\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}{정리} \newtheorem{cor}[thm]{따름정리} \newtheorem{lem}[thm]{보조정리} \newtheorem{prop}[thm]{명제} \newtheorem{cl}{Claim} {\theorembodyfont{\rm} \newtheorem{ex}{예} \newtheorem{que}{질문} \newtheorem{notation}{Notation}[section] \newtheorem{defn}{정의} \newtheorem{rem}{주} \newtheorem{note}{Note} } \renewcommand{\thenote}{} \renewcommand{\therem}{} \newenvironment{pf}{{\bf 증명}}{\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} 먼저 longitude는 $M=S^3-N(K)$에서 trivial cycle이 되도록 정의되었으므로 그림과 같이 정해 주면 된다. 이 때, $c=p\mu+ q\lam$라고 하면 solid torus로서 $M$의 meridian은 $\lam$이고 longitude는 $\mu$이므로, $M(c)=L(p,q)$임을 알 수 있다. \\ \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 %극좌표계 \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) }} %%%뚱뚱한 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} 그림에서 $\alp$는 세번 꼬여있으므로 $\alp$를 돌면서 $\mu$방향으로 세번 돌면 $M=S^3-N(K)$에서 trivial cycle이 된다. 따라서 longitude는 $\lam=\alp-3\mu$이다. 이 때 $c=p\mu+ q\lam$라 하고 $M=M(c)$의 homology를 구해보자. \\ Adjunction space의 MV-sequence에서 exact sequence \[ \xymatrix @=2em @*[c] { % \cdots \ar[r] & H_q(T^2) \ar[r] & H_q(D^2\times S^1)\bigoplus H_q(S^3-N(K)) \ar[r] & H_q(M) \ar[r] & \cdots } \] 를 얻는다. 이 때 $D^2\times S^1\simeq S^1$이고 $S^3-N(K)\simeq S^3-s^1$이다. 따라서 $H_q(M)=0, q\geq 4$이고, $H_3(M)=H_2(T^2)=\zb$이다. 또한, \[ \xymatrix @=1em @R=1ex @*[c] { % 0 \ar[r] & H_2(M) \ar[r] & H_1(T^2) \ar[r] & H_1(D^2\times S^1)\bigoplus H_1(S^3-N(K)) \ar[r] & H_1(M) \ar[r] & 0 \\ &&x\ar@{|->}[r] & (i_*x, j_*h_*x)&& \\ &&m\ar@{|->}[r] & (0, j_*(c))&& \\ &&l\ar@{|->}[r] & (1, ?)&&} \] 이다. 이 때 $\lam$의 정의에 의하여 $j_*(c)=p$이다. 따라서 $$ p\neq 0 \Rightarrow H_1(M)=\zb/p\zb, H_2(M)=0 $$ $$ p= 0 \Rightarrow H_1(M)=\zb, H_2(M)=\zb $$ 임을 알 수 있다. \\ 특별히 $p=\pm 1$의 경우에는 $S^3$와 homology type이 같다. 이런 경우에 $M(c)$를 homology sphere라고 부른다. 일반적으로 homology sphere는 sphere와 homology는 같지만 homeomorphic하지는 않다. (Fundamental group이 같지 않다.) \end{document}