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\begin{document}
 \parindent=0cm
  \section*{Contractible space and Brouwer fixed point}


  \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 Remark}\textit{  contractible $\Rightarrow$ path connected.}\\

{\bf ��} 1. $\textbf{R}^n$ is contractible.

      $\hspace{2em}$  F(x)=tx  �� �ָ� �̴� id ��  {0} ���� homotopy �� �ش�.

  $\hspace{1em}$ 2. $D^n$ is contractible. ���� F(x)=tx �� �ָ� �ȴ�.

   $\hspace{1em}$ 3. Any space which is homeomorphic to $D^n$.

   $\hspace{1em}$ 4. A "tree" is contractible.(�׸�)

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

   {\bf ���� 6.} X$\cong$Y  and X is contractible $\Rightarrow$ Y
   is also contractible.

   $\hspace{3em}$Can you show that $S^1$ is not
   contractible.($\pi_1$�� ���� �ʰ� ����������

   $\hspace{3em}$ ���� �� �ֳ�?)\\

{\bf Remark.}

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

 X�� contractible to $x_0$
�̸�  path connected �̹Ƿ�  $\forall\,\,x_1 \in$ �� ����
$x_0,x_1$ ���̿�  path $\rho$�� �����Ѵ�. F �� $id_X$ �� $c_{x_0}$
���� homotopy�� ��  �� �Ʒ��� ���� ���ǵ� H�� $id_X$�� $c_{x_0}$
���̿� ���ϴ� homotopy�� �ش�.


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



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

($\Rightarrow$ ����) $\{x_0\} \hookrightarrow X \rightarrow
\{x_0\}$����

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

$c_{x_0} \circ i = id_{x_0}$ �̰� X �� contractible �̹Ƿ� $ i
\circ c_{x_0} \simeq id_{X} $�̴�. ���� $X \simeq \{x_0\}$ �̴�.

($\Leftarrow$ ����) $X \simeq \{x_0\}$ �̹Ƿ� homotopy
equivalence f: $X \rightarrow \{x_0\},g : \{x_0\} \rightarrow X$
�� �����Ѵ�. �� �� f�� constant map $c_{x_0}$ �� �ǰ� ���� $g
\circ f$ ���� constant map�� �ȴ�. �׷��� $g \circ f \simeq
id_{X}$ �̹Ƿ� X �� contractible�ϴ�.

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

{\bf Fact.} $\pi_1(S^1) \cong \textbf{Z}$.

 ���� $S^1$ �� contractible���� �ʴ�.
 $\textbf{R}^2\backslash\{0\}$ ���� ���������̴�.

 \begin{thm}
 \textit{(Brouwer fixed point theorem)\\
 Let f : $D^2 \rightarrow D^2$ be a map. Then f has a fixed point,
  i.e., $\exists x \in D^2$ such that f(x)=x.}
 \end{thm}
\begin{proof}
Suppose not. Then $x \neq f(x), \forall x \in D^2$.

Define a function $g : D^2 \rightarrow \partial D^2$ as follows :

Let $g(x)$ be the point of intersection of the half line from
$f(x)$ to $x$ with $\partial D^2$.

i.e., $g(x) = f(x) + t(x-f(x))$ where $t$ is the unique solution
of $\|f(x) + t(x-f(x)) \|=1$. Then $g$ is continuous and $g$ is
$id$ on $\partial D^2=S^1\,\,\,$ i.e.,

$\hspace{2em}i \hspace{3em}g$

 $\,\,\,S^1 \hookrightarrow D^2 \rightarrow S^1$   and   $g \circ i =
 id$ �̹Ƿ� �����ϴ� fundamental group���� ������ ����,\\

 $\hspace{5em}i_{\sharp}\hspace{7em}g_{\sharp}$

 $\pi_1(S^1,1)\hspace{1em}\rightarrow\hspace{1em} \pi_1(D^2,1)\hspace{1em}\rightarrow\hspace{1em}\pi_1(S^1,1)$

$\hspace{2em} \textbf{Z}\hspace{6em}0 \hspace{7em}\textbf{Z}$\\

�� �ǰ� �̴� funtorial property�� ���� ����̴�. ��

$0 = g_{\sharp}\circ i_{\sharp}=(g \circ
i)_{\sharp}=id_{\sharp}=id : \textbf{Z} \rightarrow \textbf{Z}$
�̹Ƿ� �̴� ����� �ȴ�.\end{proof}


\end{document}
