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\begin{document}
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  \section*{Application}

  \textit{{\bf $1\,\,.\,\, \pi_1(S^n)$=0, $n\geq 2$}}

  \begin{proof}
 simplicial approximation ������ $S^n\setminus
 \{point\}=\mathbf{R}^N$�� contractible�̶�� ����
 �̿�����. $S^1$���� $S^n$���� ���� $\alp$�� onto�� �ƴ϶��, $S^n$�󿡼�
 $\alp$�� image�� �ƴ� �� $x_0$ �� ������ �� �ְ� $S^n\setminus
 \{x_0\}=\mathbf{R}^N$�� contractible�̹Ƿ� ������ �ϼ��ȴ�.

���� simplicial map ��ü�� $p$-skeleton ��  $p$-skeleton ����
�����Ƿ� $\alp$�� simplicial approximation $\alp '$�� onto�� �� ��
����, ���� $\alp'$�� �� ������ contractible �ϰ� $\alp \sim \alp
'$ �̹Ƿ� $\alp$ �� contractible�ϴ�.

�� �������� base point�� vertex�� �ǵ��� simplicial complex ������
$S^1$�� $S^n$�� ������ �ش�. ���� 4�� Remark�� ����
$\alp(base\,\,point)=\alp'(base\,\,point)$�� �ξ �������.

  \end{proof}

{\bf $2\,\,.\,\, i:K^1\hookrightarrow K$ induces an epimorphism
$i_*:\pi_1(|K^1|)\rightarrow \pi_1(|K|)$.}

{\bf$\hspace{1.2em}i:K^2\hookrightarrow K$ induces an isomorphism
$i_*:\pi_1(|K^2|)\rightarrow \pi_1(|K|)$.  }\\


\begin{proof}
���� epimorphism �� ���� ������. $\pi_1(|K|)$�� �� $\{\alp\}$��
����  simplicial approximation $\alp '$�� ��������. ���� $\alp
\sim \alp'$ ���� �˰� �ְ� ($\alp$�� $\alp '$ ������ homotopy
$F$�� $F(x,t)=t(fx)+(1-t)g(x)$ �����Ƿ� ���� $f(x_0)=g(x_0)$
�̶�� $F(x_0,t)$ �� t�� ���ϴ� ���� ��� $f(x_0)=g(x_0)$ ��
�����ȴ�.) $\{\alp ' \}\in\pi_1(|K^1|)$�̴�. �� $i_*(\{\alp
'\})=\{\alp\}$ �� �����ϴ� $\{\alp ' \}$�� �����Ƿ� onto ����
�����Ǿ���. ���� �ι�° ������ 1-1 ���� ���̱⸸ �ϸ� �ȴ�. 1-1
���� ���̱� ���� ������ ������ ���� note�� �̿�����.

{\bf Note.}\textit{ $\alp:S^1\rightarrow X$ represent a zero
element in $\pi_1(X)$ if and only if}

\textit{$\hspace{3em}\exists\,\,extension\,\,\overline{\alp}:D^2\rightarrow
X.$}

� $\{\alp\} \in \pi_1(|K^2|)$ �� $i_*$�� ���� identity ��
���ٸ�, $\alp$�� $|K|$�� loop�� ���� �� �� note �� ���� extension
$\overline{\alp}$ �� �����ϹǷ� ���� diagram �� �����Ѵ�.\\

$\,\,\,|K^2|\hspace{1em}\rightarrow\hspace{1em}
|K|\hspace{4em}|K^2|\hspace{1em}\rightarrow\hspace{1em} |K| $

$\alp\uparrow\hspace{6.5em}\Rightarrow
\hspace{2.5em}\alp\uparrow\hspace{4em}\uparrow
\exists\,\overline{\alp}$

$\,\,\,S^1\hspace{10.5em}S^1\hspace{1.5em}\subset\hspace{1.3em}D^2$\\

���⼭ $\overline{\alp}$�� simplicial approximation
$\overline{\alp}'$�� ������


$\,\,\,K^2\hspace{1em}\rightarrow\hspace{1em} K\hspace{4em}$

$\hspace{1em}\exists\,\overline{\alp'}\nwarrow$

$\,\,\,\hspace{4.5em}D^2$

�� �ǰ� �� �� $\{\overline{\alp}'|_{\partial D^2}\}=\{\alp\}$
�̴�. �ֳ��ϸ�, $\overline{\alp}'\simeq \overline{\alp}$ �̰�
$\overline{\alp}|_{\partial D^2}=\alp$ �̹Ƿ� $\overline{\alp
}'|_{\partial D^2}$�� $\alp$�� equivalent�ϴ�. ����
$\{\overline{\alp }'|_{\partial D^2}\}=\{\alp\}$ �̴�. �׷���
$\{\overline{\alp}'|_{\partial D^2}\}=0$ �̹Ƿ�  $\{\alp\}=0$
�̴�. ���� 1-1���� ������.









\end{proof}
\end{document}
