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\begin{document}
 \parindent=0cm
\section*{Definition of Fundamental Group}
\begin{defn}\textit{
$\Omega(X,x_0) := \{\alpha : I\rightarrow X \mid
\alpha(0)=\alpha(1)=x_0\}$ :  loop space of X based at $x_0$.}
\end{defn}

\textit{Define $\sim$ on $\Omega=\Omega(X,x_0) : \alpha \sim
\beta\Leftrightarrow \alpha\simeq \beta$ relative to $\partial I$},

i.e., $\exists$ $ F : I \times I \rightarrow X$ such that

$\hspace{2em}$     1. $F(t,0) = \alpha(t),\forall t \in I.$

$\hspace{2em}$     2. $F(t,1) = \beta(t), \forall t \in I.$

$\hspace{2em}$     3. $F(0,s) = x_0 = F(1,s), \forall s \in I.$

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In general for $f$ and $g$ : $X\rightarrow Y$, $f$ is homotophic to $g$,
denoted by $f \simeq g$,

if $\exists$ a map $F : X \times I \rightarrow  Y$ such that

$\hspace{2em}$ 1. $F(x,0)=f(x), \forall x \in X.$

$\hspace{2em}$ 2. $F(x,1)=g(x), \forall x \in X$.

{\bf Note.}  $\sim$ is an equivalence relation.

 (reflexive)  $\alpha \sim \alpha$

 (symmetric)  $\alpha \sim \beta \Rightarrow \beta \sim \alpha$ :

 $\alpha \sim \beta$를 주는  homotopy F에 대해
 G(t,s)=F(t,1-s)로 주면 이는  $\beta \sim \alpha$ 를 만족하는
 homotopy가 된다.

 (transitive)  $\alpha \sim \beta , \beta \sim \gamma \Rightarrow \alpha \sim
 \gamma$ :

F: homotopy between $\alpha$ and $\beta$, G: homotopy between
$\beta$ and $\gamma$ 라 하자. 이 때, H를

$$H(t,s)=\left \{\begin{array}{1}F(t,2s)\,\,\,\,\,\,  0\leq s \leq \frac{1}{2}\\ G(t,2s-1)\,\,\,\,\,\,  \frac{1}{2} \leq s \leq 1\end{array}
\right.$$

로 두면 H 는 $\alpha$ 와 $\gamma$ 사이의 homotopy가 된다.\\

{\bf   Introduce a group structure on $\Omega / \sim$. }

두 개의 loop 또는 일반적으로 두 개의 path $\alpha , \beta$에
대해서 product path $\alpha*\beta$를

$$\alpha * \beta (t)=\left\{\begin{array}{1} \alpha(2t),\,\,\,\,\,\, 0\leq t \leq \frac{1}{2}
 \\ \beta(2t-1),\,\,\,\,\,\,\frac{1}{2}\leq t \leq 1 \end{array} \right.$$

로 정의하자. 그러면

 \textit{(a) $*$ defines a multiplication on $\Omega/\sim$,
 i.e.,  $\{\alpha\}\{\beta\}=\{\alpha*\beta\}$} :

이 곱이 $\Omega / \sim$ 에서 잘 정의가 된다는 것을 보이자.

Show $\alpha_1 \sim \alpha_2,\beta_1 \sim \beta_2 \Rightarrow
\alpha_1 * \beta_1 \sim \alpha_2 * \beta_2$:
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F를 $\alpha_1, \alpha_2$의 homotopy, G를 $\beta_1, \beta_2$ 의
homotopy로 두면

$$H(t,s)=\left\{\begin{array}{1}F(2t,s),\,\,\,\,\,\,0\leq t \leq \frac{1}{2}\\
 G(2t-1,s),\,\,\,\,\,\, \frac{1}{2}\leq t \leq 1 \end{array}\right. $$

가 $\alpha_1*\beta_1,\alpha_2*\beta_2$ 사이의 homotopy 를 준다.

$\therefore \{\alpha\}\{\beta\} = \{\alpha*\beta\}$ is well
defined on $\Omega/\sim$. \\ \\  \textit{(b) Associativity, i.e.,
 $(\alpha * \beta)*\gamma \sim \alpha*(\beta * \gamma)$} :

$(\alpha * \beta)*\gamma$ 와  $\alpha*(\beta * \gamma)$ 는 사실상
같은 path 의 reparametrization이므로 다음 Note만 보이면 된다.
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$\\${\bf Note.} In general, if $\beta(t)=\alpha(\phi(t))$ where
$\phi : I \rightarrow I $  with $\phi(0)=0$ and $\phi(1)=1$, is a
\textit{reparametrization},
 then $\alpha \sim \beta.$
%
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(증명) F(t,s) : = $\alpha(s \phi(t) + (1-s)t)$  는 연속이고\\
 $F(t,0)=\alpha(t)$, $F(t,1)=\alpha(\phi(t))=\beta(t)$ 사이에
원하는 homotopy를 준다.\\

 \textit{(c) Existence of an identity e: I $\rightarrow \{x_0\}
\subset X.$ (a constant loop.)}

$\alpha*e$ 는  $\alpha$의 reparametrization이므로  위의  Note에
따라 $\alpha*e \sim \alpha \sim e
* \alpha.$\\

\textit{(d)Existence of an inverse.}

Given $\alpha \in \Omega$, define $\overline{\alpha}(t):=
\alpha(1-t).$ Then  $\alpha*\overline{\alpha} \sim e \sim
\overline{\alpha}*\alpha.\\F : I \times I \rightarrow X  $ 를
다음과 같이 정의하자.

$$F(t,u)=\left\{\begin{array}{1}\alpha(2t), \,\,\,\,\,\,\,\,\,\,\,\,0\leq u\leq
1-2t. \\\overline{\alpha}(2t-1),\,\,\,\,\,\, u\leq 2t-1.\\
\alpha(1-u)=\overline{\alpha}(u), \,\,\,\,\,\,u\geq
|1-2t|.\end{array} \right.$$

그러면  F는 연속이고 $\alpha * \overline{\alpha}$와 e 사이에
homotopy를 준다. 같은 방법으로  $\overline{\alpha}*\alpha \sim e$
도 역시 보일 수 있다.

\begin{defn}\textit{(The fundamental group.)}

$\pi_1(X,x_0) := \Omega(X,x_0)/\sim $을 X의  fundamental group
(based at $x_0$)라고 부른다.
\end{defn}
\end{document}