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\begin{document}
\parindent=0cm
\psset{unit=2cm}
\section*{IV.1 Descriptions of higher homotopy groups}
{\bf Notation}\\
$I^{n} = \underbrace{I \times \cdots \times I} $ = the unit
$n$-cube\\
\hspace*{4.0em}$n$ \hspace{2.0em} =$\{t=(t_1, \cdots t_n) \in
\rb^{n} | 0 \leq t_{i} \leq 1 , i=1,2, \cdots ,n\}$\\
$\pa I^{n} =$ boundary of $I^{n} =\{t=(t_1, \cdots t_n) \in
\rb^{n} | t_{i} = 0$ or $1$ for some $i\}$\\

{\bf 1st description }\\

$F_{n}\x := \{\alp : (I^{n},\pa I^{n}) \to \x\}$\\
$\alp , \bet \in F^{n}$�� ���Ͽ�, $\alp \thicksim \bet
\Leftrightarrow \alp \simeq \bet$ rel $\pa I^{n}$�̶�� �ϸ�,\\
\begin{center}
\framebox[1.1\width]{\Large $\pi_{n}\x := F_{n}\x / \thicksim$}\\
\end{center}

�� �ȴ�. ���� $\pi_{n}\x$ �� ���Ҵ� $[\alp]$(eqivalence class of
$\alp$)�̴�. \\

���� $\pi_{n}\x$�� group structure �� ����. \\
$\alp , \bet \in F_{n}\x$ �� ���ؼ� \\

\begin{displaymath}
\alp \ast \bet(t_1, \cdots, t_n) =\left\{\begin{array}{cl}
\alp(2t_{1}, t_2, \cdots ,t_{n}) & 0 \leq t_1 \leq \frac{1}{2}
\\ \bet(2t_{1}-1,t_2, \cdots ,t_{n}) & \frac{1}{2} \leq
t_1 \leq 1
\end{array}
 \right.
\end{displaymath}
 �̶�� �ϰ�, $[\alp][\bet] := [\alp \ast \bet]$ ��� �����ϸ�,
�̴� �� ���ǵȴ�.\\
��, $\alp \thicksim \alp' , \bet \thicksim \bet'$ �̸� $\alp \ast
\bet \thicksim \alp' \ast \bet'$�̴�.(check).\\
�׸��� $\pi_1$�� ���� �Ȱ��� ������ (�̶� $t_2, \cdots , t_n$ ��
����� ���)\\
$(\alp \ast \bet ) \ast \gam \thicksim \alp \ast (\bet \ast \gam),
\hspace{1.0em} \alp \ast x_0 \thicksim x_0 \ast \alp,
\hspace{1.0em} \bar{\alp} \ast \alp \thicksim \alp \ast
\bar{\alp}$ �� �����ϹǷ� $\pi_{n}\x$�� group�� �ȴ�.\\ (���⼭
$\bar{\alp}(t_1, \cdots, t_{n}) :=
\alp(1-t_1, t_2, \cdots , t_{n})$)\\

{\bf 2nd description}\\

Observe that $\pi_{1}\x = [(S^{1},1),\x]$\\
\hspace*{10.0em} = the set of homotopy classes of maps $f:(S^{1},1) \to \x$\\
Similarly,\\
\begin{center}
\framebox[1.1\width]{\Large $\pi_{n}\x = [(S^{n},e_{1}),\x]$}\\
\end{center}
�� ��, $e_{1}=(1,0, \cdots ,0) \in \rb^{n+1}$�̴�.\\

�� ��� $\alp , \bet :(S^{n},e_{1}) \to \x$�� ���ؼ� $\alp \ast
\bet$�� �Ʒ� �׸��� ���� ���ǵǴ� $(S^{n},e_{1}) \to \x$�� map ��
�����ȴ�.\\


\begin{figure}[htb]

\centerline{\includegraphics*[scale=0.4,clip=true]{pic1.eps}}

\end{figure}

{\bf 3rd description}\\

Use loop space $\Ome\x$ and view $\pi_{2}\x$ as $\pi_{1}(\Ome\x
,x_{0})$.\\
�� ��� $\pi_{1}(\Ome\x , x_{0})$�� �����ϱ� ���ؼ�
function space $\Ome\x$�� topology�� �ʿ��ϴ�.\\

{\bf Topology on a function space}\\

The most commonly used and useful topology for a function space is
the {\bf compact-open topology}.\\

���� $\yx := \{f | f : X \to Y, $ a continuous function\}���
����. �׸��� �־��� $K^{compact} \subset X$ �� $U^{open} \subset
Y$�� ���ؼ� $S(K,U) := \{f: X \to Y | f(K) \subset U\}$��� �ϸ� \\
$\frak{S} := \{S(K,U) | K^{compact} \subset X, U^{open} \subset
Y\}$�� subbasis �� ���� $\yx$�� topology�� {\bf compact-open
topology} ��� �Ѵ�.\\


{\bf ���� 11}\\
1. $Y$ is Hausdorff $\Rightarrow \yx$ is Hausdorff. \\
2. Let $(Y,d)$ be a bounded metric space. Then Show that $id :
\yx_{d} \to \yx$ is continuous.(���⼭ $\yx_{d}$�� metric
topology�� �־��� �����̰� $\yx$�� compact-open topology�� �־���
�����̴�.)\\
3. Let $X$ be a compact Hausdorff space and $Y$ be a bounded
metric space. Then $\yx_{d} = \yx$, i.e., topology of uniform
convergence is same as compact-open topology.(���� 2�� ����� ����
topology of uniform convergence �� compact-open topology����
finer�ϴٴ� ���� �ȴ�.)

\begin{thm}

Let $\varphi : Z \times X \to Y$ and let $\hat{\varphi} : Z \to
\yx$ be the induced map defined by $\hat{\varphi}(z)(x) = \varphi(z,x)$. Then\\
\hspace*{2.0em} 1) If $\varphi$ is continuous, then $\hat{\varphi}$ is continuous.\\
\hspace*{2.0em} 2) If $X$ is locally compact Hausdorff, then
$\varphi$ is continuous if and only if\\
\hspace*{4.0em}$\hat{\varphi}$ is continuous.

\end{thm}

\begin{proof}
1) $\hat{\varphi}$�� continuous �� ���� ���̴� ���� �־���
$\hat{\varphi}_{z} \in S(K,U)$�� ���ؼ� $\hat{\varphi}(V) \subset
S(K,U)$�� $z$�� neighborhood $V$ �� �������� ���̴� �Ͱ�
���� ���ε�,\\
$\varphi$ is continuous $\Rightarrow \exists V$, a neighborhood of
$z$ such that $\varphi(V \times K) \subset U$ (������ �ߴ� ��ó��
$K$�� compactness�� �̿��ؼ� ���Ѱ��� product neighborhood��
subcover�� �� $V$�� ����) $\Rightarrow \hat{\varphi}(V) \subset
S(K,U)$ �̹Ƿ� $\hat{\varphi}$�� continuous
�̴�.\\


\begin{figure}[htb]

\centerline{\includegraphics*[scale=0.4,clip=true]{pic2.eps}}

\end{figure}


2) �־��� $\varphi(z,x) \in U^{open} \subset Y$�� ���ؼ�
$\varphi(V_{z} \times W_{z}) \subset U$�� ������Ű�� $W_{z}$��
$V_{z}$�� ��������
������.\\
���� $X$�� locally compact Hausdorff �̹Ƿ�
$\hat{\varphi}_{z}(\bar{W_{z}}) \subset U$�� ������Ű�� $x$ ��
relatively compact neighborhood $W_{z}$�� �����ϴ� ���� �ȴ�. ��
$\hat{\varphi}_{z} \in S(\bar{W_{z}},U)$. ���� $\hat{\varphi}$��
continuous�̹Ƿ� $\hat{\varphi}(V_{z}) \in S(\bar{W_{z}},U)$��
������Ű�� $z$�� neighborhood $V_{z}$�� �����ϴ� ���� �˰� ����
$\varphi(V_{z}
\times W_{z}) \subset U$�̴�.\\
\end{proof}

\begin{cor}
$X$ is locally compact Hausdorff. Then $\yx \times X
\overset{ev}\longrightarrow Y$(evaluation) is continuous.\\
\end{cor}

\begin{proof}
$\hat{ev} : \yx \to \yx$ is an identity map and apply the above
theorem.
\end{proof}

{\bf exercise} $X$ is a locally compact Hausdorff space. Then
$Y^{X \times Z} \to (\yx)^{Z}$ is a homeomorphism.\\

���� �ٽ� $\pi_{n}\x$�� ���ư���.\\

Now $\Ome \x = \x^{(I,\pa I)} \subset X^{I}.$\\
Compare $\pi_{2}\x ( = F_{2}\x / \thicksim)$ and $\pi_1(\Ome\x ,
x_0) ( = \Ome(\Ome\x , x_{0}) / \thicksim)$. �׷��� \\
$\alp : I_1 \times I_2 \to X \overset{1-1}\leftrightarrow
\hat{\alp} :I_1 \to X^{I_{2}}$�� ���迡 ���ؼ� \\
\hspace*{10.0em} $\searrow \hspace{1.0em} \cup$\\
\hspace*{11.0em} $\Ome\x$\\

$\alp \in F_{2}\x \Leftrightarrow \hat{\alp} : (I,\pa I) \to
(\Ome\x , x_{0}) \Leftrightarrow \hat{\alp} \in \Ome(\Ome\x ,
x_{0})$ �� �����Ѵ�. ���� ���� ������ ���ؼ� $\hat{\alp}$��
continuous�̴�. ��\\

$\wedge : F_{2}\x \overset{1-1}\leftrightarrow \Ome(\Ome\x ,
x_{0})$�̴�.\\

���� $\alp , \bet \in F_{2}\x$�� ���ؼ� \\
\hspace*{1.0em}$\exists H : I^{2} \times I \to X$, homotopy
between $\alp$
and $\bet$ \hspace{2.0em} ($I^{2}=I_{1} \times I_{2}$)\\
$\overset{by thm}\Longleftrightarrow \hat{H} : I_{1} \times I \to
X^{I_{2}}$, homotopy between $\hat{\alp}$ and $\hat{\bet}$\\
�� �����ϹǷ� $\alp \thicksim \bet \Leftrightarrow
\hat{\alp} \thicksim \hat{\bet}$�� �ȴ�.\\

���� $\wedge : \pi_{2}\x \overset{1-1}\leftrightarrow
\pi_{1}(\Ome\x , x_{0})$�̴�.\\

����������\\
\begin{displaymath}
\hat{\alp} \ast \hat{\bet}(s)=\left\{\begin{array}{cl}
\hat{\alp}(2s) = \alp(2s,-)\\ \hat{\bet}(2s-1) = \bet(2s-1,-)
\end{array}
 \right.
\end{displaymath}

�� �ǹǷ� $\hat{\alp} \ast \hat{\bet}(s) = \alp \ast \bet(s,-) =
\widehat{\alp \ast \bet}(s)$�� �Ǿ� $\widehat{\alp \ast \bet} =
\hat{\alp}
\ast \hat{\bet}$�� �����Ѵ�.\\
\begin{center}
\framebox[1.1\width]{$\wedge$ is a canonical isomorphism between
$\pi_{2}\x$ and $\pi_{1}(\Ome\x,x_{0})$}\\
\end{center}

\newpage
In general, $\pi_{n}\x \overset{\wedge}= \pi_{n-1}(\Ome\x ,
x_{0})$\\
i.e. $\alp \in F_{n}\x \Leftrightarrow \alp : I^{n-1} \times I \to
X$ \\
\hspace*{4.0em} $\Leftrightarrow \hat{\alp} : I^{n-1} \to X^{I} \\
\hspace*{8.0em} \searrow \hspace{1.0em} \cup\\
\hspace*{9.0em} \Ome\x$\\

�̴� $\pi_2$���� ���� ������� $\wedge$�� canonical
isomorphism���� �� �� �ִ�.\\

���� $\pi_{3}(X) = \pi_{2}(\Ome(X)) =
\pi_{1}(\Ome(\Ome(X))) = \pi_{1}(\Ome^{2}X), \cdots$\\
\hspace*{3.0em} $\pi_{n}(X) = \pi_{1}(\Ome^{n-1}X)$ �� �����Ѵ�.\\
\end{document}
