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\newtheorem{thm}{정리}
\newtheorem{cor}[thm]{따름정리}
\newtheorem{lem}[thm]{보조정리}
\newtheorem{prop}[thm]{명제}
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\newtheorem{que}{질문}
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\newtheorem{defn}{정의}
\newtheorem{rem}{주}
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\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 %
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(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 %
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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 %
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\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 %
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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 %
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\begin{pspicture}(5,1.7)%
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\rput(0.5,1.4){=}%
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\psline[linecolor=blue](1.2,1.49)(1.2,1.51)%
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\psdot(0.9,1.5)%
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\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){=}%
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\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)

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\rput(1.5,0.5){$\square$마다 applying claim}%
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$\hspace{5em}= [x_0 ][x_0]\cdots[x_0 ]$\\
$\hspace*{5em}= [x_0 ] = 1$
\end{proof}
\end{document}
