸� 1�� boundary���� �׸� 2�� boundary�� e���� �� ������
������ �����Լ��� �ְ�, �̰��� �� ���� ������ �� �Ͱ��� $I^2$��
���ηε� extend�ȴ�(������ radial extension).

�� �ٸ��� ���� �����

$\{\alpha\}\{\beta\}\{\alpha\}^{-1}\{\beta\}^{-1} =
F_{\sharp}(\{\partial I^2\}) = F_{\sharp}(1) = 1.$

\end{proof}

{\bf Ecercise.} �� ������ ������ �������� ���� �Ϲ������� ������
�̿��ؼ� �����ص� �ȴ�.

 $\{\alpha\}\in \pi_1(X,x_0)$, $\alpha : S^1 = \partial D^2
\rightarrow X$  �� ����\\\textit{$\{\alpha\}=1$ if and only if
$\alpha$ can be extended to $D^2$}
\end{document}
