
\documentclass[12pt ]{article}
\setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true
cm}

\usepackage{hangul}
\usepackage{amscd,amsmath}
\usepackage{amsfonts}
\usepackage{amssymb,theorem}
\usepackage{longtable}
\usepackage{floatflt}
\usepackage{texdraw, epic, pstcol ,pstricks ,pst-3d, pst-poly,pst-grad,pst-node, pst-text}
\usepackage{floatflt}
\usepackage[all]{xy}
\usepackage{graphicx}

\newcommand{\Aff}{\mbox{\it Aff}}
\newcommand{\aff}{\mbox{\it aff}}


\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\del}{\delta}
\newcommand{\gam}{\gamma}
\newcommand{\vep}{\varepsilon}
\newcommand{\eps}{\epsilon}
\newcommand{\lam}{\lambda}
\newcommand{\kap}{\kappa}
\newcommand{\sig}{\sigma}
\newcommand{\ome}{\omega}
\newcommand{\Gam}{\Gamma}
\newcommand{\Ome}{\Omega}
\newcommand{\vph}{\varphi}
\newcommand{\Sig}{\Sigma}
\newcommand{\Del}{\Delta}
\newcommand{\Lam}{\Lambda}
\newcommand{\rb}{\mathbb{R}}
\newcommand{\zb}{\mathbb{Z}}
\newcommand{\cb}{\mathbb{C}}
\newcommand{\hb}{\mathbb{H}}
\newcommand{\bd}{\partial}
\newcommand{\key}[1]{\emph{\textbf{#1}}}
\newcommand{\hh}{\widetilde{H}}
\newcommand{\hhp}{\widetilde{H}_p}
\newcommand{\hhq}{\widetilde{H}_q}
\newcommand{\hq}{H_q}
\newcommand{\sn}{S^n}
\newcommand{\snn}{S^{n+1}}
\newcommand{\eab}{\bar{e}_\alp}
\newcommand{\ebb}{\bar{e}_\bet}

\newtheorem{thm}{����}
\newtheorem{cor}[thm]{��������}
\newtheorem{lem}[thm]{��������}
\newtheorem{prop}[thm]{����}
\newtheorem{cl}{Claim}


{\theorembodyfont{\rm}
\newtheorem{ex}{��}
\newtheorem{que}{����}
\newtheorem{notation}{Notation}[section]
\newtheorem{defn}{����}
\newtheorem{rem}{��}
\newtheorem{note}{Note}
}
\renewcommand{\thenote}{}
\renewcommand{\therem}{}

\newenvironment{pf}{{\bf ����}}{\hfill\framebox[2mm]{}\\}

\begin{document}
\parindent=0cm
\psset{unit=2cm}
\section*{II.4 Examples}

{\bf 1. Projective spaces}\\

(i) Construction\\

$\mathbb{R}P^n := \mathbb{R}^{n+1}\backslash\{0\} / \sim ,
\hspace{1em} x \sim\lambda x , \hspace{0.5em} \lambda \in
\mathbb{R}_* = \mathbb{R}\backslash \{0\}$\\
$[x ]$ = equivalence class of $x$ = the line passing through $x$,\\
or equivalently $S^n/\sim , x\sim -x, \hspace{0.5em} \forall x\in
S^n\hspace{1em}$\footnote{ \[
\xymatrix @=2em @*[c] { %
S^n \ar[r]^{i} \ar[d]^{q} \ar@{.>}
[dr]& \mathbb{R}^{n+1}\backslash \{0\}\ar[d]^{p}&\\
S^n/\sim \ar@{.>}[r]&\mathbb{R}P^n&\textrm{continuous, 1-1, onto
and compact $\Rightarrow \cong$}}
\]}

Similarly , $\mathbb{C}P^n = \mathbb{C}^{n+1} \backslash \{0\}
/\sim ,\hspace{1em} x\sim \lambda x, \hspace{0.5em} \lambda \in
\mathbb{C}_* ,\\
\hspace*{4.5em}(\mathbb{H}P^n = \mathbb{H}^{n+1}\backslash \{0\}/
\sim ,
\hspace{1em} x\sim \lambda x, \hspace{0.5em} \lambda \in\mathbb{H}^*)$\\
or equivalently $S^{2n+1}/ \sim (S^{4n+3}/\sim) , \hspace{1em}
x\sim \lambda x, \hspace{0.5em}|\lambda | =1 , \hspace{0.5em}
\lambda \in \mathbb{C}.(\mathbb{H})$
\[\xymatrix @=1em {%
\textrm{Note.} & S^0\ar[r] &S^n\ar[d] , &,&S^1\ar[r]
&S^{2n+1}\ar[d], &,&S^3 \ar[r]&S^{4n+3}\ar[d], & : & \textrm{Hopf
fibration}\\
& & \mathbb{R}P^n && &\mathbb{C}P^n &&&\mathbb{H}P^n &} \]

Note. (1) \[\xymatrix @ = 1em @*[c]{%
\mathbb{R}^1 \backslash 0\ar[d]\ar[r]^{\subset} & \mathbb{R}^2
\backslash 0 \ar[d]\ar[r]^{\subset}& \mathbb{R}^3 \backslash
0\ar[d]\ar[r]^{\subset} & \cdots\ar[r]^{\subset} & \mathbb{R}^n
\backslash 0\ar[d]\ar[r]^{\subset} & \mathbb{R}^{n+1} \backslash
0\ar[d]\ar[r]^{\subset} &
\cdots\ar[r]^{\subset} & \mathbb{R}^{\infty} \backslash 0\ar[d] \\
\mathbb{R}P^0 \ar[r]^{\subset}& \mathbb{R}P^1 \ar[r]^{\subset} &
\mathbb{R}P^2\ar[r]^{\subset} & \cdots \ar[r]^{\subset}&
\mathbb{R}P^{n-1}\ar[r]^{\subset}&\mathbb{R}P^n \ar[r]^{\subset}&
\cdots \ar[r]^{\subset}& \mathbb{R}P^{\infty} }\] Also for
$\mathbb{C}$ and $\mathbb{H}$.\\

(2) $\mathbb{R}P^n$ can be attached as an adjunction space with
$(X,A) = (E^n_+ , S^{n-1} )$ and $Y= \mathbb{R}P^{n-1} $ and $f :
S^{n-1}\rightarrow \mathbb{R}P^{n-1}$, the antipodal
identification. (=Hopf map)\\

Same thing holds for $\mathbb{C}P^n $ and $\mathbb{H}P^n$ with the
Hopf map as an attaching map.\\ (Use $2n-$disk defined by
$\{z\in\mathbb{C}^{n+1} | |z| =1 , z_{n+1}\geq 0 \}) \\
\hspace*{2.3em} 4n-
\hspace{8.7em}\mathbb{H}^{n+1}$\\
Note that $\mathbb{R}P^n \backslash \mathbb{R}P^{n-1} = $ open $n$-cell.\\
$\hspace*{4.5em} \mathbb{C}P^n \backslash \mathbb{C}P^{n-1} = $
open
$2n$-cell.\\
$\hspace*{4.5em} \mathbb{H}P^n \backslash \mathbb{H}P^{n-1} = $
open
$4n$-cell.\\
$\therefore \mathbb{R}P^{n}$ is a CW-complex with one $k$-cell in
each $k = 1 ,2 \cdots n.\\ \hspace*{1em}\mathbb{C}P^{n}$ is a
CW-complex with one $2k$-cell in each $k = 1 ,2 \cdots n.\\
\hspace*{1em}\mathbb{H}P^{n}$ is a CW-complex with one $4k$-cell
in each $k = 1 ,2 \cdots n.$\\

(ii) Homology of $\mathbb{R}P^n$\\

Consider
\[\scriptsize{
\xymatrix @ = 0.2em @*[c]{%
0\ar@{=}[d] && \mathbb{Z} = <\sigma >\ar@{=}[d]
&& \mathbb{Z}\oplus \mathbb{Z}\ar@{=}[d] &&\mathbb{Z}\ar@{=}[d] && 0\ar@{=}[d]\\
H_n(S^{n-1}) \ar[rr]\ar[dddd]^{p_*}&&
H_n(S^n)\ar[rr]^{j_*}\ar[dddd]^{p_*}& & H_n(S^n ,
S^{n-1})\ar[rr]^{\bd _*}\ar[dddd]^>>>>{p_*}& &
H_{n-1}(S^{n-1})\ar[rr]\ar[dddd]^{p_*}& & H_{n-1}(S^n)\ar[dddd]^{p_*} \\
& & & & & & & &\\
& & & H_n(E^n_+ , S^{n-1})\ar[uur]^{i_*}\ar[rr]^>>>>>{\bd ^* :
\cong} \ar[ddr]^{p|_* : \cong}
&& H_{n-1}(S^{n-1})\ar[uur]\ar[ddr]^{f_*}& & \\
& & & \mathbb{Z}\ar@{=}[u]& &\mathbb{Z}\ar@{=}[u] & & &\\
H_n(P^{n-1})\ar[rr] && H_n(P^n)\ar[rr] && H_n(P^n ,
P^{n-1})\ar[rr] && H_{n-1}(P^{n-1})\ar[rr] && H_{n-1}(P^n)\\
0 \ar@{=}[u]&H_{n+1}(S^n, E^n_+)\ar[uuurr]^<<<<<<<<<<<<{\bd_*}&& &\mathbb{Z}\ar@{=}[u] &&\\
&H_{n+1}(E^n_-, S^{n-1})\ar@{=}[u]&&&&&&& \\
& 0\ar@{=}[u]&&& & & & &\\
& & & &\alp\ar[rr]^{\bd_*}\ar[dddd] & &\bd_*\alp & &\\
&&&&&&&&\\
& & & \epsilon\ar[uur]\ar[ddr]\ar[rr]^>>>>>{\cong}& &\ar[uur]^{\cong} & &&\\
&&&&&&&&\\
& & & &\bet & & & &\\
}}\] $\bd_* \alp$ = generator of $H_{n-1}(S^{n-1})\Rightarrow
\alp$
is primitive and $H_n(S^n , S^{n-1}) = <j_*\sigma,\alp>$\\

Now consider $\gamma = \alp + (-1)^{n+1}a_*\alp
,$ where $a:S^n\rightarrow S^n$ antipodal map\\
$\Rightarrow \bd_*(\alp + (-1)^{n+1}a_*\alp ) = \bd_* \alp +
(-1)^{n+1}\bd_*a_*\alp\\
\hspace*{15em}\parallel$(by functorial property of
$\bd_*) \\
\hspace*{15em}(-1)^{n+1}a_*\bd_*\alp= (-1)^{n+1}(-1)^n\bd_*\alp\\
\hspace*{10em} = 0\\
\Rightarrow \gamma = j_* (k\sigma)=kj_*(\sigma)$\\

Claim. $\gamma$ is primitive.\\
$\Rightarrow k = \pm 1$ and may assume $k=1$ by taking $-\sigma$
as
a generator of $H_n(S^n)$ if necessary.\\
$\Rightarrow \gamma= j_*(\sigma)$\\
\newpage
\begin{pf}(Proof of Claim)\\
\[\xymatrix @= 1em @*[c]{%
&&H_n(E^n_+ , S^{n-1}) \ar[dll]\ar[d]^{i_*:\cong\textrm{excision}}&&&&\epsilon\ar[dll]\ar[d]\\
H_n(S^n , S^{n-1})\ar[rr]^{j_*}&&H_n(S^n,
E^n_-)&&\alp\ar[rr]&&i_*\epsilon}\]

Note. $j_*(a_*\alp) = 0$\footnote{\[\xymatrix @ = 1em
@*[c]{%
& H_n(E^n_+ , S^{n-1})\ar[d]^{a_*}\ar[r] &H_n(S^n ,
S^{n-1})\ar[d]^{a_*} & & &\epsilon
\ar@{|->}[r]\ar[d]&\alp\ar[d]&\\
\textrm{l.e.s} :\ar[r]& H_n(E^n_-,S^{n-1}) \ar[r]& H_n(S^n ,
S^{n-1}) \ar[r]& H_n (S^n , E^n_- ) \ar[r]&
\cdots&a_*\epsilon\ar[r] &a_*\alp\ar[r]^{j_*} &0}\] }$\Rightarrow
j_*(\gamma) = j_*\alp = i_*\epsilon$ : primitive $\Rightarrow
\gamma :$ primitive
\end{pf}\\
Key Observation. $p\cdot a = p$
\[\scriptsize{\xymatrix @= 1em @*[c]{%
S^n \ar[d]^{p}& 0\ar[r] &H_n(S^n)\ar[r]\ar[d]^{p_*} & H_n(S^n,
S^{n-1})\ar[d]^{p_*} &&\sigma\ar@{|->}[d]\ar@{|->}[r]^{j_*}
&\gamma\ar@{|->}[d]
& \gamma=\alp+(-1)^{n+1}a_*\alp\\
P^n & 0\ar[r] &H_n(P^n)\ar[r] & H_n(P^n, P^{n-1})&& p_* \sigma
\ar@{|->}[r]&(1+(-1)^{n+1})p_* \alp &(p_*\alp = \bet)}}\]
Claim.\[\xymatrix @= 1em @*[c]{%
p_* : & H_n(S^n) \ar[r]& H_n(P^n) & \textrm{if $n$ is odd.}\\
&\mathbb{Z}\ar@{=}[u]\ar[r]^{\times 2}&\mathbb{Z}\ar@{=}[u]&\\
&H_n(P^n) =0 \textrm{\hspace{1em} and}& p_* =0& \textrm{if $n$ is
even.}}\]
\begin{pf} Use induction on $n$.\\
$n$= odd : (n-1 :even)
\[\xymatrix @=1em @*[c]{%
& \mathbb{Z}\ar@{=}[d] && &\mathbb{Z}\oplus\mathbb{Z}\ar@{=}[d]& && &\\
0 \ar[r]& H_n(S^n)\ar[rrr]\ar[ddd] &&& H_n(S^n, S^{n-1})\ar[ddd]\ar[rr]& & H_{n-1}(S^{n-1})\ar[ddd]\ar[r] & 0\ar[ddd]& \\
& &\sigma\ar@{|->}[r]\ar[d] &\gamma\ar@{|->}[d]& &&&\\
&& p_*\sigma\ar@{|->}[r] & 2\bet &&&&\\
0\ar[r] & H_n(P^n)\ar[rrr] & &&H_n(P^n , P^{n-1}) \ar[rr]&&
H_{n-1}(P^{n-1})\ar[r] &
H_{n-1}(P^n)\ar[r] &0\\
&\mathbb{Z}\ar@{=}[u] &&&\mathbb{Z}\ar@{=}[u]&&0\ar@{=}[u]&
0\ar@{=}[u]^{\textrm{induction hypothesis}}&}\]$\Rightarrow p_* :
\mathbb{Z}\overset{\times 2}{\rightarrow} \mathbb{Z}$\\

$n$ = even : (n-1:odd)
\[\xymatrix @=1em @*[c]{%
& \mathbb{Z}\ar@{=}[d] && \mathbb{Z}\oplus\mathbb{Z}\ar@{=}[d]& & &\mathbb{Z}\ar@{=}[d]& &\\
0 \ar[r]& H_n(S^n)\ar[rr]\ar[ddd]& & H_n(S^n,
S^{n-1})\ar[ddd]\ar[rrr]^{\bd_*}& && H_{n-1}(S^{n-1})
\ar[ddd]^{\times 2\textrm{(induction hypothesis)}}\ar[rr]& & 0\ar[ddd]& \\
& & &&\alp\ar@{|->}[r]\ar[d] &1\ar@{|->}[d]&& &&\\
& & && \bet\ar@{|->}[r]
&2&&&&\\
0\ar[r] & H_n(P^n)\ar[dr]\ar[rr]& &H_n(P^n , P^{n-1})
\ar[rrr]^{\times 2\textrm{(injective)}}&&& H_{n-1}(P^{n-1})\ar[rr]
&&
H_{n-1}(P^{n})\ar[r] &0\\
&0\ar@{=}[u]
&0\ar[ur]&\mathbb{Z}\ar@{=}[u]&&&\mathbb{Z}\ar@{=}[u]&&
\mathbb{Z}/2\ar@{=}[u]& }\]

Conclusion. $\left(\begin{array}{ccc} H_n(P^n) =
&\left(\begin{array}
{cc} \mathbb{Z} ,& n=odd\\
0 , & n=even\end{array}\right)&\\
H_{n-1}(P^n) = & \left(\begin{array}
{cc} 0 ,& n=odd\\
\mathbb{Z}/2 , & n=even\end{array}\right)&\\
H_q(P^n) \cong & H_q(P^{n-1})& if\hspace{0.3em} q\leq
n-2\end{array}\right)$
\end{pf}\\

{\bf Cellular chain complex for $P^n$}\\
(1) $\rb P^n$ \\
$\bd : H_{k+1}(P^{k+1},P^k) \to H_k(P^k, P^{k-1})$,
($k=1,2,\cdots$)
 is $\left\{
\begin{array}{cl}
  0, & \textrm{if }k \ \textrm{  is even}  \\
  \times 2,  & \textrm{if } k\  \textrm{  is odd}
\end{array}
\right. $.

\begin{pf}
\[
\xymatrix @=2em @*[c] { %
H_{k+1}(E^{k+1}_+,S^k) \ar[r]^<<<<<{\bd_*}_<<<<<{\cong}
\ar[d]^{p_*}_{\cong}
& H_k(S^k) \ar[d]^{p_*} & \\
H_{k+1}(P^{k+1},P^k) \ar[r]^<<<<<{\bd_*} \ar@/_3ex/[rr]_{\bd}&
H_k(P^k) \ar[r]^<<<<<{i_*} & H_{k}(P^k, P^{k-1})
}\]\\
Attaching space�� isomorphism\footnote{recall
$H_q(X,A)\cong\hq(X\cup_f Y,Y)$}���� ������ $p_*$��
isomorphism�̴�. ���� $k$�� even�̸� $H_k(P^k)=0$�̹Ƿ� $\bd$��
0-map�̴�. ���� $k$�� odd�̸� $p_*:H_k(S^k)\to H_k(P^k)$�� $\times
2$-map�̰� ���� diagram���� $i_*$�� isomorphism�̴�.
\[
\xymatrix @=1.5em @*[c] { %
0=H_{k}(P^{k-1}) \ar[r] & H_k(P^k) \ar[r]^<<<<{i_*} & H_{k}(P^k,
P^{k-1}) \ar[r] & H_{k-1}(P^{k-1})=0 & \\
}\] ����, $\bd$�� $\times 2$-map�̴�.
\end{pf}

$C_k=H_{k}(P^{k},P^{k-1})$�� �θ� $H_i(P^n)=H_i(C)$�̹Ƿ�, ��
��Ƿκ��� $H_i(P^n)$�� ���غ��� ������ ��´�.

\[ H_i(\rb P^{2n+1})=\left\{
\begin{array}{cl}
  \zb /2  & 0<i=\textrm{odd}<2n+1 \\
  \zb & i=0, 2n+1 \\
  0 & \textrm{otherwise}
\end{array}
\right.
\]
\[ H_i(\rb P^{2n})=\left\{
\begin{array}{cl}
  \zb /2  & 0<i=\textrm{odd}<2n \\
  \zb & i=0 \\
  0 & \textrm{otherwise}
\end{array}
\right.
\]

Ư�� $\rb P^\infty$�� ��쿡�� ������ ����.
\[ H_i(\rb P^{\infty})=\left\{
\begin{array}{cl}
  \zb /2  & i=\textrm{odd} \\
  \zb & i=0 \\
  0 & \textrm{otherwise}
\end{array}
\right.
\]


(2) $\cb P^n $ and $\hb P^n$ \\
Recall $\cb P^n $(or $\hb P^n$) is a CW-complex with one cell in
each dimension $2j$(or $4j$), $j=0,1,\cdots, n$.\\
So, $C_k(\cb P^n)=  \left\{
\begin{array}{cl}
  \zb & \textrm{if } k=2j  \\
 0 & \textrm{otherwise}
\end{array}
\right. $
and $C_k(\hb P^n)=\left\{
\begin{array}{cl}
\zb & \textrm{if } k=4j\\ 0 & \textrm{otherwise}\end{array}
\right. $\\

���� $H_k$�� ���غ��� ������ ��´�.

\[ H_k(\cb P^n)=\left\{
\begin{array}{cl}
  \zb & \textrm{if } k=2j, j=0,\cdots,n\\
  0 & \textrm{otherwise}
\end{array}
\right.
\]
\[ H_k(\hb P^n)=\left\{
\begin{array}{cl}
  \zb & \textrm{if } k=4j, j=0,\cdots,n\\
  0 & \textrm{otherwise}
\end{array}
\right.
\]
���� $H_k(\cb P^\infty)$�� $H_k(\hb P^\infty)$�� ��쵵 ����
���������� ���� �� �ִ�.\\

\newpage
{\bf 2. Lens space revisited}\\

{\bf Second description} of Lens space $ L(p,q), p\geq 1$($p,q$: relatively prime).\\

\begin{floatingfigure}[l]{5.2cm}
\begin{pspicture}(2.3,2)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
% S^2%
\rput(1,1){\psellipse(0,0)(1,.3)
\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-1,0)(1,0)(1,0.5)(-1,0.5)(-1,0)
\psellipse[linestyle=dashed](0,0)(1,0.3) \pscircle(0,0){1}
}%
\rput(2.1,1.8){$D^3$} \psline{->}(1.95,1.7)(1.7,1.5)

\psdot(1.3,.72)\psdot(2,1)\psdot(1.3,1.28)\psdot(.4,.765)\psdot(.4,1.235)

\rput(1.4,.58){$x_0$}\rput(1.7,.9){$e$}
\end{pspicture}
\end{floatingfigure}
Lens space�� ���� �׸��� ���� $D^3$���� $\bd D^3=S^2$�� ������
������ ���� identify�Ͽ� �������. $(z,t)\in E^2_+\subset
S^2\subset \bd D^3$�� ���Ͽ� $$(z,t)\sim (e^{\frac{2\pi iq}{p}
}z,-t)$$ ��, �Ϲݱ��� ������ $q/p$���� ȸ���Ͽ� ��Ī�Ǵ� ���ݱ���
����� identify�ϰ� ������ ������ $p$����Ͽ� ��� identify��
���̴�.\\

Ư���� $q=1$, $p=2$�� ��쿡�� $\rb
P^n$�� ������� �� �� �ִ�.\\

Lens space�� �̿� ���� �����ϸ� �̴� manifold�� �ȴ�. ������
�������� locally Euclidean�� �Ǵ� ball neighborhood�� ���� �� �ְ�
������ ������ ǥ���� �������� ���ʰ� �Ʒ��ʿ��� 2��, ��������
�������� $p$���� ball neighborhood�� ������ ����µ� ������
identificaion�� ���� �ٿ��ָ� locally Euclidean�� ���� �� �� �ִ�.\\

$L(p,q)$�� cell-structure :
\begin{tabular}{l}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
one 3-cell coming from $D^3$ \\
one 2-cell coming from $E^2_+\subset \bd D^3$ \\
one 1-cell coming from $e$ \\
one 0-cell coming from $x_0$ \\
\end{tabular}\\


���� $C_k=\zb$�̰� ������ sequence�� ��´�.
\[
\xymatrix @=1.5em @R=1em @*[c] { %
0 \ar[r] & C_3 \ar[r] \ar@{=}[d]& C_2 \ar[r]  \ar@{=}[d] & C_1\ar[r] \ar@{=}[d]& C_0 \ar[r] \ar@{=}[d]&0 \\
0 \ar[r] & \zb \ar[r]^0 & \zb \ar[r]^{\times p}& \zb \ar[r]^0& \zb
\ar[r]&0 }\]

�� �� boundary map�� ������ boundary�� ����ϸ� �ǹǷ� ���� ����
�־�����. ���⼭ $\bd_2=\times p$�̹Ƿ� $im \bd _3\subset
ker\bd_2=0$���� $\bd_3=0$�� �ȴ�. ���� homology�� ����ϸ�
$$ H_3=\zb,\ H_2=0,\ H_1=\zb/p,\ H_0=\zb$$
�� ��´�.\\

{\bf Third description}\\
$S^3=\{(z_1,z_2)~|~ |z_1|^2 + |z_2|^2=1\}\subset \cb^2$\\
free $\zb/p$-action on $S^3$: $e^{\frac{2\pi i}{p}}\cdot
(z_1,z_2)=(e^{\frac{2\pi i}{p} }z_1, e^{\frac{2\pi iq}{p}
}z_2)\Rightarrow $covering action\\
Define $$S^3/(\zb/p)=L(p,q)$$

�� ���Ǵ� ���� second description�� ��ġ�Ѵ�. $S^3$���� ���� $z_1$
�������δ� $1/p$���� ȸ���Ͽ� identify�����Ƿ� $z_1$��������
$1/p$��ŭ�� ���� lune ����� fundamental domain�� ������ �̰���
$D^3$�� ������ �� �ִ�. ���� $z_2$�������� $q/p$����,
$z_1$�������� $1/p$���� ȸ���Ͽ� identify�� ���̹Ƿ� $D^3$����
�����ϸ� ���� second
identification�� ��ġ���� �� �� �ִ�.\\

���� second description�� ���������� Lens space�� ���ǿ� ��ġ����
�˾ƺ���.

\begin{center}
\psset{unit=2.5cm}
\begin{pspicture}(3.8,1.5)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
% S^2%

\rput(1,.7){


\psellipse(0,0)(1,.2)
\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-1,0)(1,0)(1,0.5)(-1,0.5)(-1,0)
\psellipse[linestyle=dashed](0,0)(1,0.2)


\psellipse(0,0)(1,.7) \rput(0,-0.5){
\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-.6,0)(.6,0)(.6,-0.3)(-.6,-0.3)(-.6,0)}

\rput(0,.56){\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-.6,0)(.6,0)(.6,0.3)(-.6,0.3)(-.6,0)}


\rput(0,.56){ \psellipse(0,0)(.6,.05)}


\rput(0,-.56){ \psellipse(0,0)(.6,.05)
\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-.6,0)(.6,0)(.6,0.3)(-.6,0.3)(-.6,0)
\psellipse[linestyle=dashed](0,0)(.6,0.05) }




\psline[linestyle=dashed](0.6,0.56)(0.6,-.56)
\psline[linestyle=dashed](-0.6,0.56)(-0.6,-.56)

\psline(-.2,.53)(-.5,-.16)\psline(-.2,-.6)(-.5,-.16)
\psline[linestyle=dashed](-.2,.53)(-.2,-.6)

\psline(.2,.53)(.5,-.16)\psline(.2,-.6)(.5,-.16)
\psline[linestyle=dashed](.2,.53)(.2,-.6)


}%

\rput(2.25,.7){$\bigcup$}

\rput(3.2,.7){

\rput(0,.56){ \psellipse(0,0)(.6,.05)
\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-.6,0)(.6,0)(.6,0.3)(-.6,0.3)(-.6,0)
\psellipse[linestyle=dashed](0,0)(.6,0.05) }


\rput(0,-.56){ \psellipse(0,0)(.6,.05)
\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-.6,0)(.6,0)(.6,0.3)(-.6,0.3)(-.6,0)
\psellipse[linestyle=dashed](0,0)(.6,0.05) }


\psellipse(0,0)(1,.7)

\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](0.6,0.56)(0.6,-.56)(1.1,-.56)(1.1,.56)
\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-0.6,0.56)(-0.6,-.56)(-1.1,-.56)(-1.1,.56)



\psline(0.6,0.56)(0.6,-.56) \psline(-0.6,0.56)(-0.6,-.56)



}%
\end{pspicture}
\end{center}

Second description���� $D^3$�� ���� �׸��� ���� �� �κ���
���������� ������ ����. ������ �������� �׸��� ���� $p$����Ͽ�
�Ѹ鿡 �־��� identification�� ���� ���̸� solid torus�� �ȴ�.
���� �����ʿ����� ����� �Ʒ����� $q/p$���� ȸ���� ��
identify�Ͽ��� �ϹǷ� ���� solid torus�� �ȴ�. \\
���� �� ���� solid torus�� ������µ� ������ �Ѹ��� ���� ����
���� �и��� ���̹Ƿ� identify�Ͽ��� �Ѵ�. ���� ������ ������
�ϰ� ���� �� description�� ��ġ���� Ȯ���� �� �ִ�.\\

{\bf ���� 10.} ������ ����� �� solid torus�� �Ѹ鿡 �־���
identification�� $m$(����)�� $qm+pl$(������)�� ������Ŵ��
Ȯ���϶�.

\end{document}
