
\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}
\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}{정리}
\newtheorem{cor}[thm]{따름정리}
\newtheorem{lem}[thm]{보조정리}
\newtheorem{prop}[thm]{명제}
\newtheorem{cl}{Claim}

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

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

\begin{document}
 \parindent=0cm
  \section*{III.2 Van Kampen theorem }

\begin{thm}\textit{\textbf{(Van-Kampen theorem)}}\\
\textcolor{blue}{Suppose $X=U\cup V$ with $x_0 \in U\cap V$ where
$U, V$ and $ U\cap V$ are open and path-connected . Then $\pi _1
(X,x_0 ) \cong \underset{\pi_1 (U \cap V, x_0 )}{\pi _1 (U,x_0 ) *
\pi _1 (V, x_0 )}$ where the homomorphism for an amalgamated
product are induced by inclusions.}
\end{thm}

\framebox{\hspace*{1em}\parbox[b]{14cm}{ $\hspace*{6em}
U\hspace{15em}\pi_1 (U, x_0 )\\
\hspace*{3em}i_1 \nearrow \hspace{3em} \searrow j_1 \hspace{9em}
i_{1\sharp} \nearrow \hspace{4em} \searrow j_{1\sharp}\\
\hspace*{1em}U\cap V \hspace{2em} \curvearrowright
\hspace{2em}X\hspace{2em}
\overset{\pi_1}{\rightsquigarrow}\hspace{1em}\pi_1 (U\cap V ,x_0 )
\hspace{3em}\curvearrowright\hspace{2em}\pi_1 (X,x_0 )\\
\hspace*{3em}i_2 \searrow \hspace{3em} \nearrow j_2 \hspace{9em}
i_{2\sharp} \searrow \hspace{4em} \nearrow
j_{2\sharp}\\
\hspace*{6em}V\hspace{15em}\pi_1 (V, x_0 )$
}}\\

\begin{proof}\\
$\underset{\pi_1 (U \cap V, x_0 )}{\pi _1 (U,x_0 ) * \pi _1 (V,
x_0 )} \cong \pi_1 (X,x_0 )$임을 보여야 한다.\\

$\exists ! h:\underset{\pi_1 (U \cap V, x_0 )}{\pi _1 (U,x_0 )
* \pi _1 (V, x_0 )} \rightarrow \pi_1 (X,x_0 )$, a homomorphism
by universal prop.\\
 Show $h$ is (i) onto and (ii) one to one:\\

(i) $h$ is onto : use 2(b), i.e., show $h_1 \pi_1 (U)$ and $h_2
\pi_1 (V)$ generate $\pi_1 (X)$:\\

$ \forall \alp \in \pi_1 (X)$, use Lebesgue lemma to obtain $ \alp
= \alp_1 * \cdots *\alp_{p},$\\s.t. $\alp_{i}
:= \alp|_{[t_{i-1} ,t_{i} ]} \subset U $ or $V$.\\
Choose a path $\rho_{i}$ from $x_0$ to $\alp_{i} (t_{i} )$ lying
completely in $U$ or $V$ \\and let $\rho_{p} = \rho_0 $ = constant
path $x_0$.\\ Let $\widetilde{\alp_{i} }
= \rho_{i-1} * \alp_{i} * \overline{\rho_{i} }.$\\
Then $[\alp] = [\alp_1 * \cdots *\alp_{p} ] = [\widetilde{\alp_1
}*\cdots *\widetilde{\alp_{p} }] =
[\widetilde{\alp_1 } ]\cdots [\widetilde{\alp_{p} }]$ and \\
each $[\widetilde{\alp_{i} }] \in h_1 \pi_1 (U) $
or $h_2 \pi_1 (V)$.\\

%   figure : Van-Kampen thm :onto %
\psset{unit=2cm}
\begin{center}
\begin{pspicture}(3,2)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(1.5,1){
    \rput(0,0){
    \rput(-1,0.8){$U$}%
    \rput(1,0.8){$V$}%
    \psellipse(-1,0)(1.5,1.0)%
    \psellipse(1,0)(1.5,1)
    \rput(0,-0.6){$x_0$}
    %\pscurve[linewidth=0.03](-1,3)(-1,1.1)(-0.9,1)(0.9,1)(1,1.1)(1,3)
    \pscurve(0,-0.5)(2,-0.5)(-2.3,0)(1.9,0.3)(2,0.6)(0,-0.5)
    \pscurve[linecolor=blue](0,-0.5)(-0.05,-0.4)(-0.1,-0.25)
    \rput(-0.7,-0.7){$\rho_1$ }
    \pscurve{->}(-0.6,-0.6)(-0.3,-0.45)(-0.05,-0.4)
    \pscurve[linecolor=blue](0,-0.5)(0.05,0.1)(0.1,0.14)
    \rput(0,0.5){$\rho_2$}
    \pscurve{->}(0,0.4)(0.05,0.25)(0.1,0.14)
    \rput(1,-0.5){$\alp_1$}
    \rput(-1,0){$\alp_2$}
    \rput(1.5,0.5){$\alp_3$}
    }
}
\end{pspicture}
\end{center}

(ii) $h$ is one-to-one i.e. show ker $h$ =0:\\

Suppose $h ( [\alp_1 ]\cdots [\alp_{p} ]) = 1$, where $[\alp_{i}]$
in either $\pi_1 (U)$ or $\pi_1 (V)$ and show $[\alp_1 ]\cdots
[\alp_{p} ] = 0 $ in $\underset{\pi_1 (U \cap V )}{\pi _1 (U )
* \pi _1 (V)}.\\ \alp_1 * \cdots *\alp_{p} \sim 1$ in $
X$ by hypothesis\\ $\Rightarrow \exists F : I \times I \rightarrow
X = U \cup V$ with $F_0 = \alp_1 * \cdots *
\alp_{p}$ and $F_1 = x_0 $.\\

%   figure : Van-Kampen thm :one-to-one %
\psset{unit=2cm}
\begin{center}
\begin{pspicture}(4,2)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(1,1){
    \rput(0,0){\psframe(-0.9,-0.7)(0.9,0.7)%
    \psdot(-0.9,-0.7)%
    \psline[linecolor=blue](-0.8,-0.7)(-0.8,0.7)%
    \psline[linecolor=blue](-0.7,-0.7)(-0.7,0.7)%
    \psdot(-0.7,-0.7)%
    \rput(-0.8,-0.9){$\alp_1$}
    \psline[linecolor=blue](-0.6,-0.7)(-0.6,0.7)%
    \psline[linecolor=blue](-0.5,-0.7)(-0.5,0.7)%
    \psline[linecolor=blue](-0.4,-0.7)(-0.4,0.7)%
    \psdot(-0.3,-0.7)%
    \rput(-0.4,-0.9){$\alp_2$}
    \psline[linecolor=blue](-0.3,-0.7)(-0.3,0.7)%
    \rput(0,-0.9){$\cdots$}
    \psline[linecolor=blue](0.5,-0.7)(0.5,0.7)%
    \psline[linecolor=blue](0.6,-0.7)(0.6,0.7)%
    \psline[linecolor=blue](0.7,-0.7)(0.7,0.7)%
    \psline[linecolor=blue](0.8,-0.7)(0.8,0.7)%
    \psdot(0.6,-0.7)%
    \rput(0.8,-0.9){$\alp_{p}$}%
    \psdot(0.9,-0.7)%
    \psline[linecolor=blue](-0.9,-0.6)(0.9,-0.6)%
    \psline[linecolor=blue](-0.9,-0.5)(0.9,-0.5)%
    \psline[linecolor=blue](-0.9,-0.4)(0.9,-0.4)%
    \psline[linecolor=blue](-0.9,-0.3)(0.9,-0.3)%
    \psline[linecolor=blue](-0.9,0.4)(0.9,0.4)%
    \psline[linecolor=blue](-0.9,0.5)(0.9,0.5)%
    \psline[linecolor=blue](-0.9,0.6)(0.9,0.6)%
    \rput(0,0.8){$x_0$}
        }}
\rput(3,1){$\longrightarrow$} %
\rput(3.5,1){$X$}%
\rput(3,1.1){$F$}%
\end{pspicture}
\end{center}

Using Lebesgue lemma, can partition $I\times I$ fine enough so that\\
(1) $F($each $\square ) \subset U $ or $V$ and \\
(2) partition contains end points of domain of each $\alp_{i}$
so that \\
$\hspace*{1em}\alp_1 = \alp_{11} * \alp_{12} *\cdots *\alp_{1i_1},\\
\hspace*{3em}\vdots \\
\hspace*{1em}\alp_{p} = \alp_{p1} * \hspace{1em}\cdots\hspace{1em} *\alp_{pi_p}.$\\

Choose and fix a path $\rho_{r,s}$ from $x_o$ to $F(t_r , t_s )$
once and for all s.t. \\
$\rho_{r,s} \subset \{x_0\}  , U\cap V , U, V$ respectively if $F(t_r ,t_s ) \in \{x_0 \}  , U\cap V , U, V$ respectively.\\

Let $\widetilde{\sigma_{r,s}} = \rho_{r,s} *\sigma_{r,s,}
*\overline{\rho_ {r+1 ,s}}$ and $\widetilde{\tau_{r,s}} =
\rho_{r,s} * \tau_{r,s} *{\overline{\rho_ {r ,s+1}}}$,\\ where
$\sigma_{r,s}$ and $\tau_{r,s}$ are short paths
given by the following picture.\\

%   figure : Van-Kampen thm :one-to-one %
\psset{unit=2cm}
\begin{center}
\begin{pspicture}(2,1)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(0.6,0.7){$\tau_{r,s}$}%
\rput(1.1,0.4){$\sigma_{r,s}$}%
\psline[linecolor=blue](0,0.2)(2,0.2)%
\psline[linecolor=blue](0,0.5)(2,0.5)%
\psline[linecolor=blue](0,0.8)(2,0.8)%
\psline[linecolor=blue](0.4,0)(0.4,1)%
\psline[linecolor=blue](0.8,0)(0.8,1)%
\psline[linecolor=blue](1.3,0)(1.3,1)%
\psline[linecolor=blue](1.7,0)(1.7,1)%
\psline{->}(0.8,0.5)(1.3,0.5)%
\psline{->}(0.8,0.5)(0.8,0.8)%
\psdot(0.8,0.5)%
\rput(0.4,0.4){$(t_r ,t_s )$}%
\psline[linecolor=blue](0.8,0.2)(0.8,0.5)%
\end{pspicture}
\end{center}
\newpage
Claim: $\widetilde{\sigma_{r,s} }*\widetilde{\tau_{r+1,s}} \sim
\widetilde{\tau_{r,s}} *\widetilde{\sigma_{r,s+1}}$ in $U$ or
$V$.\\(Proof is clear as in the following picture.)\\

%   figure : Van-Kampen thm :one-to-one %
\psset{unit=2cm}
%\begin{floatingfigure}[l]{6.5cm}
\begin{center}
\begin{pspicture}(2,2)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(0,1){\rput(0,0){\psline[linecolor=blue](0,0.2)(2,0.2)%
\psline[linecolor=blue](0,0.5)(2,0.5)%
\psline[linecolor=blue](0,0.8)(2,0.8)%
\psline[linecolor=blue](0.4,0)(0.4,1)%
\psline[linecolor=blue](0.8,0)(0.8,1)%
\psline[linecolor=blue](1.3,0)(1.3,1)%
\psline[linecolor=blue](1.7,0)(1.7,1)%
\psline{->}(0.8,0.5)(1.3,0.5)%
\psline{->}(0.8,0.5)(0.8,0.8)%
\psline{->}(0.8,0.8)(1.3,0.8)%
\psline{->}(1.3,0.5)(1.3,0.8)%
\rput(1,1){$\sigma$}%
\rput(1,0.3){$\sigma$}%
%\rput(1.2,0.3){$A$}%
%\rput(0.6,0.4){$B$}%
\rput(0.6,0.6){$\tau$}%
\rput(1.5,0.6){$\tau$}%
\psframe[fillstyle=solid,fillcolor=blue](0.8,0.5)(1.3,0.8)
\psdot(0.8,0.5)%
\psdot(0.4,-0.8)%
\pscurve(0.4,-0.8)(0.6,-0.1)(0.8,0.5)%
%\pscurve(0.4,-0.8)(1.15,-0.1)(1.4,0.63)(1.3,0.8)%
%\pscurve(0.4,-0.8)(0.7,0.8)(1.2,0.85)(1.3,0.8)%
 }}
\end{pspicture}
\end{center}
%\end{floatingfigure}

In $\underset{\pi_1 (U \cap V )}{\pi _1 (U ) * \pi _1 (V)},$\\
$[\alp_1 ] \cdots [\alp_{p}] = [\widetilde{\alp}_{11}
][\widetilde{\alp }_{12}]\cdots[\widetilde{\alp}_{1{i_1}} ]\cdots[\widetilde{\alp}_{p1} ]\hspace{1em}\cdots\hspace{1em}[\widetilde{\alp}_{p{i_p} } ]$\\

%   figure : Van-Kampen thm :one-to-one %
\psset{unit=3cm}
%\begin{floatingfigure}[l]{6.5cm}
\begin{center}
\begin{pspicture}(5,1.7)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(0.24,0.4){
\rput(0.5,1.4){=}%
\psline[linecolor=blue](0.8,1.49)(0.8,1.51)%
\psline[linecolor=blue](1,1.49)(1,1.51)%
\psline[linecolor=blue](1.1,1.49)(1.1,1.51)%
\psline[linecolor=blue](1.2,1.49)(1.2,1.51)%
\psline[linecolor=blue](2.3,1.49)(2.3,1.51)%
\psline[linecolor=blue](2.4,1.49)(2.4,1.51)%
\psline[linecolor=blue](2.2,1.49)(2.2,1.51)%
\psline(0.7,1.5)(2.5,1.5)%
\psdot(0.7,1.5)%
\rput(0.8,1.4){$\alp_1$}
\psdot(0.9,1.5)%
\rput(1.1,1.4){$\alp_2$}
\psdot(1.3,1.5)%
\rput(1.8,1.4){$\cdots$}%
\rput(2.3,1.4){$\alp_{p}$}
\psdot(2.1,1.5)%
\psdot(2.5,1.5)%

\rput(0.5,1.2){=}%
\psline(0.7,1.15)(2.4,1.15)\psline(2.4,1.15)(2.4,1.25)\psline(2.4,1.25)(2.5,1.25)
\rput(0.5,0.9){=}%
\psline(0.7,0.85)(2.3,0.85)\psline(2.3,0.85)(2.3,0.95)\psline(2.3,0.95)(2.5,0.95)

\rput(1,0.7){$\vdots$}%
\rput(1.5,0.5){$\square$마다 applying claim}%
\rput(1,0.3){$\vdots$}%
\rput(0.5,0){=}%
\psline(0.7,-0.6)(0.7,0.1)\psline(0.7,0.1)(2.5,0.1)%
\rput(3.6,0.7){
    \rput(0,0){\psframe(-0.9,-0.7)(0.9,0.7)%
    \psdot(-0.9,-0.7)%
    \psline[linecolor=blue](-0.8,-0.7)(-0.8,0.7)%
    \psline[linecolor=blue](-0.7,-0.7)(-0.7,0.7)%
    \psdot(-0.7,-0.7)%
    \rput(-0.8,-0.9){$\alp_1$}
    \psline[linecolor=blue](-0.6,-0.7)(-0.6,0.7)%
    \psline[linecolor=blue](-0.5,-0.7)(-0.5,0.7)%
    \psline[linecolor=blue](-0.4,-0.7)(-0.4,0.7)%
    \psdot(-0.3,-0.7)%
    \rput(-0.4,-0.9){$\alp_2$}
    \psline[linecolor=blue](-0.3,-0.7)(-0.3,0.7)%
    \rput(0,-0.9){$\cdots$}
    \psline[linecolor=blue](0.5,-0.7)(0.5,0.7)%
    \psline[linecolor=blue](0.6,-0.7)(0.6,0.7)%
    \psline[linecolor=blue](0.7,-0.7)(0.7,0.7)%
    \psline[linecolor=blue](0.8,-0.7)(0.8,0.7)%
    \psdot(0.6,-0.7)%
    \rput(0.8,-0.9){$\alp_{p}$}%
    \psdot(0.9,-0.7)%
    \psline[linecolor=blue](-0.9,-0.6)(0.9,-0.6)%
    \psline[linecolor=blue](-0.9,-0.5)(0.9,-0.5)%
    \psline[linecolor=blue](-0.9,-0.4)(0.9,-0.4)%
    \psline[linecolor=blue](-0.9,-0.3)(0.9,-0.3)%
    \psline[linecolor=blue](-0.9,0.4)(0.9,0.4)%
    \psline[linecolor=blue](-0.9,0.5)(0.9,0.5)%
    \psline[linecolor=blue](-0.9,0.6)(0.9,0.6)%
    %\psframe[fillstyle=solid,fillcolor=white]()()%
    \rput(-1,0){$x_0$}
    \rput(0,0.8){$x_0$}
    \rput(1,0){$x_0$}
    \psline[linecolor=red](-0.9,-0.72)(0.92,-0.72)%
    \psline[linecolor=red](0.92,-0.72)(0.92,-0.58)%
    \psline[linecolor=red](0.78,-0.72)(0.78,-0.58)%
    \psline[linecolor=red](0.78,-0.58)(0.92,-0.58)%
    }
    %\psline[linecolor=red](-0.9,0.6)(0.9,0.6)%
}
 }
\end{pspicture}
\end{center}
%\end{floatingfigure}

$\hspace{5em}= [x_0 ][x_0]\cdots[x_0 ]$\\
$\hspace*{5em}= [x_0 ] = 1$
\end{proof}
\end{document}