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\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\del}{\delta}
\newcommand{\gam}{\gamma}
\newcommand{\vep}{\varepsilon}
\newcommand{\eps}{\epsilon}
\newcommand{\lam}{\lambda}
\newcommand{\kap}{\kappa}
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\newcommand{\ome}{\omega}
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\newcommand{\rb}{\mathbb{R}}
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\newcommand{\key}[1]{\emph{\textbf{#1}}}
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\newtheorem{thm}{Á¤¸®}
\newtheorem{cor}[thm]{µû¸§Á¤¸®}
\newtheorem{lem}[thm]{º¸Á¶Á¤¸®}
\newtheorem{prop}[thm]{¸íÁ¦}
\newtheorem{cl}{Claim}


{\theorembodyfont{\rm}
\newtheorem{ex}{¿¹}
\newtheorem{que}{Áú¹®}
\newtheorem{notation}{Notation}[section]
\newtheorem{defn}{Á¤ÀÇ}
\newtheorem{rem}{ÁÖ}
\newtheorem{note}{Note}
}
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\newenvironment{pf}{{\bf Áõ¸í}}{\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 spaceÀÇ isomorphism\footnote{recall
$H_q(X,A)\cong\hq(X\cup_f Y,Y)$}¿¡¼­ ¿ÞÂÊÀÇ $p_*$´Â
isomorphismÀÌ´Ù. ¸ÕÀú $k$°¡ evenÀ̸é $H_k(P^k)=0$À̹ǷΠ$\bd$´Â
0-mapÀÌ´Ù. ¶ÇÇÑ $k$°¡ oddÀ̸é $p_*:H_k(S^k)\to H_k(P^k)$´Â $\times
2$-mapÀÌ°í ´ÙÀ½ diagram¿¡¼­ $i_*$°¡ isomorphismÀÌ´Ù.
\[
\xymatrix @=1.5em @*[c] { %
0=H_{k}(P^{k-1}) \ar[r] & H_k(P^k) \ar[r]^<<<<{i_*} & H_{k}(P^k,
P^{k-1}) \ar[r] & H_{k-1}(P^{k-1})=0 & \\
}\] µû¶ó¼­, $\bd$´Â $\times 2$-mapÀÌ´Ù.
\end{pf}

$C_k=H_{k}(P^{k},P^{k-1})$·Î µÎ¸é $H_i(P^n)=H_i(C)$À̹ǷÎ, À§
»ç½Ç·ÎºÎÅÍ $H_i(P^n)$¸¦ ±¸Çغ¸¸é ´ÙÀ½À» ¾ò´Â´Ù.

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

ƯÈ÷ $\rb P^\infty$ÀÇ °æ¿ì¿¡´Â ´ÙÀ½°ú °°´Ù.
\[ H_i(\rb P^{\infty})=\left\{
\begin{array}{cl}
  \zb /2  & i=\textrm{odd} \\
  \zb & i=0 \\
  0 & \textrm{otherwise}
\end{array}
\right.
\]


(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. $\\

µû¶ó¼­ $H_k$¸¦ ±¸Çغ¸¸é ´ÙÀ½À» ¾ò´Â´Ù.

\[ H_k(\cb P^n)=\left\{
\begin{array}{cl}
  \zb & \textrm{if } k=2j, j=0,\cdots,n\\
  0 & \textrm{otherwise}
\end{array}
\right.
\]
\[ H_k(\hb P^n)=\left\{
\begin{array}{cl}
  \zb & \textrm{if } k=4j, j=0,\cdots,n\\
  0 & \textrm{otherwise}
\end{array}
\right.
\]
¶ÇÇÑ $H_k(\cb P^\infty)$¿Í $H_k(\hb P^\infty)$ÀÇ °æ¿ìµµ À§¿Í
¸¶Âù°¡Áö·Î ±¸ÇÒ ¼ö ÀÖ´Ù.\\

\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´Â ¿ÞÂÊ ±×¸²°ú °°Àº $D^3$¿¡¼­ $\bd D^3=S^2$ÀÇ Á¡µéÀ»
´ÙÀ½°ú °°ÀÌ identifyÇÏ¿© ¾ò¾îÁø´Ù. $(z,t)\in E^2_+\subset
S^2\subset \bd D^3$¿¡ ´ëÇÏ¿© $$(z,t)\sim (e^{\frac{2\pi iq}{p}
}z,-t)$$ Áï, ºÏ¹Ý±¸ÀÇ Á¡µéÀ» $q/p$¹ÙÄû ȸÀüÇÏ¿© ´ëĪµÇ´Â ³²¹Ý±¸ÀÇ
Á¡µé°ú identifyÇÏ°í ÀûµµÀÇ Á¡µéÀº $p$µîºÐÇÏ¿© ¸ðµÎ identifyÇÑ
°ÍÀÌ´Ù.\\

Ưº°È÷ $q=1$, $p=2$ÀÇ °æ¿ì¿¡´Â $\rb
P^n$ÀÌ ¾ò¾îÁüÀ» ¾Ë ¼ö ÀÖ´Ù.\\

Lens space¸¦ ÀÌ¿Í °°ÀÌ Á¤ÀÇÇϸé ÀÌ´Â manifold°¡ µÈ´Ù. ³»ºÎÀÇ
Á¡¿¡¼­´Â locally EuclideanÀÌ µÇ´Â ball neighborhood¸¦ ÀâÀ» ¼ö ÀÖ°í
Àûµµ¸¦ Á¦¿ÜÇÑ Ç¥¸éÀÇ Á¡¿¡¼­´Â À§ÂÊ°ú ¾Æ·¡ÂÊ¿¡¼­ 2°³, ÀûµµÀ§ÀÇ
Á¡¿¡¼­´Â $p$°³ÀÇ ball neighborhoodÀÇ Á¶°¢ÀÌ »ý±â´Âµ¥ °¢°¢ÀÇ
identificaion¿¡ ÀÇÇØ ºÙ¿©ÁÖ¸é locally EuclideanÀÌ µÊÀ» ¾Ë ¼ö ÀÖ´Ù.\\

$L(p,q)$ÀÇ 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}\\


µû¶ó¼­ $C_k=\zb$ÀÌ°í ´ÙÀ½ÀÇ sequence¸¦ ¾ò´Â´Ù.
\[
\xymatrix @=1.5em @R=1em @*[c] { %
0 \ar[r] & C_3 \ar[r] \ar@{=}[d]& C_2 \ar[r]  \ar@{=}[d] & C_1\ar[r] \ar@{=}[d]& C_0 \ar[r] \ar@{=}[d]&0 \\
0 \ar[r] & \zb \ar[r]^0 & \zb \ar[r]^{\times p}& \zb \ar[r]^0& \zb
\ar[r]&0 }\]

ÀÌ ¶§ boundary mapÀº ½ÇÁ¦·Î boundary¸¦ °è»êÇÏ¸é µÇ¹Ç·Î À§¿Í °°ÀÌ
ÁÖ¾îÁø´Ù. ¿©±â¼­ $\bd_2=\times p$À̹ǷΠ$im \bd _3\subset
ker\bd_2=0$¿¡¼­ $\bd_3=0$ÀÌ µÈ´Ù. µû¶ó¼­ homology¸¦ °è»êÇϸé
$$ H_3=\zb,\ H_2=0,\ H_1=\zb/p,\ H_0=\zb$$
¸¦ ¾ò´Â´Ù.\\

{\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 $$S^3/(\zb/p)=L(p,q)$$

ÀÌ Á¤ÀÇ´Â ¾ÕÀÇ second description°ú ÀÏÄ¡ÇÑ´Ù. $S^3$¿¡¼­ ¸ÕÀú $z_1$
¹æÇâÀ¸·Î´Â $1/p$¹ÙÄû ȸÀüÇÏ¿© identifyÇßÀ¸¹Ç·Î $z_1$¹æÇâÀ¸·Î
$1/p$¸¸Å­¸¸ º¸¸é lune ¸ð¾çÀÇ fundamental domainÀÌ ³ª¿À°í ÀÌ°ÍÀ»
$D^3$·Î »ý°¢ÇÒ ¼ö ÀÖ´Ù. ¶ÇÇÑ $z_2$¹æÇâÀ¸·Î $q/p$¹ÙÄû,
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Second description¿¡¼­ $D^3$¸¦ À§ÀÇ ±×¸²°ú °°ÀÌ µÎ ºÎºÐÀÇ
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identifyÇÏ¿©¾ß ÇϹǷΠ¿ª½Ã solid torus°¡ µÈ´Ù. \\
µû¶ó¼­ µÎ °³ÀÇ solid torus°¡ ¾ò¾îÁö´Âµ¥ °¢°¢ÀÇ °Ñ¸éÀº ¿ø·¡ °°Àº
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ÇÏ°í ³ª¸é µÎ descriptionÀÌ ÀÏÄ¡ÇÔÀ» È®ÀÎÇÒ ¼ö ÀÖ´Ù.\\

{\bf ¼÷Á¦ 10.} À§¿¡¼­ ¾ò¾îÁø µÎ solid torusÀÇ °Ñ¸é¿¡ ÁÖ¾îÁø
identificationÀº $m$(¿ÞÂÊ)À» $qm+pl$(¿À¸¥ÂÊ)¿¡ ´ëÀÀ½ÃÅ´À»
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