\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*{Edge Path Group}


simplicial complex $K$�� $V(K)=\{v_0,\cdot\cdot\cdot,v_m\}$�� ����
$\Omega(|K|,v_0)$���� simplicial loop�鸸 ����. �� ������ ����
simplicial loop���� ������ ��������.

$\Omega_s(K,v_0)$=the set of simplicial
loops

$\hspace{4em}$=$\{v_0v_{i_1}\cdot\cdot\cdot
v_{i_k}v_0\,\,|\,\,v_i\in
V(K),\{v_0,v_{i_1}\},\cdot\cdot\cdot,\{v_{i_k},v_0\}\in K\}$\\

���� $\Omega_s(K,v_0)$�� equivalence relation
$\overset{s}{\sim}$�� ���� 3���� equivalence�� ���� generate��
������  ����.

$(1)\,\,\cdot\cdot\cdot v_iv_i
 \cdot \cdot \,\cdot \overset{s}{\sim} \cdot \cdot \cdot
v_i\cdot\cdot\,\cdot$

$(2)\,\,\cdot\cdot\cdot
v_iv_jv_i\cdot\cdot\,\cdot\overset{s}{\sim}\cdot\cdot\cdot
v_i\cdot\cdot\,\cdot$

$(3)\,\,\cdot\cdot\cdot
v_iv_jv_k\cdot\cdot\,\cdot\overset{s}{\sim}\cdot\cdot\cdot
v_iv_k\cdot\cdot\,\cdot$  if $\{v_i,v_j,v_k\}\in K$

�� �� $\Omega_s(K,v_0)/{\overset{s}{\sim}}:=E(K,v_0)$��
$(K,v_0)$�� edge path group�̶�� �Ѵ�. �� �� group operation��
juxtaposition �̴�.\\

$E(K,v_0)$ �� group�� ���� ������. ����

$(\{v_0v_{i_1}\cdot\cdot\, v_0\}\{v_0v_{j_1}\cdot\cdot\,
v_0\})\{v_0v_{k_1}\cdot\cdot\, v_0\}=\{v_0v_{i_1}\cdot\cdot\,
v_0\}(\{v_0v_{j_1}\cdot\cdot\, v_0\}\{v_0v_{k_1}\cdot\cdot\,
v_0\}) $

�� ��  ������ equivalence �� ���� �����ϹǷ� ���չ�Ģ�� �����ϰ�,

 $\{v_0\}\in E(K,v_0)$ �� group operation��  identity �� �ȴ�.
�׸���$\{v_0v_{i_1}\cdot\cdot\cdot v_{i_k}v_0\}\in E(K,v_0)$��
���� inverse�� $\{v_0v_{i_k}\cdot\cdot\cdot v_{i_1}v_0\}$�� �ȴ�.
���� $E(K,v_0)$ �� group �� �ȴ�.

\begin{thm}
$E(K,v_0)\cong \pi_1(|K|,v_0)$
\end{thm}

\begin{proof}
�� group������ isomorphism�� ã�� ���� ���� ���� �Լ��� ��������.

$\hspace{3em}\phi\,\,:\,\,\hspace{1em}\Omega_s(K,v_0)\hspace{1em}\rightarrow
\hspace{1em}\Omega(|K|,v_0)/\sim$

$\hspace{5em}v_0v_{i_1}\cdot\cdot\cdot
v_{i_{k-1}}v_0\hspace{1em}\mapsto\hspace{2em} \{\alp\}$\\

���⼭ $\alp:I\rightarrow |K|$��
$\alp(\frac{j}{k})=v_{ij}\,,\,j=0,1,\cdot\cdot\cdot,k\,,\,v_{i_0}=v_{i_k}=v_0$��
���ǵǴ� simplicial map�̴�. �� $\phi$�� ���� induce�Ǵ�
$\phi_{\sharp}:\Omega_s/{\overset{s}{\sim}}\rightarrow
\Omega/{\sim}$�� ���� �� �ְ�, �� �� ���� �� ������ ������.\\

$(1)\,\,\phi_{\sharp}$ is well defined :

ù��°  equivalence relation $\overset{s}{\sim}$�� ����
$\phi_{\sharp}(v_0\cdot\cdot v_iv_i\cdot\cdot
v_0)=\phi_{\sharp}(v_0\cdot\cdot v_i\cdot\cdot v_0) $ ���� �� ��
�ְ�, ������ �ΰ��� ���ؼ��� ���������� Ȯ���� �� �ִ�.\\

$(2)\,\,\phi_{\sharp}$ is a homomorphism :

$\phi(\{v_0v_{i_1}\cdot\cdot v_0\})=\alp\,$ ��
$\,\phi(\{v_0v_{j_1}\cdot\cdot v_0\})=\beta$ �� ����
$\phi(\{v_0v_{i_1}\cdot\cdot v_0\}\circ\{v_0v_{j_1}\cdot\cdot
v_0\})$ ��  juxtaposition�� ���� $\alp$�� $\beta$��
juxtaposition���� ���� �̴� $\phi(\{v_0v_{i_1}\cdot\cdot
v_0\})\circ \phi(\{v_0v_{j_1}\cdot\cdot v_0\})$ �� ����.\\\\

$(3)\,\,\phi_{\sharp}$ is onto :

������ $\alp\in \pi_1(|K|,v_0)$�� ���� $\alp$�� simplicial
approximation $\overline{\alp}$ �� �����Ѵ�.
$\phi_{\sharp}(\overline{\alp})=\alp$ �� �Ǿ� onto�̴�.\\

$(4)\,\,\phi_{\sharp}$ is 1-1 :

$\alp\,,\,\beta\,\in \Omega(K,v_0)$�� ����
$\phi(\alp)\sim\phi(\beta)$�̶�� �����ϰ�
$\alp\overset{s}{\sim}\beta$ ���� ������. ���� $\alp,\beta$ ��
���� ���� ������ ���� $\alp',\beta'$�� ����. $\phi(\alp)$��
$\phi(\beta)$ ������ homotopy
$F\,,\,F(0,t)=\alp(t),F(1,t)=\beta(t)$�� ���� $F$�� simplicial
approximation($G$�� ����)�� �����ϵ��� ������ ����
subdivision�Ѵ�.





 **�׸�10**



�� �� $G(0,t)=\alp'(t),G(1,t)=\beta'(t)$ ���� �θ� �� $\alp'$ ����
$v_0$�� base point�� �ϴ� loop�� �ȴ�. �׸���
 �� simplicial approximation ������ Remark��κ���(simplicial
 map�� approximation�� �״�� �Ȱ��� �����ǰ�,
  �׷��� ���� �κ��� approximation�� ���ϸ� ������ �κа� ���� face�� �ִ�)
  $\alp\overset{s}{\sim}\alp',\beta\overset{s}{\sim}\beta'$ �̴�.
����  $\alp'\overset{s}{\sim}\beta'$�� ���̸�
$\alp\overset{s}{\sim}\beta$���� ���� �� �ִ�.

**�׸�11**

���� ������� �ϸ� $\alp'\overset{s}{\sim}\beta'$ ���� �� �� �ִ�.
\end{proof}

\begin{cor}
$\hspace{20em}$

If $K$ is a finite simplicial complex , then $\pi_1(|K|,v_0)$ is
finitely generated.
\end{cor}
\begin{proof}
$E(K,v_0)$�� ���� �׸��� ���� ���� �⺻ ���ҵ鿡 ����
generate�ȴ�.


    \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1.4,clip=true]{graph12.eps}}

    \end{figure}




$\hspace{15em}$ �׸� 12\\

���� ���� �׸��� �޸�  self-crossing�� �Ͼ�ٸ�, �̴� �ٽ� ����
���� �⺻���ҵ��� ������ ǥ�� �����ϴ�. �׸��� �� vertex��
���Ѱ��̹Ƿ� �̷� �⺻ ���ҵ��� ���� ���� ���Ѱ��̴�.\\


    \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph13.eps}}

    \end{figure}




$\hspace{15em}$ �׸� 13\\

\end{proof}

\begin{cor}
$i_{\sharp}:\pi_1(|K^2|,v_0)\rightarrow \pi_1(|K|,v_0)$ is an
isomorphism.
\end{cor}
\begin{proof}
�տ��� �̹� �̴� ������ ���� ������ ���⼭�� �ٸ��� ������ ������.
���� 1�� ������ $\pi_1(|K^2|,v_0)\cong E(K^2,v_0)$ �̰�
$\pi_1(|K|,v_0)\cong E(K,v_0)$ �̴�. �׷��� $E(K,v_0)$��
$K^1,K^2$�� ���� �����ǹǷ� $E(K^2,v_0)=E(K,v_0)$�̴�. ����
$\pi_1(|K^2|,v_0)\cong \pi_1(|K|,v_0)$ �̴�.
\end{proof}


\end{document}
