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\begin{document}
 \parindent=0cm
  \section*{Example}

{\bf 1. Contractible space}

  \begin{defn}\textit{
  A space X is contractible to $x_0 \in X$  if  \\$id_X \simeq c,$
  where c : X $ \rightarrow \{x_0\} \subset X $is a constant
  map.}\\
  \end{defn}

{\bf ¿¹} 1. $\mathbb{R}^n$ is contractible.
 %%%%%%%%%%
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  $\hspace{2em}$  F(t,x)=tx  ·Î ÁÖ¸é ÀÌ´Â id ¿Í  constant map {0} °£¿¡ homotopy ¸¦ ÁØ´Ù.

  $\hspace{1em}$ 2. $D^n$ is contractible. ¿ª½Ã F(t,x)=tx ·Î ÁÖ¸é µÈ´Ù.

   $\hspace{1em}$ 3. Any space which is homeomorphic to a contractible space
   is contractible.

   $\hspace{1em}$ 4. A "tree" is contractible.

 %%%%%%%%%%
 %  tree  %
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   $\hspace{1em}$ 5. $S^1$ is not contractible.\\

{\bf Remark.}

1. \textit{  contractible $\Rightarrow$ path connected:}\\
ÀÓÀÇÀÇ µÎ point´Â $x_0$¸¦ ÅëÇØ path·Î ¿¬°áµÈ´Ù.\\

2. \textit{X is contractible to $x_0 \in X \Rightarrow$ X is
contractible to any other point of X }:

 X°¡ contractible to $x_0$
ÀÌ¸é  path connected À̹ǷΠ $\forall\,\,x_1 \in$ ¿¡ ´ëÇØ
$x_0,x_1$ »çÀÌ¿¡  path $\rho$°¡ Á¸ÀçÇÑ´Ù. F ¸¦ $id_X$ ¿Í $c_{x_0}$
°£ÀÇ homotopy¶ó ÇÒ  ¶§ ¾Æ·¡¿Í °°ÀÌ Á¤ÀÇµÈ H´Â $id_X$¿Í $c_{x_1}$
»çÀÌ¿¡ ¿øÇÏ´Â homotopy¸¦ ÁØ´Ù.


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

3. \textit{ X is contractible $\Leftrightarrow$ X $\simeq$
\{point\} :}

($\Rightarrow$ Áõ¸í) $\{x_0\} \hookrightarrow X \rightarrow
\{x_0\}$¿¡¼­

$\hspace{7em}i \hspace{2em}c_{x_0}$

$c_{x_0} \circ i = id_{x_0}$ ÀÌ°í X °¡ contractible À̹ǷΠ$ i
\circ c_{x_0} \simeq id_{X} $ÀÌ´Ù. µû¶ó¼­ $X \simeq \{x_0\}$ ÀÌ´Ù.

($\Leftarrow$ Áõ¸í) $X \simeq \{x_0\}$ À̹ǷΠhomotopy equivalence
f: $X \rightarrow \{x_0\},g : \{x_0\} \rightarrow X$ °¡ Á¸ÀçÇÑ´Ù.
ÀÌ ¶§ f´Â constant map $c_{x_0}$ °¡ µÇ°í µû¶ó¼­ $g \circ f$ ¿ª½Ã
constant mapÀÌ µÈ´Ù. ±×·±µ¥ $g \circ f \simeq id_{X}$ À̹ǷΠX ´Â
contractibleÇÏ´Ù.\\

\begin{thm}
X is contrantible $\Rightarrow$  $\pi_1(X) = 0$.
\end{thm}
\begin{proof}
$X \simeq \{x_0\}$ À̹ǷΠ$\pi_1(X) \cong \pi_1(\{point\})=0$.
\end{proof}\\

\newpage

{\bf 2. $\pi _1 (X\times Y)$}

\begin{thm}
If X and Y are path connected, then $\pi_1(X \times Y,(x_0,y_0))
\cong \pi_1(X,x_0) \times \pi_1(Y,y_0)$.
\end{thm}
\begin{proof}
Define $\phi : \pi_1(X \times Y, (x_0,y_0))\rightarrow
\pi_1(X,x_0)\times \pi_1(Y,y_0)$ ,

$\hspace{10em}[\alpha]\hspace{3em}\mapsto \,\,\,\,([p_1 \circ
\alpha],[p_2 \circ \alpha])$

where $p_1,p_2$ are projections to X and Y respectively.

 %%%%%%%%%%%%
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 %%%%%%%%%%%%%%%%%
 %  end{figure}  %
 %%%%%%%%%%%%%%%%%


ÀϹÝÀûÀ¸·Î µÎ homomorphism $\phi_1 : G\rightarrow H_1 ,\,\, \phi_2
: G \rightarrow H_2$ ¿¡ ´ëÇØ $\phi(g) = (\phi_1(g),\phi_2(g)) : G
\rightarrow H_1 \times H_2 $ ¿ª½Ã homomorphismÀÌ µÇ¹Ç·Î À§¿¡¼­ÀÇ
$\phi$ ´Â homomorphismÀÌ µÈ´Ù. ÀÌÁ¦ $\phi$ÀÇ ¿ªÇÔ¼ö¸¦ ã±â À§ÇØ
¾Æ·¡¿Í °°ÀÌ $\psi$¸¦ Á¤ÀÇÇÏÀÚ.

 $\,\,\,\,\,\,\,\,\,\,\,\psi : \pi_1(X,x_0)\times\pi_1(Y,y_0) \rightarrow \pi_1(X \times Y,(x_0,y_0))$

 $\hspace{5em}([\beta],[\gamma])\,\,\hspace{2em}\mapsto \hspace{2em}[\delta]$

 $\hspace{2em}$,where $\delta(t)=(\beta(t),\gamma(t))$

 ÀÌ $\psi$´Â $\phi$ ÀÇ ¿ªÇÔ¼ö°¡ µÇ°í ÀÌÁ¦ $\psi$ °¡ Àß
 Á¤ÀǵǾú´ÂÁö¸¸ »ìÆ캸¸é µÈ´Ù.

  F : $\beta \sim \beta'$ , G : $\gamma \sim \gamma'$ ÀÏ ¶§
 $H(t,s)=( F(t,s) , G(t,s))$ ·Î ÁÖ¸é ÀÌ´Â $\delta$ ¿Í $\delta'$
 »çÀÌÀÇ homotopy¸¦ ÁֹǷΠ$\psi$ °¡ Àß Á¤ÀǵǾúÀ½À» ¾Ë ¼ö ÀÖ´Ù.

\end{proof}


{\bf ¿¹:} $\pi _1 (S^1 \times S^1 ) = \pi _1 (S^1 ) \times \pi _1
(S^1 )
= \mathbb{Z} \times \mathbb{Z} \cong \mathbb{Z}^2 $\\

$\hspace{1.5em} \pi _1 (T^n ) = \mathbb{Z}^n $\\

$\hspace{1.5em}\pi (\mathbb{R}^n \times S^1 ) = \mathbb{Z}$\\

\newpage

{\bf 3. Topological group}

\begin{thm}\textit{
 G : a path connected topological group with identity
 e\\$ \hspace{3em}\Rightarrow \pi_1(G,e)$ is abelian.}
\end{thm}

\begin{proof}
$[\alpha] , [\beta] \in \pi_1(G , e)$ ¿¡ ´ëÇØ $[\alpha][\beta] =
[\beta][\alpha]$ ÀÓÀ» º¸À̱â À§ÇØ

$[\alpha][\beta][\alpha]^{-1}[\beta]^{-1} = 1$ ÀÓÀ» º¸ÀÌÀÚ.
$[\alpha][\beta][\alpha]^{-1}[\beta]^{-1}=[\alpha* \beta*
\overline{\alpha}*\overline{\beta}] \simeq 1$ À» º¸À̱â À§ÇØ
$\alpha* \beta* \overline{\alpha}*\overline{\beta}$ ¸¦ ´ÙÀ½°ú °°ÀÌ
º¸ÀÚ.

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\end{figure}


 ±×¸² 2¿¡¼­ ¸ÕÀú º¸¸é,

F : $I^2 \rightarrow G$ given by F(t,s)=$\alpha(t)\beta(s)$ ´Â
group G ¾È¿¡ °öÀÌ Á¤ÀǵǹǷΠ$I^2$»ó¿¡¼­ ¿¬¼ÓÇÔ¼ö·Î Àß Á¤Àǵǰí
$I^2$ÀÇ boundary´Â »ç½Ç»ó $\alpha* \beta*
\overline{\alpha}*\overline{\beta}$¸¦ ÁØ´Ù.

µû¶ó¼­

$[\alpha][\beta][\alpha]^{-1}[\beta]^{-1} = F_{\sharp}([\partial
I^2]) = F_{\sharp}(1) = 1.$

\end{proof}\\

{\bf ¼÷Á¦3.}  $[\alpha]\in \pi_1(X,x_0)$, $\alpha : S^1 = \partial
D^2 \rightarrow X$  ¿¡ ´ëÇØ\\\textit{$[\alpha]=1$ if and only if
$\alpha$ can be extended to $D^2$}

\end{document}