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\begin{document}
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\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}
