\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} \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} \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{proof}{{\bf 증명}}{\hfill\framebox[2mm]{}} \newenvironment{proof1}{{\bf 정리증명}}{\hfill\framebox[2mm]{}} \begin{document} \parindent=0cm \section*{Definition of Fundamental Group} A \textit{path} $\alp$ in a space $X$ is a continuous map $\alp : [0,1]\rightarrow X$. \begin{defn} Given two paths $\alp$ and $\bet$ in $X$ with the same end points, ie, $\alp (0)= \bet (0)$ and $\alp (1)= \bet (1)$ . $\alp$ is \textit{path homotopic} to $\bet$, denoted by $\alp \thicksim\bet$,\ if $\alp \overset{F}{\simeq}\bet$ rel $\partial I$ = \{0,1\} , i.e., $\exists$ $ F : I \times I \rightarrow X$ such that $\hspace{2em}$ 1. F(t,0) = $\alpha(t),\forall t \in I.$ $\hspace{2em}$ 2. F(t,1) = $\beta(t), \forall t \in I.$ $\hspace{2em}$ 3. F(0,s) = $\alp (0)$, F(1,s) = $\alp (1) , \forall s \in I.$ \end{defn} In general for $f$ and $g$ :$X\rightarrow Y$ with $f(a)$ = $g(a)$ for $a\in A\subset X$, $f$ is \textit{homotopic to } g (denoted by $f \simeq g$) relative to $A\subset X$, if $\exists$ a map $F : X \times I \rightarrow Y$ such that $\hspace{2em}$ 1. F(x,0) = $f(x) ,\forall x \in X.$ $\hspace{2em}$ 2. F(x,1) = $g(x), \forall x \in X.$ $\hspace{2em}$ 3. F(a,s) = $f(a) = g(a) , \forall a \in A.$\\ {\bf Note.} $\sim$ is an equivalence relation.\\ (reflexive) $\alpha \sim \alpha$\\ (symmetric) $\alpha \sim \beta \Rightarrow \beta \sim \alpha$ : $\alpha \sim \beta$를 주는 homotopy F에 대해 G(t,s)=F(t,1-s)로 주면 이는 $\beta \sim \alpha$ 를 만족하는 homotopy가 된다.\\ (transitive) $\alpha \sim \beta , \beta \sim \gamma \Rightarrow \alpha \sim \gamma$ : F: homotopy between $\alpha$ and $\beta$, G: homotopy between $\beta$ and $\gamma$ 라 하자. 이 때, H를 \[ H(t,s)= \begin{cases} F(t,2s) &\text{if $ 0\leq s \leq \frac{1}{2}$}\\ G(t,2s-1) &\text{if $ \frac{1}{2} \leq s \leq 1$} \end{cases} \] 로 두면 H 는 $\alpha$ 와 $\gamma$ 사이의 homotopy가 된다. \begin{picture}(270,80) \put(75,0){\makebox(0,0){$\alp$}} \put(50,10){\framebox(50,50){F}} \put(75,65){\makebox(0,0){$\bet$}} \put(135,0){\makebox(0,0){$\bet$}} \put(110,10){\framebox(50,50){G}} \put(135,65){\makebox(0,0){$\gamma$}} \put(225,0){\makebox(0,0){$\alp$}} \put(200,10){\framebox(50,25){F}} \put(200,35){\framebox(50,25){G}} \put(225,65){\makebox(0,0){$\gamma$}} \end{picture} \begin{defn} (i) 두 개의 path $\alpha , \beta$ with $\alp(1) =\bet(0)$에 대해서 \textit product path $\alpha*\beta$는 다음과 같이 정의된다. \[ \alpha * \beta (t)= \begin{cases}\alpha(2t)&\text{if $ 0\leq t \leq \frac{1}{2}$} \\ \beta(2t-1)&\text{if $\frac{1}{2}\leq t \leq 1$} \end{cases} \] (ii) $\Omega(X,x_0) := \{\alpha : I\rightarrow X \mid \alpha(0)=\alpha(1)=x_0\}$ : loop space of X based at $x_0$ \end{defn} \\ {\bf Introduce a group structure on $\Omega / \sim$:} (a) group operation은 $[\alpha ] \cdot [ \beta ] :=[\alpha*\beta ]$ \\ 이 곱이 $\Omega / \sim$ 에서 잘 정의가 된다는 것을 보이자. 즉 $\alpha \sim \alpha' ,\beta \sim \beta' \Rightarrow \alpha * \beta \sim \alpha * \beta'$임을 보이면 충분하다. F를 $\alpha, \alpha'$의 homotopy, G를 $\beta, \beta'$ 의 homotopy로 두면 \[ H(t,s)= \begin{cases}F(2t,s)& \text{if $0\leq t \leq \frac{1}{2}$}\\ G(2t-1,s)&\text{if $ \frac{1}{2}\leq t \leq 1 $} \end{cases} \] 가 $\alpha *\beta,\alpha' *\beta'$ 사이의 homotopy 를 준다.\\ 그러므로 $ [\alpha ]\cdot [\beta] = [\alpha*\beta]$는 $\Omega/\sim$에서 잘 정의되었다. \begin{picture}(270,80) \put(75,0){\makebox(0,0){$\alp$}} \put(50,10){\framebox(50,50){F}} \put(75,65){\makebox(0,0){$\alp'$}} \put(135,0){\makebox(0,0){$\bet$}} \put(110,10){\framebox(50,50){G}} \put(135,65){\makebox(0,0){$\bet'$}} \put(210,0){\makebox(0,0){$\alp$}} \put(200,10){\framebox(25,50){F}} \put(210,65){\makebox(0,0){$\alp'$}} \put(235,0){\makebox(0,0){$\bet$}} \put(225,10){\framebox(25,50){G}} \put(235,65){\makebox(0,0){$\bet'$}} \end{picture} \\ \\ (b) \textit{Associativity} $([\alpha ]\cdot [ \beta] \cdot [\gamma] = [\alpha] \cdot ([\beta ] \cdot [ \gamma])$ :\\ $(\alpha * \beta)*\gamma$ 와 $\alpha*(\beta * \gamma)$ 는 사실상 같은 path 의 reparametrization이므로 다음 Note만 보이면 된다. $\\${\bf Note.} In general, if $\beta(t)=\alpha(\phi(t))$ where $\phi : I \rightarrow I $ with $\phi(0)=0$ and $\phi(1)=1$, is a \textit{reparametrization}, then $\alpha \sim \beta.$\\ (증명) F(t,s) : = $\alpha(s \phi(t) + (1-s)t)$ 는 연속이고\\ $F(t,0)=\alpha(t)$, $F(t,1)=\alpha(\phi(t))=\beta(t)$ 사이에 원하는 homotopy를 준다.\\ \textit{(c) Existence of an identity e}:\\ Let I $\rightarrow \{x_0\} \subset X$ (a constant loop). 그러면$\alpha*e$ 는 $\alpha$의 reparametrization이므로 위의 Note에 따라 $\alpha*e \sim \alpha \sim e * \alpha.$\\ \textit{(d)Existence of an inverse:}\\ Given $\alpha \in \Omega$, define $\overline{\alpha}(t):= \alpha(1-t).$ Then we show that $\alpha*\overline{\alpha} \sim e \sim \overline{\alpha}*\alpha.$ 이것을 보이기 위해 $ F : I \times I \rightarrow X $ 를 다음과 같이 정의하자. \[ F(t,u)= \begin{cases}\alpha(2t)&\text{if $0\leq u\leq 1-2t$}\\ \overline{\alpha}(2t-1)&\text{if $u\leq 2t-1$}\\ \alpha(1-u)=\overline{\alpha}(u)&\text{if $u\geq |1-2t|$} \end{cases} \] 그러면 F는 연속이고 $\alpha * \overline{\alpha}$와 e 사이에 homotopy를 준다. 같은 방법으로 $\overline{\alpha}*\alpha \sim e$ 도 역시 보일 수 있다. \begin{picture}(270,80) \put(95,0){\makebox(0,0){$\alp$}} \put(70,10){\framebox(100,50){}} \put(120,65){\makebox(0,0){e}} \put(145,0){\makebox(0,0){$\overline{\alp}$}} \put(70,60){\line(1,-1){50}} \put(120,10){\line(1,1){50}} \put(65,35){\makebox(0,0){s}} \put(70,35){\line(1,0){100}} \put(95,35){\line(0,-1){25}} \put(145,35){\line(0,-1){25}} \end{picture} \begin{defn}\textit{(The fundamental group.)} $\pi_1(X,x_0) := \Omega(X,x_0)/\sim $을 X의 fundamental group (based at $x_0$)라고 부른다. \end{defn} \end{document}