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\begin{document}
\parindent=0cm
\psset{unit=2cm}
\section*{VI. Other duality}
\section*{VI.1 Alexander duality}
1.
\begin{thm}
Let $(X,A)$ be a compact pair. Then there exists a long exact
sequence
\[
\xymatrix @=2em { %
\cdots \ar[r] & \hcq(X-A)\ar[r]^-i
&\hq(X)\ar[r]^-j&\chq(X)\ar[r]^-{\del} &\hcqq(X-A)\ar[r]&\cdots.
 }\]
\end{thm}
���� map���� ��� ���ǵǾ����� ���캸��.\\

(1) $i$ : $U^\textrm{open}\subset X \Rightarrow \exists$a
canonical homomorphism $i: \hqc(U)\to\hq(X)$: \\
������ compact set $K\subset U$�� ���Ͽ� �Ʒ� diagram��
commute�ϹǷ� $i$�� induce�ȴ�.
\[
\xymatrix @=3em @M+1ex @R=2em { %
\hq(U,U-K)\ar[dr]^{\hspace{3em}\displaystyle{\circlearrowleft}}
\ar@{.*{\downarrow}}[d] & \hq(X,X-K)\ar[l]^\cong_{\textrm{excision}} \ar[d]^{i^*} \\
\hqc(U) \ar@{.>}[r]_-{\displaystyle{\exists ! "i"}}& \hq(X)
 }\]



(2) $j$:
\[
\xymatrix @=2em @M+1ex @*[c] { %
\hq(X)\ar[r]^{j^*, j=\textrm{inclusion}} \ar[dr]^{\hspace{3em}\displaystyle{\circlearrowleft}}_-{\displaystyle{\exists ! "j"}}& \hq(V)\ar@{.*{\downarrow}}[d]\\
& \dlim[V^\textrm{open}\supset A]\hq(V):=\chq(A)
 }\]

(3) $\del$: \\
$K=X-V$, $U=X-A$�� �θ� ���� diagram�� ��´�.
\[
\xymatrix @=2.5em @M+1ex @R=1.5em @*[c] { %
\hq(V)\ar[r]^-{\del} \ar@{.*{\downarrow}}[d]& \hqq(X,V) \ar[r]^-{\textrm{excision}}_-\cong& \hqq(U,U-K)\ar[d]\\
\chq(A) \ar@{.>}[rr]_-{\displaystyle{\exists ! "\del"}}&&\hcqq(U)
}\]

�� ��, $\del$�� natural�ϹǷ� $"\del"$�� induce�ȴ�.\\

\begin{pf}
\[
\xymatrix @=1.2em @M+.7ex @*[c] { %
&\hq(X,V)\ar[d]^\cong&&&\hqq(X,V)\ar[d]^\cong\\
\cdots\ar[r] & \hq(U,U-K)\ar[r] \ad & \hq(X)\ar[r] \ad
 & \hq(V)\ar[r] \ad & \hqq(U,U-K)\ar[r] \ad & \cdots\\
\cdots \ar[r] & \hcq(X-A)\ar[r]^-i
&\hq(X)\ar[r]^-j&\chq(X)\ar[r]^-{\del} &\hcqq(X-A)\ar[r]&\cdots
}\]

$\ndlim$�� exact functor�̹Ƿ� �� diagram�� ���� $(X,V)$��
cohomology long exact sequence�κ��� ���ϴ� exact sequence��
��´�.
\end{pf}\\


2. (Alexander duality) $X=M^n$: $R$-orientable compact manifold, $A^\textrm{cpt}\subset M$.\\
Consider \xymatrix @=2em  @C=2.5em @M+.7ex
{\hq(V)\ar[r]^-{\zeta_A\cap} \ad
\ar[dr] & \hnq(V,V-A)\ar[d]^{\textrm{excision}}_\cong \\
\chq(A) \ar[r]_-{{\exists ! D_A}} & \hnq(M,M-A)} \\
where $\zeta_A\in H_n(V,V-A)$, restriction of orientation class
$\zeta_A\in H_n(M,M-A)$.\\
Then
\[
\xymatrix @=2.5em { D_A: \chq(A) \ar[r]^-\cong & \hnq(M,M-A)
\hspace{2em} \forall q} \]

\begin{pf}
���� diagram�� 5-lemma�� �����ϸ� ������ ������.
\[
\xymatrix @=1.2em  @R=2.5em @*[c] { %
\cdots\ar[r] & \hcq(U)\ar[r]
\ar[d]^\cong_{D_U}\ar@{}[dr]|{\displaystyle{(1)}} & \hq(M)\ar[r]
\ar[d]^\cong_{D_M}\ar@{}[dr]|{\displaystyle{(2)}}
 & \chq(A)\ar[r]^\del \ar[d]^{??}_{D_A} \ar@{}[dr]|{\displaystyle{(3)}}& \hcqq(U)\ar[r] \ar[d]^\cong_{D_U} & \cdots\\
\cdots \ar[r] & \hnq(U)\ar[r] &\hnq(M)\ar[r]&\hnq(M,U)\ar[r]^\bd
&H_{n-q-1}(U)\ar[r]&\cdots }\]

$U$�� �ռ��� ���������� $M-A$�̰� $D_U$�� $D_M$�� Poincar\'{e} duality map���̴�. \\
���� $(1),(2),(3)$�� commute�Ѵٴ� �͸� ���̸� �ȴ�. (1)�� cap
product�� naturality�� ���Ͽ� commute�Ѵ�. \\
(2)�� ���� diagram����
\[
\xymatrix @=2em @M+.7ex @C=3em @*[c] { %
\hq(M)\ar[r] \ar[d]^\cong_{D_M}
\ar@{}[dr]|{\displaystyle{\circlearrowleft}}& \hq(V)
\ar[d]^{\zeta_A\cap} \ar@{.*{\rightarrow}}[r]
 & \chq(A) \ar[d]_{D_A} \\
\hnq(M)\ar[r]&\hnq(V,V-A)\ar[r]^\cong_{\textrm{excision}}&\hnq(M,U)
}\] cap product�� naturality�� ���Ͽ� ���� �簢���� commute�ϰ�
�̰��� direct limit�� (2)�̹Ƿ� commute�Ѵ�. \\

(3)�� up to sign���� commute�ϴµ� ���� diagram���� Ȯ���� ��
�ִ�.
\[
\xymatrix @=1.2em @M+.7ex @C=-1em @*[c] { %
\hspace*{10em}&&&\hqq(M,V) \ar[dr]^-{\textrm{excision}}_-\cong& \\
&&\hq(V) \ar[ur]^\del \ar[ddll]_{\zeta_A\cap} \ad & & \hqq(U,U-K) \ar@/^3em/[dd]^{\zeta_K\cap} \ad \\
&&\chq(A) \ar[d]_{D_A}\ar[rr]^\del & &\hcqq(U) \ar[d]_{D_U}& \\
\hnq(V,V-A)\ar[rr]_-{\cong} &&\hnq(M,U) \ar[rr]^\bd &&
H_{n-q-1}(U) }
\]
���⼭ $V=M-K$�̴�. chain level���� $\bd(\zeta\cap
a)=(-1)^q(\bd\zeta\cap a+\zeta\cap \del a)=(-1)^q(\zeta\cap \del
a)$�̹Ƿ� up to sign���� commute���� Ȯ���� �� �ִ�.
\end{pf}\\


3.(Taut embedding) When is $\chq(A)=\hq(A)$?\\

e.g. non-taut embedding
$$ A=\{(x,\sin\frac{1}{x}\}\cup \{(1,y)~|~ |y|\leq 1 \}$$
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(0.77,0.963209971858135) (0.78,0.958601978091349)
(0.79,0.95385489739527) (0.8,0.948984619355586)}
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$A$�� $\rb^2$�� closed subset�̰� ������ �����Ѵ�.
\begin{list}{}{
\setlength{\leftmargin}{1.5cm}}
\item
(1) number of path components of $A=2$\\
(2) number of connected components of $A=1$
\end{list}
(1)�� ���Ͽ� $H^0(A)=\zb^2$�̴�. �׷��� (2)���� ������ open set
$U\supset A$�� ���Ͽ� $A$�� $U$�� �� component�� ���������Ƿ�
$\check{H}^0(A)=\ndlim H^0(U)=\zb$�̴�. ���� $A$�� taut
embedding�� �ƴϴ�. \\

\newpage
��, $A=$
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�̶�� $\check{H}^1(A)=\zb$, $H^1(A)=0$���� Ȯ���� �� �ִ�.\\

Let $A^\textrm{closed}\subset X^\textrm{normal}$ and suppose $A$
is ANR. Then there exists $U^\textrm{open}$ such that \xymatrix{A
\ar[r]_i & U \ar@/_/[l]_r} with $r i=id$.\\

$\Longrightarrow$ \xymatrix @M+.7ex {\hq(A) \ar[r]_{r^*} & \hq(U)
\ar@/_/[l]_{i^*}\ad \\& \chq(A)\ar@{.>}[ul]^{\displaystyle{\exists
!\kappa}} }\hspace{3em} and $\kappa$ is onto. \\

Furthermore, let $X$ be a binormal ANR. (i.e. $X\times I $ is also
normal.)\\

$A\subset U^\textrm{open}\subset X^\textrm{normal} \Rightarrow
\exists U'$ open such that $A\subset U'\subset
\overline{U'}\subset U $.\\
Consider
\[
\xymatrix @=3em @M+.7ex @C=6em @*[c]{ \mbox{
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%\psgrid %
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%\pscurve[linestyle=dashed](0.7,0)(1,-.1)(1.3,0)
} \rput(1,0.2){$A$} \rput(1.7,0.05){$\overline{U'}$}
\end{pspicture}}
\ar@(u,u)@<2ex>[r]^-r \ar[d]^{\textrm{inclusion}}
\ar@<1.5ex>[r]^{F} \ar@(dr,d)[r]_-{\textrm{inclusion}}="a" &
\mbox{
\begin{pspicture}(2,0)
%\psgrid %
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} \rput(1.2,-.2){$\overline{U'}$}
\psarc(.2,0){.2}{150}{210}\rput(2.2,0){$U$}
\psarc(1.8,0){.2}{-30}{30}}
\end{pspicture}}\\
\mbox{
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%\psgrid %
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%\pscurve[linestyle=dashed](0,0)(1,-.25)(2,0)
%\pscurve[linestyle=dashed](0.7,0)(1,-.1)(1.3,0)
} \rput(1,0.1){$V$}\rput(1.7,0.1){$N$}
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,where $N$ is an open neighborhood of \mbox{\psset{unit=.5cm}
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$F$�� ���� �׸��� ���� $C$���� $U$�� ���� map���μ� 0-level������
inclusion���� 1-level������ $r$��, $A\times I$ ������ $A$����
projection���� �����Ѵ�. $\overline{U'}\times I$�� normal space
$X\times I$�� closed subset�̹Ƿ� normal�̰� $C$��
$\overline{U'}\times I$�� closed subset�̴�. $X$�� ANR�̹Ƿ� $U$��
ANR�̴�. ���� $N$���� ���ǵ� $F$�� extension $\overline{F}$�� �����Ѵ�.\\

%Choose $V^\textrm{open}\supset A$ such that $V\times I \subset N$.
%Then $$ \overline{F}|_{V\times I} : \{j:V\hookrightarrow
%U\}\simeq\{ ir' : V\overset{r'}{\to} A\overset{i}{\hookrightarrow}
%U, r'=r|_V\}$$

$V^\textrm{open}\supset A$�� $V\times I \subset N$�� �ǵ��� ������
$ \overline{F}|_{V\times I}$�� $j:V\hookrightarrow U$�� $ir' :
V\overset{r'}{\to} A\overset{i}{\hookrightarrow} U, r'=r|_V $
���̿� homotopy�� �ش�. ���� $r'^*i^*=j^*$�̴�.\\

���� $\kappa$�� 1-1���� ������. $\kappa(x)=0$�̶� �ϸ�, � $U$��
���Ͽ� $x=\{x_u\},x_u\in \hq(U)$�̰�, $i^*(x_u)=0$�̴�. ����
$j^*(x_u)=r'^*i^*(x_u)=0$�̰� �ᱹ $x=\{j^*(x_u)\}=0$���� �� ��
�ִ�.\\

Note. The product of Paracompact space and compact Hausdorff is
paracompact.\\
(Munkres, p.259)\\

In particular $\kappa$ becomes an isomorphism for
$A^\textrm{closed, ANR}$ in paracompact manifold $X$.  (eg. $A$ is
a
closed submanifold of $X$.)\\

{\bf Remark.} The proof of 3 shows in particular that if $\exists
V\supset A$ such that $A$ is a deformation retract of $V$ then $A$
is taut.\\


4. If $X$ is compact and $\ch^q(A)\cong H^q(A)$, then we have
\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
&&H^q(U,U-K)\ar[r]^{\cong}\ar[d]&H^q(X,X-K)\ar[ddl]^<<<{i^*}&\save[]+<1cm,0cm>*\txt<8pc>{where
$X-K=V$ and $ X-A = U$}
\restore \\
\cdots \ar[r]&\ch^{q-1}(A)\ar[r]^{\delta}\ar[d]^{\kappa:\cong}&
H^q_c(X-A)\ar[r]\ar@{.>}[d]^{\exists
!}&H^q(X)\ar[r]\ar[d]^{id:\cong}&\ch^q(A)\ar[d]^{\kappa:\cong}
\ar[r]&\cdots\\
\cdots\ar[r]
&H^{q-1}(A)\ar[r]^{\delta}&H^q(X,A)\ar[r]&H^q(X)\ar[r]&H^q(A)\ar[r]&\cdots
}
\]
Diagram commutes since all the maps involved are induced by
inclusion and $\delta$ is natural. Then by 5 lemma,
$H_c^q(X-A)\cong H^q(X,A).$\\

\textbf{���� 25.} Show that $H_c^q(\mathbb{R}^n)\cong
H^q(S^n,\infty)=\widetilde{H^q}(S^n)$.\\

5. Let $M$ be a compact $n$-manifold, $A^{closed\hspace{0.2em}
ANR}\subset M$. Then
\[\xymatrix@M=1ex @C=1em @R=1em @*[l]{%
 H^q_c(M-A) \ar[rr]^{\cong}\ar[d]^{P.D :\cong}
 && H^q(M,A)\ar@{.>}[lld]^{\textrm{"Lefschetz duality"}:\cong}\\
 H_{n-q}(M-A) &}\]

e.g. $H^q(M,\bd M)\cong H_{n-q}(M-\bd M)\cong H_{n-q}(M)\hspace{0.5em}(\bd M$ has a collar)\\

\textbf{Remark.} $(K,L)$: compact pair in $M$, a compact
manifold.\\
relative A.D. :
$\ch^q(K,L)\overset{\cong}{\rightarrow}H_{n-q}(M-L,M-K)$\\
In particular, if $M=K$, then
$\ch^q(M,L)\overset{\cong}{\rightarrow}
H_{n-q}(M-L)$.\textbf{(���� 26.)}\\

\underline{\textbf{Application of Alexander Duality}}\\

\textbf{6.} $A^{compact}\subset \mathbb{R}^n\subset S^n\\
\Rightarrow (1) \ch^q(A)\cong \widetilde{H}_{n-q-1}(\mathbb{R}^n-A)\\
\hspace*{1.3em}(2)\check{\widetilde{H}^q}(A)\cong
\widetilde{H}_{n-q-1}(S^n-A)$, where $\check{\widetilde{H}^q}(A) =
\underset{\underset{V}{\rightarrow}}{lim}\tilde{H}^q(V)$\\
\begin{pf}(1) and (2) :
\[\xymatrix@M=1ex @C=2em  @*[c]{%
\ch^q(A)\ar[r]^-{A.D :\cong}& H_{n-q}(S^n, S^n-A)\ar[r]^{excision
: \cong }\ar[d]^{(*):\cong}& H_{n-q}(\mathbb{R}^n,
\mathbb{R}^n-A)\ar[r]^{\cong}& \widetilde{H}_{n-q-1}(\mathbb{R}^n-A)\\
&\widetilde{H}_{n-q-1}(S^n-A)&&
 }\]
(*) : reduced long exact sequence of pair where $q \neq 0$\\

$q=0$ case $ : \\
\rightarrow \widetilde{H}_n(S^n-A)\rightarrow
\widetilde{H}_n(S^n)\rightarrow H_n(S^n,
S^n-A)\overset{\bd}{\rightarrow}\widetilde{H}_{n-1}(S^n-A)\rightarrow
\widetilde{H}_{n-1}(S^n)\\
\hspace*{2em}(\overset{open}{=}0)\hspace{3em}(=R)\hspace{20em}(=0)\\
\Rightarrow \widetilde{H}_{n-1}(S^n-A)\cong H_n(S^n, S^n-A)/R$.\\

Now note that
$\ch^0(A)/R\cong\check{\widetilde{H}^0}(A)$.\\
Then by A.D.,$H_n(S^n, S^n-A)\cong \ch^0(A)$ and
$\widetilde{H}_{n-1}(S^n-A)\cong \breve{\widetilde{H}^0}(A)$.\end{pf}\\

\textbf{7.} $A\subset\mathbb{R}^n\subset{S^n},\hspace{0.5em}A:$
compact $(n-1)$-manifold.\\
$\Rightarrow$ the number of components of $\mathbb{R}^n-A$= (the
number of components of $A$)+1
\begin{pf} $\widetilde{H}_0(\mathbb{R}^n-A;\mathbb{Z}/2)\cong
H^{n-1}(A;\mathbb{Z}/2)\overset{P.D.}{=}H_0(A,\mathbb{Z}/2) =
(\mathbb{Z}/2)^k$  where $k$= the number of components of $A$
\end{pf}\\

\textbf{8.} A non-orientable closed $M^n$ can not be embedded in
$\mathbb{R}^{n+1}.$\\
\begin{pf} May assume $M$ is connected.\\ $M$ is non-orientable.\\
$\Rightarrow H_n(M) =0 \Rightarrow rk(H^n(M;\mathbb{Z}))=0$ and
hence $
H^n(M;\mathbb{Z})\cong\widetilde{H}_0(\mathbb{R}^{n+1}-M;\mathbb{Z})=0\\
\therefore \mathbb{R}^{n+1}-M$ has 1 component. This is a
contradiction to 7.\end{pf}\\

\textbf{9.} $A$ = a link with $k$ components in $\mathbb{R}^3\\
\Rightarrow H_*(\mathbb{R}^3-A)=H_*(\mathbb{R}^3$-
trivial link with $k$ component)\\
\begin{pf} $\widetilde{H}_{n-q-1}(\mathbb{R}^3-A)=H^q(A) =
H^q($trivial link)=$\widetilde{H}_{n-q-1}(\mathbb{R}^3-$trivial
link with $k$ component). \end{pf}\\

\end{document}


%\ar@{.*{\rightarrow}}[r] & \chq(A)
%\ar@{}[dr]|{\displaystyle{\circlearrowleft}}
