\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} \newcommand{\mo}{\mathcal{O}} \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*{III. Orientation of Manifolds} \textbf{III.1 Orientation}\\ \textbf{1.} $M : n$-manifold, $x \in M , U$: open neighborhood of $x$\\ Let $V$ be a coordinate chart s.t. $(V,U,x)\cong (\mathbb{R}^n,D^n,0)$.\\ $H_n(M,M-x)\overset{\cong}{\underset{i_* : excision}{\leftrightarrow}}H_n(V, V-x) \cong H_n(\mathbb{R}^n,\mathbb{R}^n - 0)\cong H_{n-1}(\mathbb{R}^n- 0)= \mathbb{Z}$\\ A choice of a generator in $H_n(M, M-x)\cong \mathbb{Z}$ is called an orientation at $x$. \[ \xymatrix @C=1em @*[c]{% H_n(M,M-x) & H_n(V, V-x) \ar[l]^{i_* : \cong}\ar[r]^{\cong} & H_n(\mathbb{R}^n, \mathbb{R}^n- 0 )\ar[r]^{\cong} & H_{n-1}(\mathbb{R}^n- 0)\ar[r]^>>>{\cong} &\mathbb{Z}\\ H_n(M,M-D) \ar[u]^{j_*} & H_n(V, V-U)\ar[u]^{i_*} \ar[l]^{i_* : \cong}\ar[r]^{\cong}& H_n(\mathbb{R}^n, \mathbb{R}^n- D ) \ar[u]\ar[r]^{\cong} & H_{n-1}(\mathbb{R}^n- D)\ar[u] \ar[r]^>>>{\cong}& \mathbb{Z} } \] $\rho ^U _x := j_* $: "restriction to $x$" and denote by $\rho ^U _x (\alp ) = \alp |_x$\\ Note. $j_* : \cong \\ \Rightarrow (1) $(uniqueness)$ \forall \alp ,\bet \in H_n(M , M-U) ,\hspace{0.5em} \alp |_x = \bet |_x \Rightarrow \alp =\bet \\ \hspace*{1.3em} (2) (\exists$ of continuation) $\forall\bet _x \in H_n(M, M-x) , \exists \bet\in H_n(M, M-U)\\ \hspace*{2em}$ s.t. $\bet |_x = \bet_x$\\ In general, for $A\subset B\subset C\subset M ,$\\ we have $(M,M-C)\hookrightarrow (M, M-B)\hookrightarrow (M,M-A).\\ \Rightarrow \rho_A^B\cdot \rho_B^C = \rho_A^C$ or $\alp_C|_B|_A = \alp_C|_A$ and restriction homomorphism is natural with respect to homeomorphism.\\ \textbf{2.} An orientation on $M$ is a "continuous" choice $\{\alp_x\}$ of a generator $\alp_x$ of $H_n(M, M-x)$ at each $x\in M$, i.e., $\forall x\in M , \exists U$, a ball neighborhood of $x$ and a generator $\alp\in H_n(M, M-U)$ s.t. $\rho_y^U(\alp) = \alp_y,\hspace{0.5em} \forall y\in U$.\\ $M$ is ($R$-) $orientable$ if $\exists$ an orientation on $M$.\\ (1) $M'\subset M,$ an open submanifold. $M$ orientable $\Rightarrow M' :$ orientable.\\ ($H_n(M',M'-x)\overset{\cong}{\underset{i_*}{\rightarrow}}H_n(M, M-x))\\ (2) \forall M$ is $\mathbb{Z}/2$-orientable. (A choice of generator is unique.)\\ We can make continuity clear by viewing an orientation as a section.\\ \textbf{Sheaf topology on $M_\mo = \{ \bet_x\in H_n(M,M-x)| x\in M\}$}\\ Basis for the topology : Given $\bet_U\in H_n(M,M-U),\hspace{0.5em} U^{open}\subset X,\\ \hspace*{11em}$ let $<\bet_U>=\{\bet_x \in M_\mo | \hspace{0.5em}\bet_U|_x = \rho_x^U(\bet) = \bet_x\}$\\ Check. \\ (1) $\forall \bet_x \in M_\mo ,\hspace{0.5em} \exists$ coordinate ball neighborhood $U$ and $\bet_U\in H_n(M, M-U)$ s.t. $\bet_U|_x = \bet_x$.\\ (2) $\bet_x \in <\bet_U>\cap <\bet_V> \\ \Rightarrow \exists W \subset U\cap V$ coordinate ball of $x$ and $\bet_W$ s.t. $\bet_W|_x =\bet_x$.\\ Show $<\bet_W>\subset <\bet_U>\cap <\bet_V>:\\ \bet_y\in <\bet_W>\Rightarrow \bet_W|_y = \bet_y \Rightarrow \bet_U|_W|_x = \bet_U|_x = \bet_x = \bet_W|_x \Rightarrow \bet_U|_W = \bet_W$\\ $\Rightarrow\bet_y = \bet_W|_y = \bet_U|_W|_y = \bet_U|_y \in <\bet_U>\hspace{2em}\square$\\ $M_\mo$ with this topology is called the orientation sheaf of $M$.\\ \textbf{3. Properties of $M_\mo$}\\ (1) $p : M_\mo\rightarrow M$ is a covering.($M_\mo$ is not connected in general.)\\ $\hspace*{3em}\bet_x \mapsto x$\\ \begin{pf} $p$ is continuous : $\forall \bet_x \in M_\mo $ and $V , $ a neighborhood of $x$, $\exists U$, a coordinate ball $\subset V$ s.t. $p(<\bet_U>) = U\subset V$.\\ $p$ is open : $p$ sends basic open sets $<\bet_U>$ to open sets $U$.\\ $\forall x\in M,$ choose a coordinate ball neighborhood $U$, then $\{<\bet_U>| \bet_U\in H_n(M, M-U)\}$ evenly covers $U : \\ $ disjoint: uniqueness로부터 $\alp_U|_x = \bet_U|_x\Rightarrow \alp_U =\bet_U \Rightarrow <\alp_U> = <\bet_U> \\ $ open :clear \end{pf}\\ (2) $|\hspace{0.5em}| = \nu :M_\mo \rightarrow \mathbb{Z}_{\geq 0}$ defined by $\bet_x = \nu (\bet_x )\cdot$a generator in $H_n(M, M-x)\cong \mathbb{Z}$ is continuous.\\ \begin{pf} $\forall \bet_x , \exists \bet_U (U:$ coordinate ball) s.t. $\bet_U|_x = \bet_x.\\ $Suppose $\bet_U = n\cdot \alp_U , \hspace{0.5em}\alp_U = $ a generator of $H_n(M, M-U), n\geq 0.$\\ Thm. $\Rightarrow \bet_y \in <\bet_U>\Rightarrow \bet_y = \bet_U|_y = n\cdot \alp_U|_y \\ \therefore \nu(\bet_y ) = n ,\forall y\in U$.\end{pf}\\ (3) A section $s$ of $p: M_\mo \rightarrow M$ on $A\subset M$ is continuous iff $s$ is locally constant, i.e., $\forall x\in A, \hspace{0.3em} \exists U$ and $\bet_U$ s.t. $s(x) = \bet_U|_x,\hspace{0.3em},\forall x\in A\cap U$.\\ \begin{pf} 숙제 10.\end{pf}\\ From now on, sections are always continuous.\\ (4) $s , s'$: sections on a connected $A\subset M$\\ $s(a) = s'(a)$ for some $a\in A \Rightarrow s \equiv s'$\\ \begin{pf} $\hspace{4em}M_\mo\\ \hspace*{4em}s \nearrow \hspace{1.2em}\downarrow$(Uniqueness of Lifting)\\ $\hspace*{3em} A_{cnt} \hspace{0.2em}\underset{i}{\hookrightarrow}\hspace{0.2em} M$\end{pf}\\ Note. $\bet_U|_{A\cap U}$ can be viewed as a section on $A\cap U$ and denote it by $\bet_{A\cap U}$.\\ \textbf{4.} We can rephrase the orientability of $M$ as follows:\\ $M$ is orientable if $\exists$ a global section $s : M\rightarrow M_\mo$ with $\nu(s(x)) = 1$ and $s$ is called an orientation.\\ More generally, $M$ is orientable along $A\subset M$ if $\exists$ a section $s :A\rightarrow M_\mo$ with $\nu(s(x)) = 1$.\\ (1) $M$ is orientable iff $\exists$ a nowhere vanishing global section $s$:\\ Note. $s ,s' \in \Gamma M =$ sections over $M\\ \Rightarrow s +s' \in \Gamma M\\ \hspace*{2em}ns\hspace{1em}$ (and $rs \in \Gamma M,\hspace{0.3em} r\in R)$\\ \begin{pf} May assume $M$ is connected.\\ Suppose $\nu(s(x)) = |s(x)| = n \neq 0$. Then $s(x) = n\alp_x$ for some generator $\alp_x$.\\ $M_\mo \hspace{0.2em}\overset{\nu}{\rightarrow} \hspace{0.2em} \mathbb{Z}_{\geq 0 }\\ s \uparrow\hspace{0.7em}\nearrow\nu\cdot s\hspace{1em} $is continuous and $M$ is connected. $\Rightarrow \nu(s(y)) = n,\forall y\in M.\\ M\\ \Rightarrow "\frac{1}{n} s"$ is a well-defined section and locally constant. (Use $\bet_U = s|_U$)\end{pf}\\ (2) $M_\mo - \nu^{-1} (0)(\cong M)$ is orientable :\\ \begin{pf} $x\in U{=\frac{1}{2} -ball} \subset V$ = coordinate unit ball. \[\scriptsize{ \xymatrix @C=0.3em @*[c]{% H_n(M_\mo , M_\mo - <\bet _U>) & H_n(<\bet_V> ,<\bet_V> -<\bet_U>)\ar[r]^{\cong}\ar[l]^{\cong} \ar[d]^{p_* :\cong}& H_n(<\bet_V>,<\bet_V> - \bet_x)\ar[r]^{\cong} \ar[d]^{p_* : \cong} & H_n(M_\mo, M_\mo-\bet_x)\\ H_n(M,M-U)&H_n(V, V-U)\ar[r]^{\cong}\ar[l]^{\cong} & H_n(V, V-x)\ar[r]^{\cong}& H_n(M, M-x)\\ "\bet_U " \ar[rrr]&\ar[d]&\ar[d] & "\bet_x "\neq 0\\ \bet_U\ar[rrr]&& & \bet_x } }\]$\Rightarrow$ locally constant $\Rightarrow (1)$로부터 clear. \end{pf}\\ (3) Let $M$ be connected and let $\bar{M} $ be a componenet of $M_\mo - \nu^{-1}(0).\\ \Rightarrow p : \bar{M} \rightarrow M$ is a covering. (at most two-fold)\\ $p$ is a 1-fold covering (i.e. homeomorphism ) iff $M$ is orientable.\\ (This follows from a general fact from Covering Space Theory.)\\ \begin{pf} $ ( \Rightarrow )\hspace{0.5em}$ Since $p$ is a homeomorphism and $\bar{M}$ is orientable, $M$ is orientable. In fact, $p^{-1}$ is a non-vanishing section on $M$.\\ $ (\Leftarrow )\hspace{0.5em} M$ : orientable $\Rightarrow \exists $ section $s$ with $\nu (s(x)) = 1.$ \\ Then for $ \bet _x \in \bar{M} , \bet_x = n_0 s(x).\\ \Rightarrow s' = n_0 s$ is a section and hence $p$ is a homeomorphism. Note that since $s'(M)$ is a connected set intersecting a component , $s'(M)\subset \bar{M}.$\end{pf}\\ Remark. The same argument shows that $M$ : orientable $\Rightarrow \bar{M} = n_0 s(M)$ and hence $M_\mo = \underset{n\in \mathbb{Z}}{\coprod} ns(M)$, i.e., $M_\mo \cong M\times \mathbb{Z}.$\\ \begin{cor} $p$ is a 2-fold covering iff $M$ is non-orientable.\\ $\bar{M}$ is an orientable double covering of non-orientable $M$.\end{cor} (4) $\pi_1 M $ does not have a subgroup of index 2. $\Rightarrow M$ is orientable. In particular, $\pi_1 M = 0 \Rightarrow M$ is orientable.\\ 5 . $M$ is $orientable\hspace{0.5em} along\hspace{0.5em} A\subset M$ if $\exists $ a section $s : A\rightarrow M_\mo$ with $\nu(s(x))=1$. Let $\Gamma A = \{ $ sections on $A \}$ : a group (or $R-$ module)\\ (1) $M$ : orientable along $A \Rightarrow$ \[\xymatrix @C=1em @*[l]{% p^{-1}(A) \ar[rr]^{\phi :\cong}\ar[dr]^{p}& & A\times \mathbb{Z}\ar[dl]^{p_1}\\ &A& } \] \begin{pf} $p^{-1}(A)\underset{s}{\overset{p}{\rightleftarrows}} A$ is a covering.\\ $\bet_x \in p^{-1}(A) \Rightarrow \bet_x = ns(x)$ and define $\phi (\bet_x ) = (x, n). $\\ $ \phi $ is 1-1 and onto. : clear\\ $\forall x\in A , 3(3) \Rightarrow \exists\hspace{0.5em} U$, a coordinate ball neighborhood and $\alp_{U}\in H_n (M, M-U ),$ \\ $\hspace*{7em}$ s.t. $\alp_{U} = s$ on $A\cap U.$\\ $\forall \bet_U$, if $\bet_U|_x = ns(x) = n\alp_U|_x$ for some $n$, then $\bet_U = n\alp_U =ns$ and \[ \xymatrix @C=1em @*[c]{% <\bet_U > \cap p^{-1}(A)\ar@{=}[r]& <\bet_U |_{A\cap U}> \ar@{=}[r]\ar[dr]^{p : \cong}& <\bet_{A\cap U}>\ar[r]^{\phi} &(A\cap U, n)\ar[dl]^{p_1: \cong}&\textrm{commute.} \\ & & A\cap U&&& } \]$\Rightarrow \phi$ is a local homeomorphism.\\ $\therefore \phi$ is a homeomorphism. \end{pf}\\ 따라서 다음 사실들이 성립한다.\\ (2) $M$ : orientable along $A$ and $A$ : connected $\Rightarrow \Gamma A \cong \mathbb{Z}($or $R).$\\ In general, $\Gamma A \cong \mathbb{Z}^k , \hspace{0.5em} k = $ the number of components of A.\\ (3) $M$ : orientable $\Rightarrow M$ is orientable along $\forall A\subset M$. \\In this case, $A^{connected} \Rightarrow \Gamma A \cong \mathbb{Z}$.\\ (4) $\bar{A}$ : a component of $p^{-1}(A) - \nu^{-1} (0) \Rightarrow p : \bar{A} \rightarrow A$ is 1 or 2-fold covering and orientable iff $p$ is homeomorphism. (same proof as 4(3))\\ $M$ : non-orientable along $A \Rightarrow \Gamma A = 0$.\\ \end{document}