\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[arrow,matrix]{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{\Sig}{\Sigma}
\newcommand{\Del}{\Delta}
\newcommand{\Lam}{\Lambda}
\newcommand{\disi}{\displaystyle{\sum_{i=1}^n}}
\newcommand{\disj}{\displaystyle{\sum_{j=1}^m}}
\newcommand{\rb}{\mathbb{R}}
\newcommand{\zb}{\mathbb{Z}}
\newcommand{\bd}{\partial}
\newcommand{\key}[1]{\emph{\textbf{#1}}}
\newcommand{\rh}[1]{\widetilde{H}(#1)}
\newcommand{\hh}{\widetilde{H}}
\newcommand{\hhp}{\widetilde{H}_p}
\newcommand{\ssx}[1][p]{\triangle^{#1}}
\newcommand{\iosh}{i_{0\sharp}}
\newcommand{\ish}{i_{1\sharp}}
\newcommand{\ios}{i_{0*}}
\newcommand{\is}{i_{1*}}
\newcommand{\sm}{S^n-s^r}

\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
\section*{I.4 Poincare Homology 3-sphere}

Dehn Surgery : \\
Let $M$ be a 3-manifold with $\bd M = T^2 .$\\
Given an isotopy class of simple closed curve $c$ on $\bd M$,\\
consider an adjunction space, $M(c) = D^2\times S^1
\underset{h}{\cup} M$ for $h : \bd(D^2 \times S^1 ) \rightarrow
\bd M$. If $K$ is a knot in $S^3$ and $M = S^3 - N(K)$ for $N(K)=
$ tubular neighborhood of $K$, then $M(c)$ is called a Dehn
surgery along $K$ with slope $c$. $N(K)$ has an obvious meridian
$\mu$, but the choice of longitude $\lambda$ is not clear and
defined as
follows: \\
\[
\xymatrix @M=1ex @C=2em @R=0.5em @*[c]{%
MV: 0 \ar[r]& H_1 (\bd N)\ar@{=}[d]\ar[r]^<<<<{\phi}
& H_1(N) \oplus H_1(M)\ar@{=}[d]\ar[r] & H_1(S^3 )\ar@{=}[d]\ar[r]&\cdots\\
& \mathbb{Z}\oplus\mathbb{Z} \ar[r]^{\cong}&
\mathbb{Z}\hspace{1em}\oplus\hspace{1em}\mathbb{Z}\ar[r] & 0
&\\
& \mu \ar[r]& (0,1)\ar[l] & &\\
& \lambda\ar[r] &(1,0)\ar[l]&& }
\]\\



{\bf Examples}

\psset{unit=2cm}
1. $K$= trivial not.\\

\begin{center}
\begin{pspicture}(2.2,1.3)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(1,.7){\psellipse(0,0)(1,.5) \psellipse(0,0)(.7,.35)
\rput(-.5,-.35){
 \psellipse[linestyle=dashed](0,0)(.08,.09)}
\psellipse[linestyle=dashed](0,0)(.85,.42)

\rput(1.1,.4){$\lam$} \rput(-.6,-.55){$\mu$}}

\psline{->}(2,1)(1.8,.85)
\end{pspicture}
\end{center}

���� longitude�� $M=S^3-N(K)$���� trivial cycle�� �ǵ���
���ǵǾ����Ƿ� �׸��� ���� ���� �ָ� �ȴ�. �� ��, $c=p\mu+
q\lam$��� �ϸ� solid torus�μ� $M$�� meridian�� $\lam$�̰�
longitude�� $\mu$�̹Ƿ�, $M(c)=L(p,q)$����
�� �� �ִ�. \\

\newpage

2. $K=$trefoil knot\\

\begin{center}
\begin{pspicture}(2.2,2.2)
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]

\SpecialCoor  %����ǥ��
\degrees[3]

%%KNOT%%
% \multido{\ni=0+1}{3} {
% \rput{\ni}(1,1){\pscurve(.35;0.35)(.6;0.5)(1;.75)(.55;1)(.15;1.98) }}

%%%�׶��� KNOT%%
\multido{\ni=0+1}{3} {
 \rput{\ni}(1,1){\pscurve[linewidth=.2](.35;0.35)(.6;0.5)(1;.75)(.55;1)(.15;1.98)
 \pscurve[linewidth=.15,linestyle=solid,linecolor=white](.35;0.35)(.6;0.5)(1;.75)(.55;1)(.15;1.98)
 \pscurve[linestyle=dashed](.35;0.35)(.6;0.5)(1;.75)(.55;1)(.15;1.98)}}

\rput(1,1){ \rput(-.5,-.55){
 \psellipse(0,0)(.05,.1)}}

\rput(.5,.2){$\mu$} \rput(1.6,1.8){$\alp$}

\psline{->}(1.55,1.75)(1.3,1.6)

\end{pspicture}
\end{center}

�׸����� $\alp$�� ���� ���������Ƿ� $\alp$�� ���鼭 $\mu$��������
���� ���� $M=S^3-N(K)$���� trivial cycle�� �ȴ�. ����
longitude�� $\lam=\alp-3\mu$�̴�. �� �� $c=p\mu+ q\lam$�� �ϰ�
$M=M(c)$��
homology�� ���غ���. \\
Adjunction space�� MV-sequence���� exact sequence
\[
\xymatrix @=2em @*[c] { %
\cdots \ar[r] & H_q(T^2) \ar[r] & H_q(D^2\times S^1)\bigoplus
H_q(S^3-N(K)) \ar[r] & H_q(M) \ar[r] & \cdots }
\]
�� ��´�. �� �� $D^2\times S^1\simeq S^1$�̰� $S^3-N(K)\simeq
S^3-s^1$�̴�. ���� $H_q(M)=0, q\geq 4$�̰�,
$H_3(M)=H_2(T^2)=\zb$�̴�. ����,
\[
\xymatrix @=1em @R=1ex @*[c] { %
0 \ar[r] & H_2(M) \ar[r] & H_1(T^2) \ar[r] & H_1(D^2\times
S^1)\bigoplus H_1(S^3-N(K)) \ar[r] & H_1(M) \ar[r] & 0 \\
&&x\ar@{|->}[r] & (i_*x, j_*h_*x)&& \\
&&m\ar@{|->}[r] & (0, j_*(c))&& \\
&&l\ar@{|->}[r] & (1, ?)&&}
\]
�̴�. �� �� $\lam$�� ���ǿ� ���Ͽ� $j_*(c)=p$�̴�. ����
$$ p\neq 0 \Rightarrow H_1(M)=\zb/p\zb, H_2(M)=0 $$
$$ p= 0 \Rightarrow H_1(M)=\zb, H_2(M)=\zb $$
���� �� �� �ִ�. \\
Ư���� $p=\pm 1$�� ��쿡�� $S^3$�� homology type�� ����. �̷�
��쿡 $M(c)$�� homology sphere��� �θ���. �Ϲ������� homology
sphere�� sphere�� homology�� ������ homeomorphic������ �ʴ�.
(Fundamental group�� ���� �ʴ�.)





\end{document}
