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\begin{document}
\parindent=0cm
\section*{IV.3 Some basic properties of $\pi_n$.}
\begin{thm}
$\pi_n(X\times Y, (x_0,y_0)) \cong \pi_n\x \times\pi_n(Y,y_0)$
\end{thm}
\begin{proof} Exactly same as $\pi_1$-case.\end{proof}
\begin{thm}
Let $p : \tx \to \x $ be a covering space. Then $p_* : \pi_n\tx
\to \pnx$ is an isomorphism for $n\geq 2$. \end{thm}
\begin{proof}
(1) $p_*$ is one to one :\\
$p_*[\alp]=0$�̶�� $p\circ \alp \sim x_0$�̹Ƿ�, covering
homotopy property�� ���Ͽ� $\alp \sim \xt_0$�� �Ǿ�
$[\alp]=0$�̴�. (Lifting property section�� �������� 5 ����)
���� $p_*$�� one to one�̴�. \\
(2) $p_*$ is onto :\\
������ $\alp :(S^n, e_1)\to \x$�� ���Ͽ� $\pi_1(S^n)=0$�̹Ƿ�
($n\geq 2$), general lifting theorem�� ������ �����Ѵ�. ����,
$\alp$�� lifting $\widetilde{\alp}:(S^n, e_1)\to \tx$�� �����Ѵ�.
�̶�, $p_*[\widetilde{\alp}]=[\alp]$�̹Ƿ�, $p_*$�� onto�̴�.
\end{proof}\\

{\bf Examples.}\\
1. $\pi_n(S^1)=\pi_n(\rb)=0$.\\
$\rb$�� contranctible space�̹Ƿ� $\pi_n(\rb)=0$�̰�, $\rb$��
$S^1$�� universal covering space�̹Ƿ� ���� ������ ���Ͽ� �ڸ��ϴ�.\\

2. $\pi_n(T^2)=\pi_n(S^1\times S^1)=\pi_n(S^1)\times \pi_n(S^1)
=0$.\\

3. $\pi_n($Closed surface with genus $g\geq 1)=\pi_n(\triangle
)=0$ ($\triangle=$Open unit disk).\\
2.���� ���캻 Torus�� ���� ������ ���� ������ε� ������ ��
�ִ�. Torus�� 4������ ���ֺ��� ���� ���� identify�� ������ ��Ÿ��
�� �ִµ� �� identification���� ���� ����, ���� ���� �̵��� ����
realize�ǰ�, �� �簢���� �����̵��ϸ� �����ü�� tileó�� ���� ��
�ִ� ���� "tessellation"�� �ְԵǾ�, $\rb^2$�� �� �����̵�����
deck transformation group���� ������ Torus�� covering space��
�ȴ�.
\begin{center}
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����, $\pi_n(T^2)=\pi_n(\rb^2)=0$���� �ٽ� Ȯ���� �� �ִ�. \\
�Ϲ������� genus�� $g$�� surface�� $4g$�������� ��Ÿ�� �� �ִµ�,
$g>1$�� ��쿡�� �������� $\frac{2\pi}{4g}$�� �ƴϹǷ� ��
$4g$������ �����̵��Ͽ� ����� tessellation�� �� �� ����. ����,
������ ���� Poincar\'{e} disk����
�����Ѵ�. \\
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Poincar\'{e} disk�� ������ open disk�� ���� �׸��� ���� ����
�������� ������ ��ȣ(�Ǵ� �߽��� ����ϴ� ����)�� �������� �ǵ���
metric�� �� ���̴�. ��, 8����($g=2$)�� ��츦 ���� ��� ������
�׸��� ����
��� Poincar\'{e} disk������ 8������ �ȴ�. \\
�� ��, �ٱ��� 8�������� �������� $0^\circ$�̰�, ��������
�����Ҽ��� �������� ������ ������ ��������Ƿ� ��������
��鿡���� ���� $135^\circ$�� �����Ѵ�. ���� �׸��� ���� �߰���
�������� ũ�Ⱑ $45^\circ$�� �Ǵ� 8������ ���� �� �ִ�. �� 8������
"�����̵�"�ϸ� 8���� �������� ��� ������ ���̹Ƿ� Poincar\'{e}
disk�� tessellation�� �� �� �ִ�. \\ ���� ������� �ٸ� $4g$������
���ؼ��� ��� �����̵��� ���Ͽ� Poincar\'{e} disk�� tessellation��
�� �� �����Ƿ�, Poincar\'{e} disk�� genus $g$�� surface�� covering
space�� �ȴ�.
\\
�׷���, Poincar\'{e} disk ���� contractible space�̹Ƿ�
\begin{center}$\pi_n($Closed surface with genus $g\geq 1)=\pi_n(\triangle
)=0$\end{center}�� ��´�.\\
�̿� ���� $\pi_1(X)=\Pi$, $\pi_n(X)=0, n\geq 2$�� �Ǵ� space��
$K(\Pi,1)$-space��� �θ���. \\

\newpage
\begin{thm}
If $n\geq 2$, then $\pi_n\x$ is abelian.
\end{thm}
\begin{proof}
Define
\[
\alpha \circ \beta (t_1, t_2,\cdots,t_n)=
\begin{cases}\alpha(t_1, 2t_2,\cdots,t_n)&\text{if $ 0\leq
t_2 \leq \frac{1}{2}$} \\
\beta(t_1,2t_2-1,\cdots,t_n)&\text{if $\frac{1}{2}\leq t_2 \leq
1$}
\end{cases}
\]
��, $\alp*\bet$�� ù��° ������ ���Ͽ� ���� ���ε�
$\alp\circ\bet$�� �ι�° ������ ���Ͽ� ���� ���̴�. ���Ƿ� ����
������ �ٷ� �˼� �ִ�. (�׸����� �������� $I_1$, �������� $I_2$��
��Ÿ���� $I_3,\cdots,I_n$�� �����Ǿ� �ִ�.)

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����, $\pi_n\x$�� abelian�̴�.
\end{proof}\\

{\bf \large $\Omega$-version} \\
�� ������ loop space $\Omega$�� �̿��Ͽ� ������ ����.
\begin{defn} $(X, \mu, x_0)$ is an H-space if\\
\hspace*{2ex} (1) $\mu: X\times X \to X$ is continuous. Write
$\mu(x,y)=xy$.\\
\hspace*{2ex} (2)
%diagram �׸���
$%
\xymatrix @M=1ex @C=1.5cm @R=4ex @*[c] { %
X \ar[r]^{(id,x_0)} \ar[dr]_{id} & X\times X \ar[d]^\mu & X \ar[l]_{(x_0,id)} \ar[dl]^{id} \\
&X&} $ \raisebox{-.5cm}{\parbox{5cm}{commute up to homotopy
relative to $x_0$}}
\end{defn}

\bigskip
��, ���� (2)�� ������ ��ġ�̴�. \\
$L_{x_0}: X \to X$ defined by $L_{x_0}(x)=x_0x$ is homotopic to
$id$ (rel $x_0)$ and \\
$R_{x_0}: X \to X$ defined by $R_{x_0}(x)=xx_0$ is homotopic to
$id$ (rel $x_0)$.\\
H-space�� ���δ� Topological group, loop space $\Ome$���� �ִ�.

\begin{thm}
$\Ome$ is an H-space.
\end{thm}
\begin{proof}
(1) Define $\mu:\Ome\times\Ome\to\Ome$ by
$\mu(\alp,\bet)=\alp*\bet$ \\[3mm]
{\bf ���� 13.(1)}  Show
\parbox{5.5cm}{$(X,x)^{(I,1)} \times
(X,x)^{(I,0)}\to X^I\\
\hspace*{3em}(\alp, \ \bet)\hspace{1.5cm}\mapsto \alp\bet$} is
continuous.\\[3mm]
({\bf ���� 13.(2)}�� �Ʒ���) \\

(2) Show $L_{x_0} \simeq id$ (rel $x_0$) : (Similarly for
$R_{x_0}$)\\
$H: \Ome \times I \to \Ome$�� ������ ���� �����Ѵ�. \\
$H(\alp,s)=\alp_s$, $\alp_s(t):=\alp\circ F(s,t)$ where
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��, $F$�� �� $s$-level���� ����ģ �κ��� $0$���� ������ �κ���
linear�ϰ� $I$�� �����ִ� �Լ��̰�, ���� $\alp_s$�� $0\leq
t\leq\frac{1}{2}-\frac{1}{2}s$�϶� $x_0$��
�ӹ����ִٰ� $\alp$�� ���� curve�� �ȴ�. \\
�� ��, $H$�� $L_{x_0}$�� $id$ ������ homotopy���� ������.\\ ����
$H_0=L_{x_0}$,$H_1=id$���� �ڸ��ϴ�. \\
$G:\Ome\times I_s\times I_t\to X$�� $G(\alp, s, t)=\alp(F(s,t))$��
������ $\widehat{G}=H$�̹Ƿ�, $H$�� �������� ���̱� ���ϼ��� $G$��
�������� ���̸� ����ϴ�. �׷���, $G$��
\begin{center}
$G:\Ome\times I_s\times I_t\overset{id\times F}{\longrightarrow}
\Ome\times I \overset{evaluation}{\longrightarrow} X$\\
$\hspace*{5em}(\alp,s,t)\longmapsto (\alp,F(s,t))\longmapsto
\alp(F(s,t))$
\end{center}
�� ���� �����Լ����� �ռ��̹Ƿ� �����̴�. ���� $L_{x_0} \simeq
id$.
\end{proof}

\begin{thm}
Suppose $(X,e)$ is an H-space. Then $\pi_1(X,e)$ is abelian.
\end{thm}
\begin{proof}
{\bf ���� 13.(2)} (Hint. Almost same as the case of topological
group.)
\end{proof}\\

���� ���� �� �����κ��� ������ ���������� ��´�.
\begin{cor}
If $n\geq 2$, $\pi_n\x$ is abelian.
\end{cor}

\end{document}
