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\begin{document}
\parindent=0cm
\section*{VIII. 3 Hopf theorem and $\pi_n(S^n)$}
%1. Suspension
\begin{defn}
(1) Let $X$ be a space. The \key{suspension} of $X$, $\Sigma X$ is
the quotient space of $X\times[-1,1]$ obtained by identifying
$X\times
\{1\}$ to a point and $X\times \{-1\}$ to another point.\\
(2) Given a map $f : X\to Y$, the \key{suspension} of $f$, $\Sigma
f : \Sigma X \to \Sigma Y $ is defined by $\Sigma
f(x,t)=(f(x),t)$.
\end{defn}

\textbf{��.} $\Sigma \sn=\snn$.\\
\textbf{Note.} $f\simeq g : X\to Y \Rightarrow \Sigma f\simeq
\Sigma g :  \Sigma X\to \Sigma Y $.
\begin{thm}
$f: \sn \to \sn \Rightarrow deg f = deg(\Sigma f)$
\end{thm}
\begin{pf}
In general, let $f: X\to Y $ so that $\Sigma f : (\Sigma X, X_+,
X_-)\to (\Sigma Y, Y_+, Y_-)$.\\
From MV-sequence, we have a commutative diagram,
\[
\xymatrix @=2em @*[c] { %
\hh_{p+1}(\Sigma X) \ar[r]^{\bd_*} \ar[d]_{(\Sigma f)_*} & \hhp(X)\ar[d]_{f_*}\\
\hh_{p+1}(\Sigma Y) \ar[r]^{\bd_*} & \hhp(Y)}
\]\\[1mm]
�̶�, ${\bd_*}$�� isomorphism�̹Ƿ� $deg f = deg(\Sigma f)$���� ��
�� �ִ�.
\end{pf}
\begin{thm} $f: \sn \to \sn(=\Sigma S^{n-1}))$, $n\geq 2$ $ \Rightarrow
\exists g : S^{n-1} \to S^{n-1}$ such that $f\simeq \Sigma g$.
\end{thm}
\begin{lem}
Let $f: \sn \to \sn$, $n\geq 2$. Then there exist $g: \sn\to \sn $
and $q\in \sn $ such that $f\simeq g $ and $g^{-1}(q)=\emptyset$
or one point.
\end{lem}
\begin{pf}
(1) may assume $\exists q$ such that $f^{-1}(q) $ is $\emptyset$
or finite set $\{p_0, \cdots ,p_k\}$ by simplicial approximation
theorem.\\
$f$�� simplicial approximation�� �����Ƿ�, $f$�� simplicial
map���� �����ص� �ȴ�. ���� $f$�� simplex�� simplex��
�����ֹǷ�, $\sn$�� ������ triangulation�� �ְ� ��
triangle(simplex) ������ ���� �����ϸ� �� inverse image�� isolated
set�̴�. ����
������������ �� �� �ִ�. \\

(2) may assume$\{p_0, \cdots ,p_k\}\subset \overset{\circ}{E^n_+}$
by an isotopy of $\sn$ by "pulling pants".\\
���⼭ isotopy�� homotopy $F:X\times I \to X$�̸鼭 �� $t\in I$��
���Ͽ� $F_{t} : X\to X$, $F_t(x)=F(x,t)$�� homeomorphism�� map��
���Ѵ�.\\

(3) Let $L$ be the union of line segments from $p_0$ to $p_i$.
Then may assume $f(L)\neq \sn$. ($\because$ $f$�� simplicial
map���� ���������Ƿ�, image�� $\sn$ ��ü�� ������ �ʴ´�.)\\
Since $f(L)$ is compact, there exists $ U $, an
$\eps$-neighborhood of $L$ contained in $\enp$ such that
$f(\overline{U})\neq \sn$. So, may assume $f:(\sn,
\overline{U})\to (\sn,\enp) $ and $q$ is the north pole
by rotation and pulling pants.\\

(4) There exists $f_1\simeq f$ such that $f_1^{-1}(q)=L$. \\
Let $\mu: \sn \to [0,1]$ be defined by $\mu(x)=\frac{1}{\eps} min
\{ d(x,L),\eps\}$ and $$f_1(x):=
\frac{\mu(x)f(x)+(1-\mu(x))q}{|\mu(x)f(x)+(1-\mu(x))q|}$$

��, $f_1$�� $U$ �ٱ������� $f_1=f$, $L$�󿡼��� $f_1=q$�� map�̴�.
����
$f$�� $f_1$�� ��� ������ antipodal�� ���� �ʰ� ������ ������ ���� $f\simeq f_1$�̴�. \\

(5) There exists $h: (\sn,L)\to (\sn, p)$ such that $h: \sn
\setminus L \to \sn\setminus p$ is a homeomorphism.\\
Let $$h(x) = \frac{\mu(x)x+(1-\mu(x))p}{|\mu(x)x+(1-\mu(x))p|}.$$
Futhermore $h$ and identity are never antipodal, so $h\simeq
id$.\\

(6)\hspace{9em}
\xymatrix @=2em @R=.5em @*[c] { %
& (\sn, L ) \ar[r]^{f_1} \ar[dd]^h \ar[dl]_q & (\sn, q)\\
\sn /L \ar[dr]_{\overline{h}} & & \\
&(\sn,p) \ar@{.>}[uur]_{\exists g}&&} \\[1mm]
(5)�� ���Ͽ� $\overline{h}$�� homeomorphism�̴�. ���� $h$��
quotient map���� ������ �� �ְ� $g^{-1}(q)=p$�� �Ǵ� $g$��
�����Ѵ�. ���� $$f\simeq f_1=g\circ h \simeq g$$�� �����ϹǷ�
������ ������.
\end{pf}

{\bf ���� 2�� ����} $n$�� north pole, $s$�� south pole�̶�� ����.
���� ���������� ���Ͽ� $f^{-1}(n)=\emptyset$ or $\{n\}$�̶��
�����ص� �ȴ�.\\
���� $f^{-1}(n)=\emptyset$�̸� $f\simeq c\simeq \Sigma c$ ($c$:
constant map)�̹Ƿ�, $f^{-1}(n)=\{n\}$�� ��츸 �����ϸ� �ȴ�. \\
$s$�� ball neighborhood $B_s$�� �����ϸ�, $f(D)\subset B_s^c$��
$n$�� neighborhood $D$�� �����Ѵ�. �̶�, $f(D^c)$�� compact
set�̰� $n$�� �������� �����Ƿ�, $f(D^c)\subset B_n^c$�� $n$��
neighborhood $B_n$�� �����Ѵ�.\\
$\sn$�� isotopy�� �̿��Ͽ� $D=\enp$�� �����ص� �����ϴ�. ����,
$\bd D=S^{n-1}$�̰�, $f(S^{n-1})\subset (B_n\cup B_s)^c$��
�����Ѵ�.\\
$\tau$�� "squeezing deformation" $\tau : (\sn,\bd B_n,\bd
B_s)\to(\sn,S^{n-1},S^{n-1})$�� ����. $\tau((B_n\cup
B_s)^c)\subset S^{n-1}$�̹Ƿ�,
$$ g:= \tau\circ f|_{S^{n-1}} :S^{n-1}\to S^{n-1}$$
�� �� ���ǵȴ�. ����
$$\tau\circ f(E^n_+)\subset E^n_+, \ \ \tau\circ f(E^n_-)\subset
E^n_-$$ �� �����ϹǷ�, $\Sigma g$�� $\tau\circ f$�� ��� ������
antipodal�� ���� �ʰ� ���� $\Sigma g\simeq \tau\circ f \simeq f$
���� �� �� �ִ�.\hfill\framebox[2mm]{}

\begin{thm} $deg : \pi_1(S^1)\to \zb$ is an isomorphism.\end{thm}
\begin{pf}
$\alp_n: S^1 \to S^1$�� $\alp_n(z)=z^n$���� �����ϸ�
$deg(\alp_n)=n$�̰�, $\pi_1(S^1)=\{[\alp_n]\}$���� �� �� �ִ�.
$\pi_1(S^1)$���� $[\alp_n] +[\alp_m]=[\alp_{n+m}]$�� �����ϹǷ�
$deg$�� isomorphism���� Ȯ���� �� �ִ�.
\end{pf}
\begin{thm}[Hopf]\hfill\\
$n\geq 1, f\simeq g : \sn \to \sn \Leftrightarrow deg f =deg g$
\end{thm}
\begin{pf}
Induction on $n$.\\
$n=1$: may assume $f,g : (S^1,1)\to (S^1,1)$ by rotations so that
$[f],[g]\in \pi_1(S^1)$. Then $deg f=deg g\Rightarrow [f]=[g]
\Rightarrow f\simeq g$.\\
$n>1$: $f,g : \sn\to\sn$$\Rightarrow \exists f_1,g_1: S^{n-1}\to
S^{n-1}$ such that $f\simeq \Sigma f_1$ and $g\simeq \Sigma g_1$.\\
Then $ deg f_1 =deg(\Sigma f_1)=deg f = deg g =deg(\Sigma g_1)=deg
g_1$. By induction, $f_1\simeq g_1$. So, $f\simeq g$. \end{pf}

\begin{thm} $deg : \pi_n(S^n)\to \zb$ is an isomorphism.\end{thm}
\begin{pf} Induction on $n$.
\[
\xymatrix @=1.5em @R=2em @*[c] { %
\pi_{n-1}(S^{n-1}) \ar[rr]^{\Sigma_*} \ar[dr]_{deg_{n-1}} & & \pi_n(\sn) \ar[dl]^{deg_n}\\
& \zb&}
\]
���� 1�� ���Ͽ� �� diagram�� commute�ϰ�, ���� 2�� ���Ͽ�
$\Sigma_*$�� onto�̴�. Induction�� ���Ͽ� $deg_{n-1}$��
isomorphism�̹Ƿ� $\Sigma_*$�� injection�̰� ���� $\Sigma_*$,
$deg_n$�� isomorphism�� �ȴ�.
\end{pf}

{\bf Remark.} (Freudental suspension theorem)\\
$\Sigma_*: \pi_k(\sn) \to \pi_{k+1}(\snn)$ is an isomorphism for
$k<2n-1$ and onto for $k=2n-1$.\\

\underline{General version of Hopf's theorem} \\
Let $M^n$ be a closed orientable $n$-manifold. Then $H_n(M^n)\cong
\zb$. So for a map $f: M^n\to \sn$, $deg f$ is well-defined.

\begin{thm}
Let $f: M^n \to \sn$. Then there exist $g: M^n \to \sn $ and $q\in
\sn $ such that $f\simeq g $ and $g^{-1}(q)=\emptyset$ or one
point.
\end{thm}
\begin{pf} Same as �������� 3. \end{pf}
\begin{thm}[Hopf]\hfill\\
$f\simeq g : M^n \to \sn \Leftrightarrow deg f =deg g$
\end{thm}
\begin{pf}
{\bf ���� 3} \end{pf}
\begin{cor}
$[M^n,S^n]\cong \zb$ \end{cor}

\end{document}
