\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}
\usepackage{floatflt}
\usepackage{texdraw, epic, pstcol ,pstricks ,pst-3d, pst-poly,pst-grad,pst-node, pst-text}
\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*{Application}

 \textit{{\bf $1\,\,.\,\, \pi_k(S^n)$=0, if $k < n$.}}

  \begin{proof}
 simplicial approximation ������ $S^n\setminus
 \{point\} \simeq \mathbb{R}^n$�� contractible�̶�� ����
 �̿�����. $S^k$���� $S^n$���� ���� $\alp$�� onto�� �ƴ϶��, $S^n$�󿡼�
 $\alp$�� image�� �ƴ� �� $x_0$ �� ������ �� �ְ� $S^n\setminus
 \{x_0\} \simeq \mathbb{R}^n$�� contractible�̹Ƿ� ������ �ϼ��ȴ�.

���� simplicial map ��ü�� $p$-skeleton ��  $p$-skeleton ����
�����Ƿ� $\alp$�� simplicial approximation $\alp'$�� onto�� �� ��
����, ���� $\alp'$�� �� ������ contractible �ϰ� $\alp \simeq
\alp '$ �̹Ƿ� $\alp$ �� contractible�ϴ�.

�� �������� base point�� vertex�� �ǵ��� simplicial complex ������
$S^k$�� $S^n$�� ������ �ش�. ���� ������ Remark�� ����
$\alp(base\,\,point)=\alp'(base\,\,point)$�� �ξ �������, ����
���� 5�������� homotopy $F$�� base point�� fix�Ѵٰ� �����Ͽ���
����.

  \end{proof}


{\bf $2\,\,.\,\, i:K^{k+1}\hookrightarrow K$ induces an
isomorphism $i_{\ast}:\pi_{k}(|K^{k+1}|)\rightarrow \pi_k(|K|)$.}\\


\begin{proof}
���� simplicial approximation thoerem�� �̿��ϸ� epimorphism ��
�Ǵ� ���� �� �������� ó�� �ڸ��ϴ�. ���� 1-1 ���� ������. 1-1
���� ���̱� ����
������ ������ ���� note�� �̿�����.\\


{\bf Note.}\textit{ $\alp:S^{k}\rightarrow X$ represent a zero
element in $\pi_{k}(X)$ if and only if}

\textit{$\hspace{3em}\exists\,\,extension\,\,\overline{\alp}:B^{k+1}\rightarrow
X.$}\\


� $\{\alp\} \in \pi_k(|K^{k+1}|)$ �� $i_{\ast}$�� ���� identity
�� ���ٸ�, ���� note �� ���� extension
$\overline{\alp}$ �� �����ϰ� ���� diagram �� �����Ѵ�.���⼭ ó������ $\alp$�� simplicial�̶� �����ص� �������.\\

$\,\,\,|K^{k+1}|\hspace{1em}\rightarrow\hspace{1em}
|K|\hspace{4em}|K^{k+1}|\hspace{1em}\rightarrow\hspace{1em} |K| $

$\alp\uparrow\hspace{7.5em}\Rightarrow
\hspace{3.5em}\alp\uparrow\hspace{4.5em}\uparrow
\exists\,\overline{\alp}$

$\,\,\,S^{k}\hspace{11.5em}S^{k}\hspace{2.0em}\subset\hspace{1.8em} B^{k+1}$\\

���⼭ $\overline{\alp}$�� simplicial approximation
$\overline{\alp}'$�� ������


$\,\,\,K^{k+1}\hspace{1em}\rightarrow\hspace{1em} K\hspace{4em}$

$\hspace{1em}\exists\,\overline{\alp'}\nwarrow$

$\,\,\,\hspace{4.5em}B^{k+1}$

�� �ǰ� �� �� ���� ������ remark�� ���ؼ�
$\{\overline{\alp}'|_{S^{k}}\}=\{\alp\}$ �� �ξ �ȴ�. �׷���
$|K^{k+1}|$���� $\{\overline{\alp}'|_{S^{k}}\}=0$ �̹Ƿ�
$\{\alp\}=0$ �̴�.
���� 1-1���� ������.\\


\textit{{\bf 3. Edge-path group}}

$v_{0} \in V(K)$�� ���ؼ�\\

$\Omega_s(K,v_0)$=the set of closed edge-paths based at $v_{0}$

$\hspace{4em}$=$\{v_0v_{i_1}\cdot\cdot\cdot
v_{i_k}v_0\,\,|\,\,v_i\in
V(K)$ and  $\{v_0,v_{i_1}\},\cdot\cdot\cdot,\{v_{i_k},v_0\}$ : 1-simplices 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 $<v_i,v_j,v_k>$ is a 2-simplex 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��� ��������.

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

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

where $\alp:I\rightarrow |K|$ is a piecewise linear map
representing simplicial loop corresponding to $v_{0}, v_{i_{1}},
\cdots v_{i_{k}}, v_{0}$.\\

�� $\phi$�� ���� 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 :

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

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

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


\begin{center}
\psset{unit=1.5cm}
\begin{pspicture}(0,-1.0)(6.0,2.5)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\psline{|-|}(0,0)(0.4,0)\psline{|-|}(0.4,0)(0.6,0)\psline{|-|}(0.6,0)(1.1,0)
\psline{|-|}(1.1,0)(1.4,0)\psline{|-|}(1.4,0)(1.6,0)\psline{|-|}(1.6,0)(2.0,0)
\psline{|-|}(0,2.0)(0.2,2.0)\psline{|-|}(0.2,2.0)(0.5,2.0)\psline{|-|}(0.5,2.0)(0.9,2.0)
\psline{|-|}(0.9,2.0)(1.4,2.0)\psline{|-|}(1.4,2.0)(1.5,2.0)\psline{|-|}(1.5,2.0)(1.7,2.0)
\psline{|-|}(1.7,2.0)(2.0,2.0)\psline(0,0)(0,2.0)\psline(2.0,0)(2.0,2.0)
\rput(1.0,-0.3){$\alp$} \rput(1.0,2.3){$\bet$}

\rput(2.5,0){\psline{|-|}(0,0)(0.4,0)\psline{|-|}(0.4,0)(0.6,0)\psline{|-|}(0.6,0)(1.1,0)
\psline{|-|}(1.1,0)(1.4,0)\psline{|-|}(1.4,0)(1.6,0)\psline{|-|}(1.6,0)(2.0,0)
\psline{|-|}(0,2.0)(0.2,2.0)\psline{|-|}(0.2,2.0)(0.5,2.0)\psline{|-|}(0.5,2.0)(0.9,2.0)
\psline{|-|}(0.9,2.0)(1.4,2.0)\psline{|-|}(1.4,2.0)(1.5,2.0)\psline{|-|}(1.5,2.0)(1.7,2.0)
\psline{|-|}(1.7,2.0)(2.0,2.0)\psline(0,0)(0,2.0)\psline(2.0,0)(2.0,2.0)
\rput(1.0,-0.3){$\alp'$} \rput(1.0,2.3){$\bet'$}
\psline(0.4,0)(0.4,2.0)\psline(0.6,0)(0.6,2.0)\psline(1.1,0)(1.1,2.0)\psline(1.4,0)(1.4,2.0)
\psline(1.6,0)(1.6,2.0)\psline(0.2,0)(0.2,2.0)\psline(0.5,0)(0.5,2.0)\psline(0.9,0)(0.9,2.0)
\psline(1.5,0)(1.5,2.0)\psline(1.7,0)(1.7,2.0)
\psline(0.2,0)(0.2,2.0)\psline(0.8,0)(0.8,2.0)\psline(1.8,0)(1.8,2.0)
\psline(0,0.2)(2.0,0.2)\psline(0,0.4)(2.0,0.4)\psline(0,0.6)(2.0,0.6)
\psline(0,0.8)(2.0,0.8)\psline(0,1.0)(2.0,1.0)\psline(0,1.2)(2.0,1.2)
\psline(0,1.4)(2.0,1.4)\psline(0,1.6)(2.0,1.6)\psline(0,1.8)(2.0,1.8)}


\rput(4.7,1.0){$\overset{G}{\longrightarrow}$}

\rput(5.0,0){\pspolygon(0,1.0)(0.3,0.4)(0.7,0.2)(1.1,0.5)(1.5,0.9)(1.7,1.4)(1.4,1.7)(1.0,1.7)
(0.6,1.5)(0.2,1.3)(0,1.0)(0.4,0.8)(0.8,0.5)(1.2,1.0)(0.7,1.4)(0.4,1.2)(0,1.0)}


\end{pspicture}

\end{center}


���⼭ $G(0,t)=\alp'(t),G(1,t)=\beta'(t)$ ���� �ξ��� ��, $\alp
\overset{s}{\sim} \alp' , \bet \overset{s}{\sim} \bet'$��
�����Ͽ��� �����ϴ�. �ֳ��ϸ�, $\alp$�� �� �ұ����� 1-simplex
$<v,w>$�� ���ٸ�, �̰��� subdivision�� ������ $<v,w>,<w,w>,\cdots
<w,w>$�� ������ $\alp \overset{s}{\sim} \alp'$�̰� $\phi(\alp) =
\phi(\alp')$�� �ȴ�. $\phi(\alp')$�� $\phi(\bet')$������ homotopy
$F$�� simplicial approximation $G$�� $\alp' , \bet'$�� �̹�
simplicial �̹Ƿ� $G_{0}=\alp'$�̰� $G_{1}=\bet'$��� �����ص�
����. ���� $\alp'\overset{s}{\sim}\beta'$�� ���̸�
$\alp\overset{s}{\sim}\beta$���� ���� �� �ִ�.


\begin{center}
\psset{unit=2.5cm}
\begin{pspicture}(-0.5,-0.5)(2.5,2.5)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]

\pspolygon(0,0)(2.0,0)(2.0,2.0)(0,2.0)(0,0)
\psline(0.2,0)(0.2,2.0)\psline(0.4,0)(0.4,2.0)\psline(0.6,0)(0.6,2.0)
\psline(0.8,0)(0.8,2.0)\psline(1.0,0)(1.0,2.0)\psline(1.2,0)(1.2,2.0)
\psline(1.4,0)(1.4,2.0)\psline(1.6,0)(1.6,2.0)\psline(1.8,0)(1.8,2.0)

\psline(0,0.2)(2.0,0.2)\psline(0,0.4)(2.0,0.4)\psline(0,0.6)(2.0,0.6)
\psline(0,0.8)(2.0,0.8)\psline(0,1.0)(2.0,1.0)\psline(0,1.2)(2.0,1.2)
\psline(0,1.4)(2.0,1.4)\psline(0,1.6)(2.0,1.6)\psline(0,1.8)(2.0,1.8)

\psline(0,0)(2.0,2.0)\psline(0,0.2)(1.8,2.0)\psline(0,0.4)(1.6,2.0)
\psline(0,0.6)(1.4,2.0)\psline(0,0.8)(1.2,2.0)\psline(0,1.0)(1.0,2.0)
\psline(0,1.2)(0.8,2.0)\psline(0,1.4)(0.6,2.0)\psline(0,1.6)(0.4,2.0)
\psline(0,1.8)(0.2,2.0)\psline(0.2,0)(2.0,1.8)\psline(0.4,0)(2.0,1.6)
\psline(0.6,0)(2.0,1.4)\psline(0.8,0)(2.0,1.2)\psline(1.0,0)(2.0,1.0)
\psline(1.2,0)(2.0,0.8)\psline(1.4,0)(2.0,0.6)\psline(1.6,0)(2.0,0.4)
\psline(1.8,0)(2.0,0.2)

\pspolygon[fillstyle=solid,fillcolor=lightgray](1.8,0)(2.0,0)(2.0,0.2)(1.8,0)

\rput(1.0,-0.2){$\alp'$} \rput(1.0,2.2){$\bet'$}
\rput(-0.3,1.0){$v_0$} \rput(2.2,1.0){$v_0$}

\end{pspicture}

\end{center}



�� �׸����� ����ģ �ﰢ������ �ϳ��� $\overset{s}{\sim}$��
�����ϸ�, $\alp'\overset{s}{\sim}\beta'$ ���� �� �� �ִ�.

\end{proof}\\



\textit{{\bf 4. Graph}}


A 1-dimensional simplicial complex is called a graph.\\
A simply connected simplicial complex is called a tree.\\

\begin{center}
\psset{unit=2.5cm}
\begin{pspicture}(-0.5,-0.5)(5.5,2.5)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]

\pspolygon(0.5,1.0)(0.75,0.5)(1.25,1.0)(1.0,1.5)(0.5,1.0)
\psline(0.5,1.0)(1.25,1.0)
\pspolygon(1.25,1.0)(1.5,0.5)(1.5,1.25)(1.25,1.0)
\pspolygon(0.5,1.0)(1.0,1.5)(0.5,1.75)(0.5,1.0)
\psline(0.75,0.5)(0.5,0)\psline(0.5,1.0)(0.25,0.5)\psline(0.5,1.0)(0,1.20)
\psline(1.0,1.5)(1.3,2.0)\psline(1.0,1.5)(1.6,1.75)\psline(1.5,0.5)(1.5,0)
\psline(1.5,0.5)(2.0,0.4)\psline(1.5,1.25)(2.0,1.5)

\rput(3.0,0){ \pscircle(0.4,1){0.4} \pscircle(1.6,1){0.4}
\psline(0.8,1.0)(1.2,1.0)\psdots(0.8,1.0)\psdots(1.2,1.0)
\psdots(0.3,1.4)\psdots(0.3,0.6)\psdots(1.7,1.4)\psdots(1.7,0.6)}

\end{pspicture}

\end{center}


\begin{lem}

A graph $K$ is a tree if and only if it is contractible.

\end{lem}

\begin{proof}

($\Leftarrow$) clear.\\
($\Rightarrow$) Fix $v_{0} \in V(K). \forall v \in V(K)$, choose a
path $\alp_{v}$ from $v$ to $v_{0}$.\\
Define $F : V \times I \rightarrow |K|$ by
$F(v,t)=\alp_{v}(t)$.Then\\
For each edge $\sigma$, define $F:|\sigma| \times I \rightarrow
|K|$ as the following picture.\\

\begin{center}
\psset{unit=2.0cm}
\begin{pspicture}(-0.5,-0.5)(3.5,2.5)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]

\pspolygon[fillstyle=solid,fillcolor=lightgray](0,0)(2.0,0)(2.0,2.0)(0,2.0)(0,0)
\rput(-0.2,1.0){$I$} \rput(-0.1,-0.1){$v_{i}$}
\rput(2.1,-0.1){$v_{i+1}$} \rput(1.0,-0.2){$\sigma$}
\rput(1.8,1.0){$\alp_{v_{i+1}}$} \rput(0.2,1.0){$\alp_{v_{i}}$}
\rput(1.0,2.2){$v_{0}$}

\rput(2.5,1.0){$\overset{F}\longrightarrow$} \rput(3.2,1.0){$|K|$}

\end{pspicture}

\end{center}

���⼭ $|K|$�� simply connected�̰� �̹� boundary condition�� �˰�
�����Ƿ� $|\sigma| \times I$�� ���η� extend��ų �� ������
��������.\\
�׷��� $F|_{|\sigma| \times I}$�� �����̹Ƿ� $F:|K| \times I
\rightarrow |K|$�� �����̰� $F_{0} = id , F_{1} = v_{0}$�̹Ƿ�
contractible�� ���� Ȯ���Ͽ���.

\end{proof}

\begin{lem}

Let $K$ be a connected simplicial complex. Then\\
(1) $K$ contains a maximal tree.\\
(2) Every maximal tree contains all the vertices of $K$.\\

\end{lem}


\begin{proof}

(1) Use Zorn's lemma.\\

Let ${\bf \mathcal{T}}$ be the collection of all the trees in $K$.
Then
${\bf \mathcal{T}}$ is a partially ordered set with respect to inclusion.\\
Let $\{T_{j}\}$ be a simply ordered set of trees in $K$. And $T =
\bigcup T_{j}$. Then $T$ is simply connected.\\
$\because$ For given $\alp : S^{1} \rightarrow T, \alp(S^{1})$ is
compact. Hence it is contained $T_{j}$ for some $j$. Consequently,
$\alp \simeq const.$ in $T_{j}$, hence in $T$.\\

By Zorn's lemma, there exists a maximal element in ${\bf \mathcal{T}}$.\\

(2) Let $T$ be the maximal tree. Suppose that $T$ does not contain
all the vertices of $K$. Then $\exists v_{1}, v_{2} \in V(K)$ s.t
$v_{1} \in T$ and $v_{2} \notin T$ and $\{ v_{1},v_{2} \}$ is
 a 1-simplex in $K$. \\

$T_{1} := T \cup \{ \{ v_{1},v_{2} \} , \{ v_{2} \} \}$. Then
$|T|$ is a strong deformation retract of $|T_{1}|$. Hence $T_{1}$
is simply connected and hence a tree containing $T$ properly. This
contradicts to the normality of $T$.

\end{proof}

{\bf Note} (1) In fact, given a tree $T$, $\exists$ a maximal tree
containing $T$.\\
(2) A tree containing all the vertices of $K$ is maximal.\\

(1)�� ������ lemma�� ������ ���� ���̰�, (2)�� edge-path group��
�����ϸ� ���� Ȯ���� �� �ִ�.(exercises)\\


Let $T$ be a maximal tree in $K$.\\
Let $G$ be the group generated by the oriented edges $(v,v')$ of
$K$ with the following relations.\\
\hspace*{2.0em} (a) $(v,v') \in T \Rightarrow (v,v')=1$\\
\hspace*{2.0em} (b) $(v,v')(v',v) =1$\\
\hspace*{2.0em} (c) $v_1 , v_2 , v_3$ are vertices of a 2-simplex
in $K$.Then\\
\hspace*{4.0em} $(v_{1},v_{2})(v_{2},v_{3}) = (v_{1},v_{3})$\\

\begin{thm}

$E(K,v_{0}) \cong G = F/R$

\end{thm}

\begin{proof}

\hspace*{2.0em}$\phi : E(K,v_0) \longrightarrow G$��� �����ϸ�, \\
$[v_{0}v_{i_{1}}v_{i_{2}} \cdots v_{i_{k}} v_{0}] \mapsto
(v_{0}v_{i_{1}})(v_{i_{1}}v_{i_{2}}) \cdots (v_{i_{k}}v_{0})$\\
�̴� junxtaposition�� equivalence relation�� �����ϹǷ� ��
���ǵȴ�.��\\

\hspace*{2.0em}$\psi : F \longrightarrow E(K,v_{0})$�� ��������.\\
\hspace*{2.0em}$(v,v') \mapsto \alp_{v}vv'\alp_{v'}^{-1}$\\
�� �� �� vertex $v$���� $v_{0}$���� $v$�� ���� simplicial path
$\alp_v$�� $T$���� �����ϰ� ������Ű��. $T$�� ��� vertex�� ��
�����ϹǷ� $\psi$�� $R$�� ���ҵ��� ��� 0���� �����Ƿ� \\
$F/R \overset{\bar{\psi}}\longrightarrow E(K,v_{0})$��
induce�Ѵ�.\\
$\phi \circ \bar{\psi} =id. \hspace{1.0em} \bar{\psi} \circ \phi =
id.$���� ���� �� �� �����Ƿ� ������ �����Ǿ���.

\end{proof}

\begin{cor}

Let $K$ be a finite connected simplicial complex. Then
$\pi_{1}(|K|,v_{0})$ is finitely presented.

\end{cor}

\begin{proof}

$\pi_{1}(|K|,v_{0}) = E = G$�ε� edge�� ������ �����̹Ƿ�
generator�� ������ �����̰�, $G$�� relation�� �����̹Ƿ�
relation�� ������ �����̴�. ���� finitely presented�̴�.

\end{proof}

\begin{cor}

Let $K$ be a connected graph. If $T$ is a maximal tree in $K$,
then $E(K,v_{0})$ is a free group generated by the 1-simplices in
$K-T$.

\end{cor}

\begin{proof}
�̴� $G$�� ���ǿ� ���ؼ� �翬�ϴ�.
\begin{center}
\psset{unit=1.0cm}
\begin{pspicture}(-0.5,-0.5)(14.5,3.5)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]

\psline(0,2.0)(0.6,1.0) \psline(0.6,1.0)(0.6,0.4)
\psline(0.6,0.4)(1.2,0.8) \psline(1.2,0.8)(1.3,0)
\psline(1.3,0)(1.6,0.2) \psline(1.6,0.2)(1.8,0)
\psline(1.2,0.8)(1.5,1.0) \psline[linecolor=red](1.5,1.0)(1.3,0)
\psline(0.6,1.0)(1.3,2.0)

\psline(1.3,2.0)(1.8,1.5) \psline(1.8,1.5)(1.8,0.9)
\psline(1.8,0.9)(2.5,0.4) \psline[linecolor=red](1.8,0.9)(2.3,1.7)
\psline(2.3,1.7)(1.8,1.5) \psline(2.3,1.7)(2.7,1.0)

\psline(1.3,2.0)(2.2,2.5) \psline(2.2,2.5)(2.4,3.0)

\psline(2.2,2.5)(2.5,2.0) \psline(2.5,2.0)(2.7,1.7)
\psline(2.7,1.7)(3.4,1.8) \psline(3.4,1.8)(3.6,1.2)
\psline[linecolor=red](3.6,1.2)(3.8,1.9) \psline(3.8,1.9)(3.4,1.8)
\psline[linecolor=red](2.7,1.7)(3.2,2.8) \psline(3.2,2.8)(2.5,2.0)
\psline(3.2,2.8)(3.5,2.5) \psline(3.5,2.5)(3.6,2.4)
\psline(3.5,2.5)(3.7,2.9)

\rput(5.0,0){ \psline(0,2.0)(0.6,1.0) \psline(0.6,1.0)(0.6,0.4)
\psline(0.6,0.4)(1.2,0.8) \psline(1.2,0.8)(1.3,0)
\psline(1.3,0)(1.6,0.2) \psline(1.6,0.2)(1.8,0)
\psline(1.2,0.8)(1.5,1.0) \psline(0.6,1.0)(1.3,2.0)

\psline(1.3,2.0)(1.8,1.5) \psline(1.8,1.5)(1.8,0.9)
\psline(1.8,0.9)(2.5,0.4) \psline(2.3,1.7)(1.8,1.5)
\psline(2.3,1.7)(2.7,1.0)

\psline(1.3,2.0)(2.2,2.5) \psline(2.2,2.5)(2.4,3.0)

\psline(2.2,2.5)(2.5,2.0) \psline(2.5,2.0)(2.7,1.7)
\psline(2.7,1.7)(3.4,1.8) \psline(3.4,1.8)(3.6,1.2)
\psline(3.8,1.9)(3.4,1.8) \psline(3.2,2.8)(2.5,2.0)
\psline(3.2,2.8)(3.5,2.5) \psline(3.5,2.5)(3.6,2.4)
\psline(3.5,2.5)(3.7,2.9) \rput(2.0,-0.3){$T$}}

\rput(10.0,0){\psline[linecolor=red](1.5,1.0)(1.3,0)
\psline[linecolor=red](1.8,0.9)(2.3,1.7)
\psline[linecolor=red](3.6,1.2)(3.8,1.9)
\psline[linecolor=red](2.7,1.7)(3.2,2.8) \rput(2.0,-0.3){$K-T$}}

\end{pspicture}

\end{center}

\end{proof}

{\bf Note} Let $K$ be a finite connected graph, $n_{1}$ be the
number of 1-simplices in $K$ and $n_{0}$ be the number of
0-simplices in $K$. Then \\
The number of 1-simplices in $K-T = n_{1} - (n_{0} - 1) =
1-(n_{0}-n_{1}) = 1-\chi(K)$.

\begin{thm}

Let $p : \widetilde{X} \rightarrow X$ be a covering. If $X=|K|$
for some simplicial complex $K$, then the simplicial complex
structure of $X$ can be lifted to $\widetilde{X}$ in such a way
that $p$ becomes a simplicial map.

\end{thm}

\begin{proof}

simplex�� fundamental group�� trivial�̴�. ���� lifting�� ��
�ְ� $\widetilde{X}$������ locally �Ȱ��� �� �� �ִ�.\\

{\bf ���� 18} (Prove in detail.)

\end{proof}

\begin{cor}

1. Any subgroup of a free group is free.\\
2. Let $F$ be a free group on $n$ generators and $F'$ be a
subgroup of $F$ of index $m$. Then $F'$ is a free group in
$1-m+mn$ generators.

\end{cor}

\begin{proof}

1. $\exists K$, a connected graph such that $\pi_{1}(|K|) = F$.\\
$F'<F \Rightarrow \exists$ a covering $K'$ of $K$ such that
$\pi_{1}(K')=F'$ and $K'$ is also a graph. $\Rightarrow F'$ is
free.\\

2. Can take $K$ as a finite graph.\\
$n=1-\chi(K)$ and $\chi(K')=m\chi(K)$ ���� $F'$�� generator��
������ $1-\chi(K') = 1-m\chi(K) = 1-m(1-n) = 1-m+mn$�̴�.

\end{proof}

\end{document}
