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\begin{document}
 \parindent=0cm
\section*{Definition of Fundamental Group}
A \textit{path} $\alp$ in a space $X$ is a continuous map $\alp :
[0,1]\rightarrow X$.

\begin{defn}
Given two paths $\alp$ and $\bet$ in $X$ with the same end points, ie, $\alp (0)= \bet (0)$ and $\alp (1)= \bet (1)$ .

 $\alp$ is \textit{path homotopic} to $\bet$, denoted by $\alp
\thicksim\bet$,\ if $\alp \overset{F}{\simeq}\bet$ rel $\partial
I$ = \{0,1\} ,

 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) = $\alp (0)$, F(1,s) = $\alp (1) , \forall s \in I.$

\end{defn}


In general for $f$ and $g$ :$X\rightarrow Y$ with $f(a)$ = $g(a)$
for $a\in A\subset X$,

$f$ is \textit{homotopic to } g (denoted by $f \simeq g$) relative
to $A\subset X$,

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) = $g(x), \forall x \in X.$

$\hspace{2em}$     3. F(a,s) = $f(a) = g(a) , \forall a \in A.$\\

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

 (reflexive)  $\alpha \sim \alpha$\\

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

 $\alpha \sim \beta$�� �ִ�  homotopy F�� ����
 G(t,s)=F(t,1-s)�� �ָ� �̴�  $\beta \sim \alpha$ �� �����ϴ�
 homotopy�� �ȴ�.\\

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

F: homotopy between
$\alpha$ and $\beta$, G: homotopy between
$\beta$ and $\gamma$ �� ����. �� ��, H��

\[
H(t,s)=
\begin{cases}
F(t,2s) &\text{if $ 0\leq s \leq \frac{1}{2}$}\\
G(t,2s-1) &\text{if $ \frac{1}{2} \leq s \leq 1$}
\end{cases}
\]
�� �θ� H �� $\alpha$ �� $\gamma$ ������ homotopy�� �ȴ�.


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\put(75,0){\makebox(0,0){$\alp$}}

\put(50,10){\framebox(50,50){F}}

\put(75,65){\makebox(0,0){$\bet$}}

\put(135,0){\makebox(0,0){$\bet$}}

\put(110,10){\framebox(50,50){G}}

\put(135,65){\makebox(0,0){$\gamma$}}

\put(225,0){\makebox(0,0){$\alp$}}

\put(200,10){\framebox(50,25){F}}

\put(200,35){\framebox(50,25){G}}

\put(225,65){\makebox(0,0){$\gamma$}}
\end{picture}


\begin{defn}
(i) �� ����  path $\alpha , \beta$ with $\alp(1) =\bet(0)$��
���ؼ� \textit product path $\alpha*\beta$�� ������ ���� ���ǵȴ�.

\[
\alpha * \beta (t)=
\begin{cases}\alpha(2t)&\text{if $ 0\leq
t \leq \frac{1}{2}$} \\
\beta(2t-1)&\text{if $\frac{1}{2}\leq t \leq 1$}
\end{cases}
\]

(ii) $\Omega(X,x_0) := \{\alpha : I\rightarrow X \mid
\alpha(0)=\alpha(1)=x_0\}$ :  loop space of X based at $x_0$
\end{defn}
\\
{\bf Introduce a group structure on $\Omega / \sim$:}

(a) group operation��  $[\alpha ] \cdot [ \beta ] :=[\alpha*\beta ]$ \\

�� ���� $\Omega / \sim$ ���� �� ���ǰ� �ȴٴ� ���� ������.

�� $\alpha \sim \alpha' ,\beta \sim \beta' \Rightarrow \alpha
* \beta \sim \alpha * \beta'$���� ���̸� ����ϴ�.

F�� $\alpha, \alpha'$�� homotopy, G�� $\beta, \beta'$ ��
homotopy�� �θ�

\[
H(t,s)=
\begin{cases}F(2t,s)& \text{if $0\leq t \leq \frac{1}{2}$}\\
G(2t-1,s)&\text{if $ \frac{1}{2}\leq t \leq 1 $}
\end{cases}
\]

�� $\alpha *\beta,\alpha' *\beta'$ ������ homotopy �� �ش�.\\

�׷��Ƿ� $ [\alpha ]\cdot [\beta] = [\alpha*\beta]$��
$\Omega/\sim$���� �� ���ǵǾ���.

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\put(75,0){\makebox(0,0){$\alp$}}

\put(50,10){\framebox(50,50){F}}

\put(75,65){\makebox(0,0){$\alp'$}}

\put(135,0){\makebox(0,0){$\bet$}}

\put(110,10){\framebox(50,50){G}}

\put(135,65){\makebox(0,0){$\bet'$}}

\put(210,0){\makebox(0,0){$\alp$}}

\put(200,10){\framebox(25,50){F}}

\put(210,65){\makebox(0,0){$\alp'$}}

\put(235,0){\makebox(0,0){$\bet$}}

\put(225,10){\framebox(25,50){G}}

\put(235,65){\makebox(0,0){$\bet'$}}
\end{picture}
\\
\\

(b) \textit{Associativity} $([\alpha ]\cdot [ \beta] \cdot
[\gamma] = [\alpha] \cdot ([\beta ] \cdot [ \gamma])$ :\\

$(\alpha * \beta)*\gamma$ ��  $\alpha*(\beta * \gamma)$ �� ��ǻ�
���� path �� reparametrization�̹Ƿ� ���� Note�� ���̸� �ȴ�.


$\\${\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 $\alpha \sim \beta.$\\

(����) F(t,s) : = $\alpha(s \phi(t) + (1-s)t)$  �� �����̰�\\
 $F(t,0)=\alpha(t)$, $F(t,1)=\alpha(\phi(t))=\beta(t)$ ���̿�
���ϴ� homotopy�� �ش�.\\

 \textit{(c) Existence of an identity e}:\\

Let I $\rightarrow \{x_0\} \subset X$ (a constant loop).
�׷���$\alpha*e$ ��  $\alpha$�� reparametrization�̹Ƿ�  ����
Note�� ���� $\alpha*e \sim \alpha \sim e
* \alpha.$\\

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

Given $\alpha \in \Omega$, define $\overline{\alpha}(t):=
\alpha(1-t).$ Then we show that $\alpha*\overline{\alpha} \sim e
\sim \overline{\alpha}*\alpha.$
 �̰��� ���̱� ���� $ F : I \times
I \rightarrow X $ �� ������ ���� ��������.

\[
F(t,u)=
\begin{cases}\alpha(2t)&\text{if $0\leq
u\leq 1-2t$}\\
\overline{\alpha}(2t-1)&\text{if $u\leq 2t-1$}\\
\alpha(1-u)=\overline{\alpha}(u)&\text{if $u\geq
|1-2t|$}
\end{cases}
\]

�׷���  F�� �����̰� $\alpha * \overline{\alpha}$�� e ���̿�
homotopy�� �ش�. ���� �������  $\overline{\alpha}*\alpha \sim e$
�� ���� ���� �� �ִ�.

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

$\pi_1(X,x_0) := \Omega(X,x_0)/\sim $�� X��  fundamental group
(based at $x_0$)��� �θ���.
\end{defn}
\end{document}
