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\begin{document}
\parindent=0cm

\section*{VI.2 Lefschetz Duality}

\textbf{Induced orientation on the boundary}\\

\textbf{1.} Let $M$ be an $R$-orientable manifold with boundary
$\bd M \neq\phi\\
x\in\bd M, W$ a coordinate neighborhood of $x\cong
\mathbb{R}^n_+=\{x\in\mathbb{R}^n\hspace{0.2em}|\hspace{0.2em}x_n\geq
0\}\\
\hspace*{3em}\supset V $  a coordinate neighborhood of $x\cong
D^n_+=\{x\in D^n\hspace{0.2em}|\hspace{0.2em}x_n\geq 0\}.$\\

Want to extent the orientation on $\overset{\circ}{V}$ to $\bd V$
locally first in
a compatible way to obtain a global extension :\\

orientation $\in \Gamma\co
\overset{\cong}{\underset{j_{\co}}{\longleftarrow}} H_n(M, M-\co)\\
\overset{Want\hspace{0.2em} "\bd"}{\longrightarrow} $ orientation
$\in \Gamma(\bd
V)\overset{\cong}{\underset{j_{\bd\co}}{\longleftarrow}}H_{n-1}(\bd
M, \bd M-\bd V)$:\\

Note that $j_{\co}$ is $\cong$ since
\[
\xymatrix @C=2em @*[c]{%
\textrm{orientation}\in \Gamma \co & H_n(M, M-\co)\ar[l]^{j_{\co}}
 &\\H_n(\mathbb{R}^n,\mathbb{R}^n-\co)
\ar[u]^{j :\cong}& H_n(\rb^n_+,\rb^n_+ -
\co)\ar[u]^{excision:\cong}\ar[l]^{excision :\cong} }\] and
similarly $ \Gamma(\bd V) \overset{\cong}{\underset{j_{\bd V}
}{\longleftarrow}} H_{n-1}(\bd M, \bd M-\bd V)$\\

Consider the long exact sequence of the triple $(M,M-\co , M-V)$
and note that $M-V$ is a strong deformation retract of $M$.
\[\scriptsize{
\xymatrix @M=1ex @C=2em @R=1em @*[c]{%
& & \Gamma\co \ar@{.>}[dddl]& &\\
0\ar@{=}[r] & H_n(M, M-V)\ar[r] &
H_n(M,M-\co)\ar[r]^{\bd:\cong}\ar[u]^{\cong} & H_{n-1}(M-\co,
M-\co-\bd V) \ar[r]& 0\\
& & & H_{n-1}(U,U-\bd V)\ar[u]^{excision:\cong}\ar[d]^{\bd W \textrm{is a s.d.r. of} U}\\
 & R\cong\Gamma(\bd V)&\ar[l]^{\cong} H_{n-1}(\bd M, \bd M-\bd V)&
H_{n-1}(\bd W, \bd W-\bd V)\ar[l]^{excision:\cong} }}
\]
Here $U=(W-\co)\cap "\{x\in \mathbb{R}^n | 0\leq x_n < 1\}."$\\
Note the construction of
$\bd_V : \Gamma\co \rightarrow \Gamma(\bd V)$ is compatible:\\
i.e., let $x\in U\subset V$, where $x\in \bd M$ and $U$ is a half
ball neighborhood of $x$ contained in $V$ , then
\[\scriptsize{
\xymatrix @M=1ex @C=0.7em @R=0.7em @*[c]{%
&&& &\Gamma\co\ar[ddllll]^{\bd_V:\cong}\ar[dddd]^{r}& &&
 &\ar[llll]\ar[dddd]\ar[ld]^{\bd:\cong}H_n(M,M-\co)\\
&& && & && H_{n-1}(M-\co, M-V) \ar[dddd]& \\
\Gamma(\bd V)\ar[dddd]^{r:retraction}&&& & & &H_{n-1}(\bd M, \bd
M-\bd V)
\ar[llllll]^{\cong}\ar[dddd]\ar[ru]^{excision:\cong}&  & \\
& & \textrm{want commutitivity } & &&  & &\\
&& \textrm{of left face}&&\Gamma\cu \ar[ddllll]^{\bd_U:\cong}& &&
& H_{n-1}(M, M-\cu)
\ar[llll]\ar[ld]^{\bd:\cong}\\
& && & &&&H_n(M-\cu, M-U) &\\
\Gamma(\bd U)&& & & &&H_{n-1}(\bd M, \bd M-\bd U)
\ar[ur]^{excision:\cong}\ar[llllll]& &} }\] $\Rightarrow $
commutes(�ٸ� ���� ��� commute�ϹǷ�)\\

$\therefore$ we have a well-defined map $\bd :\Gamma M\rightarrow
\Gamma(\bd M)\\
\therefore M:R$-orientable $\Rightarrow \bd M : R$-orientable.\\

{\bf 2. Double of $M$}

Consider the double of $M$, $DM=M_{1} \coprod \bd M \times [0,1]
\coprod M_{2}/\sim$, where $M_{1}=M_{2}=M$ and $x \in \bd M_{1}
\Rightarrow x \sim (x,0) \in \bd M \times [0,1], x \in \bd M_{2}
\Rightarrow x \sim (x,1) \in \bd M \times [0,1]$.

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{\bf Note} If $M$ is $R$-orientable, $DM$ is $R$-orientable.
(unique orientation compatible with that of $M$)

\begin{pf2}

$U$: "coordinate neighborhood across the boundary" can be obtained
as follow:

Given $x \in V \subset M$,

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[0,1]$} \rput(-0.5,4){$M_{1}$} \rput(14.5,4){$M_{2}$}



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$\Rightarrow \xymatrix @M=1ex @C=1em @R=1em @*[c]{%
H_{n}(M_{2},M_{2}-x)= H_{n}(V_{2},V_{2}-x)
\ar[r]^-{r_{*}}_-{\times
(-1)} \ar[d]_{\cong} & H_{n}(V^{'},V^{'}-x^{'}) \ar[d]_{\cong} \\
H_{n-1}(S^{n-1}) \ar[r]^{r_{*}}_{\textrm{deg}\,r=-1} &
H_{n-1}(S^{n-1})}$

$\Rightarrow \xymatrix @M=1ex @C=1em @R=1em @*[c]{%
H_{n}(M_{1},M_{1}-x) \ar[r]^{id.} & H_{n}(DM,DM-x) \ar[dr]^{id.}
& \\
&&H_{n}(DM,DM-U) = R \\
H_{n}(M_{2},M_{2}-x) \ar[r]_{ \times (-1)} & H_{n}(DM,DM-x^{'})
\ar[ur]_{id.} & }$

$\Rightarrow (M_{1},s)\,\, \textrm{and}\,\, (M_{2},-s)$ can be
spliced together to give an orientation of $DM$.

\end{pf2}


{\bf 3. (Fundamental orientation class for $\bd M$)}

Let $M(\bd M \neq \emptyset)$ be compact, $R$-orientable.
(i.e.,$\overset{\circ}{M}$ is $R$-orientable.)

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
H_{n}(M, M-\overset{\circ}{M}) \ar[r]^-{j}
\ar[d]_{\cong}^{\textrm{excision}}
\ar@{}[dr]|<<<{\displaystyle{\circlearrowleft}}
& \Gamma \overset{\circ}{M}  \\
H_{n}(DM,DM-\overset{\circ}{M}) \ar[ur]_-{j_{\overset{\circ}{M}}}
\ar@{}[dr]|{\displaystyle{\circlearrowleft}} &
\\
H_{n}(DM,DM-M) \ar[u]^{i_{*}}_{\cong} \ar[r]^-{j_{M}}_-{\cong} &
\Gamma M \ar[uu]_{r:\textrm{restriction}}}
\]

Here $i_{*}$ is an isomorphism since $DM-\overset{\circ}{M}$ and
$DM-M$ are homotopically equivalent to $M_{2}$, and $j_{M}$ is
isomorphism since $M$ is compact.

Note that $r$ is an isomorphism since an orientation on
$\overset{\circ}{M}$ can be extended uniquely on $DM$.({\bf by
2.})

Hence $j$ is an isomorphism.

$\therefore$ An orientation of $\overset{\circ}{M}, s \in \Gamma
\overset{\circ}{M}$ determines a unique class $\zeta \in
H_{n}(M,\bd M)$ via $j(\cong)$, so that $\zeta|_{x \in
\overset{\circ}{M}} = s(x) \in
H_{n}(\overset{\circ}{M},\overset{\circ}{M}-x)$.\\

Now consider

%Diagram �׸���%
\[
\xymatrix @M=1ex @C=1ex @R=3ex @*[c] { %
% ����
& H_{n}(M,M-\overset{\circ}{V}) \ar[rr]_{\cong}
\ar[dd]_<{\cong}^<{\textrm{exc}^{-1} \circ \bd} & &
\Gamma\overset{\circ}{V}
\ar[dd]_{\cong}^{\bd_{V}}\\
% ��� ��1
\zeta \in H_{n}(M,\bd M) \ar[ur] \ar[rr]_<<<<<{\cong}^<<<<<{j}
\ar[dd]^{\bd} &
 & \Gamma M  \ar[ur]_{r} \ar[dd]^{\bd} & \\
% ��� �� 2
&  H_{n-1}(\bd M, \bd M- \bd V) \ar[rr]^{\cong} & &
\Gamma(\bd V) \\
% �Ʒ���
\bd \zeta = \zeta_{\bd M} \in H_{n-1}(\bd M) \ar[ur]
\ar[rr]^{j}_{\cong} & & \Gamma(\bd M) \ar[ur]_{r}& }
\]

diagram���� ����� ���Ʒ����� commutativity�� �ڸ��ϰ�, �޸���
commutativity�� 1�� �������� ���� �����Ѵ�. ���� �ո鵵
commute�ϴ�.($r$�� �ۿ����� �� ��ġ�ϸ� section�� uniqueness�� ���� ��ġ�Ѵ�.)\\


{\bf 4. Lefschetz Duality}

Let $M$ be compact, $R$-orientable manifold with $\bd M \neq
\emptyset$. Consider

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & H^{q-1}(M) \ar[r] \ar[d]_{\zeta_{\cap}}
\ar@{}[dr]|{\Large{(1)}} & H^{q-1}(\bd M) \ar[r]^-{\del}
\ar[d]_{\cong ,\textrm{P.D} }^{\bd \zeta_{\cap}}
\ar@{}[dr]|{\Large{(2)}} & H^{q}(M,\bd M) \ar[r] \ar[d]^{\zeta
\cap} \ar@{}[dr]|{\Large{(3)}} & H^{q}(M) \ar[r]
\ar[d]^{\zeta_{\cap}} & \cdots \\
\cdots \ar[r] & H_{n-q+1}(M,\bd M) \ar[r]_-{\bd} & H_{n-q}(\bd M)
\ar[r] & H_{n-q}(M) \ar[r] & H_{n-q}(M,\bd M) \ar[r] & \cdots}
\]
,where $\zeta \in H_{n}(M,\bd M)$ is the fundamental orientation
class.


(1) is commutative up to sign $(-1)^{q-1}$ : Note that we have the
following on the chain level.

$\bd(\zeta \cap a) = (-1)^{q-1}(\bd \zeta \cap a - \zeta \cap \del
a). \qquad \qquad \qquad (*)$

(2) $\bd \zeta \cap a = \zeta \cap \del a  : \qquad (*)\,\,
\textrm{on the chain level}\, \Rightarrow !$.

(3) Clear.\\

Show $\zeta_{\cap} : H^{q}(M) \rightarrow H_{n-q}(M, \bd M)$ is an
isomorphism :

\begin{center}
$H^{q}(M) = \check{H}^{q}(M) \overset{\cong,
\textrm{A.D.}}{\rightarrow} H_{n-q}(DM,DM-M)
\overset{\textrm{excision, cf.3}}{\cong} H_{n-q}(M,\bd M)$
\end{center}

and note $\zeta \in H_{n}(M, \bd M) = \Gamma\overset{\circ}{M}$ is
a restriction of $\zeta_{M} \in H_{n}(DM,DM-M)$ and A.D. map is
essentially $\zeta_{M \cap}$.

Now 5-lemma $\Rightarrow \zeta_{\cap} : H^{q}(M,\bd M) \rightarrow
H_{n-q}(M)$ is also an isomorphism.\\

{\bf Application}\\
{\bf 5.} Let $M$ be a compact manifold with $\bd M$.

(1) $\chi(DM) = 2 \chi(M) - \chi(\bd M)$.

(2) $(M : R-\textrm{orientable} \Rightarrow) \chi(\bd
M)=\textrm{even}$.

\begin{pf2}

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Use MV-sequence : "$N_{1} \cap N_{2} \rightarrow N_{1} \bigoplus
N_{2} \rightarrow DM$".

$\Rightarrow \chi(N_{1}) + \chi(N_{2}) = \chi(N_{1} \cap N_{2}) +
\chi(DM) \Rightarrow (1)$


(2) $n$: even $\Rightarrow$ dim($\bd M$) = odd $\Rightarrow \chi
(\bd M) =0$.

$n$ : odd $\Rightarrow \chi(DM) = 0 \Rightarrow \chi(\bd M) = 2
\chi(M).$

\end{pf2}

{\bf Example} \,\,\, $\rb P^{2n} (\chi = 1),\,\, \mathbb{C}P^{2n}
(\chi = 2n+1)\,\,\, \textrm{and}\,\,\, \mathbb{H}P^{2n} (\chi
=2n+1)$\,\, can not be a boundary of some manifolds.\\


{\bf 6. (Signature of $M^{4k}$)}

Let $M^{4k}$ be $R$-orientable, closed (and connected).

Consider $I : H^{2k}(M) \bigotimes H^{2k}(M) \rightarrow R \\
\Rightarrow I$ is a symmetric bilinear pairing.


Let $R = \rb \Rightarrow I$ is a non-degenerated symmetric
bilinear form.

Signature of $M = \sigma(M) :=$ signature of $I$.\\


{\bf Linear Algebra}

Let $V^{n}$ be a vector space over $\rb$.

$ b : V \bigotimes V \rightarrow \rb$, a symmetric bilinear form.

Choose $e_{1} \in V$ such that $b(e_{1},e_{1}) \neq 0$. (If not,
$b \equiv 0$). May assume $b(e_{1},e_{1}) = \pm 1$.

Let $\eps^{1} : V \rightarrow \rb , \eps^{1}(x) = b(x,e_{1})$.
Then $\eps^{1}$ is onto. $\Rightarrow V_{1} =\ker \eps^{1} =
\left< e_{1} \right>^{\perp}$.

Apply the same argument to $V_{1}$ to get $e_{2}$ with
$b(e_{2},e_{2}) = \pm 1$.\\


%���
$\Rightarrow b = \begin{pmatrix} 1 &&&&&&&& \\ & \ddots &&&&&&& \\
&& 1&&&\textrm{{\huge 0}}&&& \\ &&& -1 &&&&& \\ &&&& \ddots &&&& \\ &&&&& -1 &&& \\
&&&\textrm{{\huge 0}}&&& 0 && \\ &&&&&&& \ddots & \\ &&&&&&&&0
\end{pmatrix} \qquad b_{ij} = b(e_{i},e_{j})$\\


sign $b$ := $\sharp$ of positive eigenvalues of $b - \sharp$ of
negative eigenvalues of $b = r-s$ and rk $b = r+s$.\\



{\bf Witt index}\,\, Let $b$ be a non-degenerated symmetric
bilinear form on $V$.

$\nu(b):=$ Witt index of $b$ = dim(maximal totally isotropic
subspace $U$ of $V$ i.e., $\forall x \in U \Rightarrow b(x,x) = 0
( b|_{U} \equiv 0 )$).

Then $\nu(b) = \frac{1}{2}(n-|\textrm{sign} b|) = \sharp$ of
(1,-1) pairs in $\begin{pmatrix} 1 &&&&& \\ & \ddots &&&\textrm{{\huge 0}}& \\
&&1&&& \\ &&& -1 && \\ & \textrm{{\huge 0}}&&& \ddots & \\ &&&&&-1
\end{pmatrix}$.

\begin{pf2}
Let $U$ be a maximal totally isotropic subspace and $\{x_{1},
\cdots, x_{\nu}\}$ be a basis for $U$.

Let $\alp_{1} : U \rightarrow \rb$ be the dual of $x_{1}$, i.e.,
$\alp_{1}(x_{1}) =1, \alp_{1}(x_{i}) = 0 , i \geq 2$ and extend
$\alp_{1} : V \rightarrow \rb$ trivially ($ V = U \bigoplus U^{'}
\,\,\textrm{and}\,\, \alp_{1}(U^{'}) = 0$)

$b$: non-degenerate $\Rightarrow \exists y_{1} \in V$ such that
$\alp_{1} = b( \,\, , y_{1})$ so that $b(x_{1}, y_{1}) =1,
\,\,b(x_{i},y_{1})=0, i \geq 2$ and may assume $b(y_{1},y_{1})=0$.

$(b(y_{1}+ax_{1},y_{1}+ax_{1}) = b(y_{1},y_{1}) + 2ab(x_{1},y_{1})
+a^{2}b(x_{1},x_{1}) = b(y_{1},y_{1})+2a \,\,\textrm{and let}\,\, a=-\frac{1}{2}b(y_{1},y_{1}))$.\\

Let $H_{1}= \textrm{span} \{x_{1},y_{1}\}$ and consider
$\theta_{1} : V \rightarrow \rb^{2}$ given by $\theta_{1}(v) =
(b(v,x_{1}),b(v,y_{1}))$. Then $\theta_{1}(x_{1}) = (0,1),
\,\,\theta_{1}(y_{1}) = (1,0)$ , so $\ker \theta_{1} =
H_{1}^{\perp}$ and $\{x_{2}, \cdots, x_{\nu}\} \subset \ker
\theta_{1}$.\\

Apply the same argument to $x_{2}$ in $H_{1}^{\perp}$, etc.,
finally to get $V = H_{1} \bigoplus \cdots \bigoplus H_{\nu}
\bigoplus W$(orthogonal direct sum) for some $W$.

Now $W$ does not have an isotropic vector by the maximality of
$U$.

$\Rightarrow b|_{W}$ is definite and $b|_{H_{i}} = \begin{pmatrix}
1 & 0 \\ 0 & -1 \end{pmatrix}$ with respect to
$\{\frac{x_{i}+y_{i}}{\sqrt{2}} ,\frac{x_{i}-y_{i}}{\sqrt{2}}\}$.
\end{pf2}

\begin{thm}
Let $M^{4k+1}$ be compact and $\rb$-orientable. Then

(1) dim $H_{2k}(\bd M)$ = 2 dim(im\,$j_{*}$) = 2dim($\ker\,
j_{*}$) = 2dim(im\, $j^{*}$), where $j_{*} : H_{2k}(\bd M)
\rightarrow H_{2k}(M)$.

(2) $\sigma(\bd M) = 0 $.

\end{thm}

\begin{pf2}

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
H_{2k+1}(M,\bd M) \ar[d]_{\bd}
\ar@{}[dr]|{\displaystyle{\circlearrowleft}} & H^{2k}(M)
\ar@{}[dr]|{\displaystyle{\circlearrowleft}}
\ar[l]^-{\textrm{L.D}}_-{\cong} \ar[r]^-{\cong} \ar[d]^{j^{*}} &
\htx(H_{2k}(M),\rb) \ar[d]^{\widetilde{j_{*}}} \\
H_{2k}(\bd M) \ar[d]^{j_{*}} & H^{2k}(\bd M)
\ar[l]^{\textrm{P.D}}_{\cong} \ar[r]^-{\cong} & \htx(H_{2k}(\bd
M),\rb) \\
H_{2k}(M) &&}
\]

dim $H_{2k}(\bd M)$ = dim $H^{2k}(\bd M)$

dim(im $j^{*}$) = dim (im $\bd$) =dim ($\ker\, j_{*}$) and

dim(im $j^{*}$) = dim(im $\widetilde{j}_{*}$) = dim(im $j_{*}$)

$\therefore$ dim $H_{2k}(\bd M)$ = dim(im $j_{*}$) + dim ($\ker
j_{*}$) = 2dim(im $j_{*}$) = 2dim(im $j^{*}$)


Now note $I \equiv 0$ on im $j^{*} \subset H^{2k}(\bd M)$ since

$I(j^{*}a,j^{*}b) = \left<\zeta_{\bd M}, j^{*}a \cup j^{*}b
\right> = \left< \zeta_{\bd M}, j^{*}(a \cup b)\right> =
\left<j_{*}\zeta_{\bd M}, a \cup b \right> = \left<j_{*}\bd\zeta,
a \cup b \right> = 0$

Hence $\nu(I) = \frac{1}{2}\textrm{dim}H^{2k} \Rightarrow
\sigma(\bd M) = \sigma(I) = 0$.

\end{pf2}

{\bf Cobordism theory Reference}

Milnor, "Characteristic classes"

Stong, "Cobordism theory"










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