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\begin{document}
\parindent=0cm
\section*{VII.4 Mayer-Vietoris Sequence}

\textbf{$\mathcal{U}$-small homology and sd-operator}\\

1. Define a chain map $sd : S_p(X) \rightarrow S_p(X)$ naturally :
use induction\\
$p=0 \hspace{3em}\cdots \rightarrow
S_0(X)\overset{\epsilon}{\rightarrow}
\mathbb{Z}\\
\hspace*{8em} \downarrow sd_0 = id \parallel\\
\hspace*{7em}\rightarrow S_0(X)\overset{\epsilon}{\rightarrow} \mathbb{Z}\\
p>0 : $ 1st for $i_p \in S_p(\bigtriangleup^p )
\hspace{1em},\forall p$\\

Define $\bet_p : {\bigtriangleup '}_k (\bigtriangleup^p
)\rightarrow{\bigtriangleup '}_{k+1} (\bigtriangleup^p )$ as
$\bet(v_0,\cdots,v_k ) = (b,v_0,\cdots,v_k)$ where $b$ is a
barycenter of $ \bigtriangleup^p$ and ${\bigtriangleup '}_k
(\bigtriangleup^p )$ is a subgroup of $S_k(\bigtriangleup^p)$
generated by affine singular $k$-simplices denoted by $(v_0 , \cdots , v_k)$.\\
Then $\bd\bet + \bet\bd =1 :$\\
Given $\sigma = (x_0,\cdots,x_k)\in {\bigtriangleup
'}_k(\bigtriangleup^p ),\\
\hspace{2em} \partial \sigma = \partial (\sigma_\sharp i_k ) =
\sigma_\sharp (\partial i_k ) = \sigma_\sharp (\Sigma (-1)^i
(e_0,\cdots, \hat{e_i},\cdots,e_k)) = \Sigma (-1)^i
(x_0,\cdots,\hat{x_i},\cdots,x_k)$\\

Now apply the formula for join operator in the proof of Lemma9.(3.Homotopy invariance of homology)\\
1 st : Define $sd $ for $i \in \bigtriangleup_p
(\bigtriangleup^p)$ as follows.\\
\[
\xymatrix @=3em @*[r] { %
i \ar[r]^{\bd} \ar@{.>}[d]^{sd} & \bd i \ar[d]^{sd}\\
\bet sd \bd i \ar[r]^{\bd} & sd \bd i \ar[l]^{\bet}\in
\bigtriangleup_{p-1}(\bigtriangleup^p)}
\]\\
$sd i := \bet sd \bd i$\\
Then $\bd sd i = \bd\bet sd \bd i = sd \bd i - \bet \bd sd \bd i =
sd \bd
i- \bet sd \bd\bd i = sd \bd i$,\\ i.e., $sd i $ is a chain map. \\

2nd on $X$ :
\[
\xymatrix @M=1ex @C=1ex @R=3ex @*[c] { %
% ����
& i_p \ar[dl]^{\sigma_{\sharp}} \ar[rr]^{\bd} \ar[dd]^<{sd} & &
\bd i
\ar[dl]^{\sigma_{\sharp}} \ar[dd]^{sd}\\
% ��� ��1
\sg{p} \ar[rr]  \ar@{.>}[dd]^{sd} & & \sg{p-1}  \ar[dd]& \\
% ��� �� 2
&  \ar[dl]^{\sigma_{\sharp}} \ar[rr] & &
\ar[dl]^{\sigma_{\sharp}} \\
% �Ʒ���
S_{p}(X) \ar[rr] & & S_{p-1}(X)& }
\]
���� ���� ���� ���� commute�ϰ� $sd_X$�� ����\\
�׷��� �ո鵵 commute�ϰ� �ǰ� ���� chain map���� ���ǵȴ�.\\
Naturality of $sd_p$ : same as before\\

2. $sd \simeq id $: chain homotopic.\\
\begin{proof} use AMT.\\
$sd, id :S(X)\rightarrow S(X)$ natural chain maps\\ $\Rightarrow
\exists $ natural chain homopoty $D: sd \simeq id.$\end{proof}\\

\textbf{Note.} Cor. of AMT $\Rightarrow sd$ is a natural chain
homopoty equivalence.\\

3. Let $\mathcal{U} = \{A\}$ be s.t. $X = \bigcup \{\AA | A\in
\mathcal{U}\}$. Then $\forall \sigma :$ singular $p$-simplex in
$X, \exists m>0 $ s.t. $sd^m(\sigma) \in S_p^{\mathcal{U}} (X) $
where $S_p^{\mathcal{U}} (X) = \{\sigma \in S_p(X) | \sigma
(\bigtriangleup^p ) \subset \AA$ for some
$A\subset\mathcal{U}\}$\\
\begin{proof} $X=\bigcup\{\AA| A\in \mathcal{U}\}\Rightarrow
\bigtriangleup^p = \bigcup \{\sigma^{-1}(\AA)| A\in
\mathcal{U}\}$\\
Let $\epsilon$ be a Lebesque number for this open cover. \\Choose
$m$ s.t. $(\frac{p}{p+1})^m mesh (\bigtriangleup^p ) <
\epsilon$\end{proof}\\

4. Let $\mathcal{U} = \{A\}$ be s.t. $X = \bigcup \{\AA | A\in
\mathcal{U}\}.$ Then $i : S^{\mathcal{U}}(X) \hookrightarrow S(X)$
is a chain homotopy
equivalence.\\
\begin{proof} Construct a chain homotopy  inverse $\tau: S(X)
\rightarrow S^{\mathcal{U}}(X)$ s.t. $\tau\circ i = id$ and
$i\circ\tau\simeq
id$:\\
Let $m(\sigma)$ be the smallest integer s.t.
$sd^{m(\sigma)}({\sigma})\in S_p^{\mathcal{U}}(X)$\\
Note $m(\sigma) \geq m(\sigma^{(i)})$\\
Want $T$ s.t. $\partial T + T\partial = i\circ \tau - id.\\
\overset{2.}{\Rightarrow} \exists D$ s.t. $\partial D + D\partial = sd-id\\
\Rightarrow \partial D sd + D sd \bd = sd^2 - sd\\
\hspace*{2em}\vdots\\
\partial D sd^{k-1} + D sd^{k-1}\partial = sd^k - sd^{k-1}$\\

��� ���ϸ�\\
$\partial D(1+\cdots + sd^{k-1}) +D(1+\cdots +sd^{k-1})\partial =
sd^k - id$\\

Define $T(\sigma) = D(1+\cdots +sd^{m(\sigma)-1})(\sigma)$\\
Then $(\partial T + T\partial )(\sigma) =\partial T(\sigma) +
T\partial(\sigma)\\ = sd^{m(\sigma)}(\sigma)-\sigma-D(1+\cdots +
sd^{m(\sigma)-1})\partial \sigma + \Sigma(-1)^i D(1+\cdots
+sd^{m(\sigma^{(i)})})^{-1}\sigma^{(i)}\\
= sd^{m(\sigma)}(\sigma) -\sigma -\Sigma(-1)^i
D(sd^{m(\sigma^{(i)})}+\cdots + sd^{m(\sigma)-1})\sigma^{(i)}$\\

$\therefore$ Define $\tau(\sigma) = sd^{m(\sigma)}(\sigma) -
\Sigma (-1)^i D (sd^{m(\sigma^{(i)})} + \cdots +
sd^{m(\sigma)-1})(\sigma)$\\

Then $(\partial T + T\partial ) = i \circ \tau - id$.\\

Now note that $\tau\circ i = id$ on $S^{\mathcal{U}}(X)\\
(\because$ �̹� $\mathcal{U}$-small �ϴϱ� $m(\sigma) = 0,
m(\sigma^{(i)})=0$)\end{proof}

\newpage

{\bf Snake lemma(Zig - Zag lemma)}\\

(1) Given a short exact sequence of chain complexes,\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & \cc^{'} \ar[r]^i & \cc \ar[r]^q & \cc^{''} \ar[r] & 0}
\]

\hspace*{1.0em} where $\cc = \{C_{p},\bd\}, \cdots $ etc. and $i$
and $q$ are chain maps. Then we can obtain a long exact sequence\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r]^{\bd_{*}} & H_{p}(\cc^{'}) \ar[r]^{i_{*}} &
H_{p}(\cc) \ar[r]^{q_{*}} & H_{p}(\cc^{''}) \ar[r]^{\bd_{*}} &
H_{p-1}(\cc{}') \ar[r] & \cdots}
\]

���⼭ $\bd_{*}$�� ������ ���� ���ǵȴ�. �켱 $c^{''} \in
Z_{p}^{''}$�� ���ؼ� $\exists c \in Z_{p}$ such that $q(c)=c^{''}$
�ϰ� $q$�� chain map �̹Ƿ� $q \circ \bd(c)= \bd \circ
q(c)=\bd(c^{''})=0$�� �����ؾ� �Ѵ�. ���� exactness�� ���ؼ�
$i(c^{'})= \bd(c)$�� �����ϴ� $c^{'} \in Z_{p-1}^{'}$�� �����ؾ�
�ϰ� $\bd_{*}[c^{''}] :=
[c^{'}]$�� �����Ѵ�.\\

Well-definedness�� check�ϱ� ���ؼ� ���� �ΰ����� ���캸��. ����
$c$ ��ſ� �ٸ� $c_1 \in Z_{p}$�� �����ߴٰ� �����ϸ�, $\exists a
\in Z_{p}^{'}$ such that $i(a)=c-c_{1}$�� �Ǿ� $c_{1}$�� ���ؼ�
����� $c_{1}^{'} \in Z_{p-1}^{'}$�� ���ؼ� $\bd a = c^{'} -
c_{1}^{'}$�� �����Ѵ�. �� $\bd_{*}$�� ���Ǵ� $c$�� ���ð� �����ϰ�
�� ���ǵȴ�. ���� $c^{''}$ ��ſ� $c^{''} + \bd d^{''}$�� �����ص�
�� ���ǵȴ�. ���⼭ $\bd d^{''}$�� $[c^{''}] = [c^{''} + \bd
d^{''}]$�� �����ϴ� $Z_{p}^{''}$�� �ٸ� �����̴�.\\


���� long exact sequence�� exactness�� check �غ���.\\
���� $H_{p}(\cc)$���� exact������ ���캸��, $q \circ i = 0$�̹Ƿ�
$q_{*} \circ i_{*} =0$�� �����ϰ� ���� $im \,\, i_{*} \subset
ker \,\, q_{*}$�� �����Ѵ�. ���� $q(c)=\bd d^{''}$�� $c$�� ���ؼ�
$q(d)=d^{''}$�� �θ�, $q(c- \bd d)=q(c)-\bd q(d) = q(c)-\bd d^{''}
= 0$ �̹Ƿ� $i(c')=c- \bd d$�� $c' \in Z_{p}^{'}$�� �����Ѵ�.
���� $[c]=i_{*}[c^{'}]$�� �����ؼ� $ker \,\, q_{*} \subset im
\,\, i_{*}$ �� �Ǿ� $H_{p}(\cc)$������ exact�ϴ�. ���� �������
diagram chasing�� ���ؼ� $H_{p}(\cc^{''})$��
$H_{p}(\cc^{'})$���� exact���� ���� �� �� �ִ�.\\



(2) ���� ���� ����� long exact sequence�� functorial�̴�. ��, �־��� commutative diagram�� ���ؼ� \\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & \cc{}' \ar[r]^i \ar[d]^{\alp'} & \cc \ar[r]^q \ar[d]^{\alp} & \cc^{''} \ar[r] \ar[d]^{\alp^{''}} & 0\\
0 \ar[r] & \dd{}' \ar[r]_j & \dd \ar[r]_p & \dd^{''} \ar[r] & 0 }
\]


�Ʒ� diagram�� commute�Ѵ�.\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r]^{\bd_{*}} & H_{p}(\cc{}') \ar[r]
\ar[d]^{\alp_{*}^{'}} & H_{p}(\cc) \ar[r] \ar[d]^{\alp_{*}} &
H_{p}(\cc^{''}) \ar[r]^{\bd_{*}} \ar[d]^{\alp_{*}^{''}} &
H_{p-1}(\cc{}') \ar[r] \ar[d] & \cdots\\
\cdots \ar[r]^{\bd_{*}} & H_{p}(\dd{}') \ar[r] & H_{p}(\dd) \ar[r]
& H_{p}(\dd^{''}) \ar[r]^{\bd_{*}} & H_{p-1}(\dd{}') \ar[r] &
\cdots }
\]



\newpage

{\bf Mayer-Vietoris sequence}\\


1. Let $\mathcal{U}=\{A,B\}$ be such that $\overset{\circ}{A} \cup
\overset{\circ}{B} = X$. Consider $S_{p}(A), S_{p}(B) \subset
S_{p}^{\mathcal{U}}(X)$. Then we have a short exact sequence\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
&& \ar[d]^{\bd} & \ar[d]^{\bd \oplus \bd} && \\
(\star) & 0 \ar[r] & S_{p}(A \cap B) \ar[r]^>>>{\phi} \ar[d]^{\bd}
& S_{p}(A) \bigoplus S_{p}(B)  \ar[r]^>>>{\psi} \ar[d]^{\bd \oplus
\bd}
& S_{p}^{\mathcal{U}}(X) \ar[r] & 0\\
&&&&&&  }
\]

\hspace*{2.0em} where $\phi(c) = (c,-c)$ and $\psi(a,b) = a+b$.\\

(Check) $\phi$ is one-to-one.\\
\hspace*{3.2em} $\psi$ is onto.\\
\hspace*{3.2em} $\psi \circ \phi = 0$ and $ker \,\, \psi \subset
im \,\, \phi$\\


{\bf Note} $\phi$ and $\psi$ are chain maps. Furthermore,
$(\star)$ is functorial,\\ \hspace*{1.0em} i.e., for any $f :
(X,A,B) \to (X^{'},
A^{'}, B^{'})\\
\Rightarrow$
\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & S(A \cap B) \ar[r]^>>>{\phi} \ar[d]^{f_{\sharp}} & S(A)
\bigoplus S(B) \ar[r]^>>>{\psi} \ar[d]^{f_{\sharp}
\oplus f_{\sharp}} & S^{\mathcal{U}}(X) \ar[r] \ar[d]^{f_{\sharp}} & 0 &\\
0 \ar[r] & S(A^{'} \cap B^{'}) \ar[r]_>>>{\phi} & S(A^{'})
\bigoplus
S(B^{'}) \ar[r]_>>>{\psi} & S^{\mathcal{U}}(X^{'}) \ar[r] & 0 & \textrm{commutes}\\
}
\]

Hence, by snake lemma, $(\star)$ induces a functorial long exact
sequence\\
\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & H_{p}(A \cap B) \ar[r]^>>>{\phi_{*}}
\ar[d]^{f_{*}} & H_{p}(A) \bigoplus H_{p}(B) \ar[r]^>>>{\psi_{*}}
\ar[d]^{f_{*} \oplus f_{*}} & H_{p}^{\mathcal{U}}(X)
\ar[r]^>>>{\bd_{*}} \ar[d]^{f_{*}} & H_{p-1}(A \cap B) \ar[r] \ar[d]^{f_{*}} & \cdots \\
\cdots \ar[r] & H_{p}(A^{'} \cap B^{'}) \ar[r] & H_{p}(A^{'})
\bigoplus H_{p}(B^{'}) \ar[r] & H_{p}^{\mathcal{U}}(X^{'})
\ar[r] & H_{p-1}(A^{'} \cap B^{'}) \ar[r] & \cdots \\
}
\]

���⼭ �� long exact sequence �� {\bf Mayer-Vietoris sequence}
��� �θ���.\\

{\bf Note} $\phi_{*}(\gam) = (\gam, - \gam)$ and $\psi_{*}(\alp,
\bet) = \alp + \bet$.\\

���� $\bd_{*}\alp$�� ����غ���. $\alp \in H_{p}(X) \cong
H_{p}^{\mathcal{U}}(X)$�� $\alp = \{a\} = \{a_{1}+a_{2}\}$, for
$a_{1} \in S_{p}(A)$ and $a_{2} \in
S_{p}(B)$�� ������ �� �ְ�, �Ʒ��� diagram�� ����,\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r]^{\phi} & (a_1 , a_2) \ar[r]^{\psi} \ar[d] &
a_{1}+a_{2} \ar[d]^{\bd}\\
\bd a_{1} \ar[r] & (\bd a_{1}, \bd a_{2}) \ar[r] & \bd a_{1} + \bd
a_{2} = 0 }
\]

���� $\bd_{*}\alp = \bd_{*}\{a\} = \bd_{*}\{a_1 + a_2\} = \{\bd
a_{1}\}$�� �ȴ�.\\


2. Mayer-Vietoris sequence holds also for reduced homology if $A
\cap B \neq \emptyset$ ; It suffices to check that $(\star)$ holds
for
augmented chain complex.\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & S_{0}(A \cap B) \ar[r]^>>>{\phi} \ar[d]^{\eps} &
S_{0}(A) \bigoplus S_{0}(B) \ar[r]^>>>{\psi} \ar[d]^{\eps \oplus
\eps} &
S_{0}^{\mathcal{U}}(X) \ar[r] \ar[d]^{\eps} & 0\\
0 \ar[r] & \zb \ar[r]^{\phi} & \zb \bigoplus \zb \ar[r]^{\psi} &
\zb \ar[r] & 0 \\
}
\]

���� diagram���� diagram commutativity �� check �� �� �����Ƿ�
�츮��
reduced homology�� funtorial long exact sequence�� ��� �ȴ�.\\


3.{\bf(Remark)} Mayer-vietoris sequence holds in general for an
excisive couple $A, B \subset X (A \cup B = X)$.\\
\hspace*{1.0em} $\{A,B\}$ is an $excisive \,\, couple$ if $S(A) +
S(B) \hookrightarrow
S(X)$ induces an isomorphism in homology.\\


4. {\bf Mayer-Vierotis sequence for simplicial homology}\\
$\bigtriangleup(K_{1}) + \bigtriangleup(K_{2}) =
\bigtriangleup(K_{1} \cup K_{2})$ and $C(K_{1}) + C(K_{2}) =
C(K_{1} \cup K_{2})$�� �����ϹǷ� simplicial homology�� ����
Mayer-Vietoris sequence��
�����Ѵ�.\\


5.{\bf Example}\\
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%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
% S^2%
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\pspolygon[fillstyle=solid,fillcolor=white,linestyle=none](-1,0)(1,0)(1,0.5)(-1,0.5)(-1,0)
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\psellipse[linecolor=blue](0,-0.3)(.6,.12)
\psellipse[linecolor=blue](0,0.3)(.6,.12)
\pscurve[linecolor=blue](0.69,0.3)(0.72,-0.2)(0.69,-0.7)\rput(1,-0.2){$B$}
\pscurve[linecolor=blue](-0.69,-0.3)(-0.72,0.2)(-0.69,0.7)\rput(-1.2,0.2){$A$}

}%
\end{pspicture}
\end{floatingfigure}


(1) {\bf $H_{*}(S^{n})$}\\
�׸��� ���� $A$�� �Ϲݱ��� ���ݱ��� �ణ�� ������ ������ �ϰ�
$B$�� ���ݱ��� �Ϲݱ��� �ణ�� ������ ������ �ϸ�, $A$�� $B$��
$\overset{\circ}{D^{n}}$ ��, point�� homotopy type�� ����, $A \cap
B$�� $S^{n-1}$�� homotopy type�� ����. ���� Mayer-vietoris
sequence�� �����غ���,

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & \widetilde{H_{p}}(A \cap B) \ar[r] &
\widetilde{H_{p}}(A) \bigoplus \widetilde{H_{p}}(B) (=0) \ar[r] &
\widetilde{H_{p}}(S^{n}) \ar[r]^>>>{\bd} & \widetilde{H_{p-1}}(A
\cap B) \ar[r] & \cdots }
\]

$\therefore \widetilde{H_{p}}(S^{n}) \cong
\widetilde{H_{p-1}}(S^{n-1}) \cong \cdots \cong $�� �����ؼ�
�����ϸ� ������ ����.\\

$n>0 \hspace{1.0em}  p=n>0 \Rightarrow \widetilde{H_{n}}(S^{n})
\cong \cdots \cong \widetilde{H_{0}}(S^{0}) \cong  \zb \\
\hspace*{2.5em} p>n \Rightarrow  \widetilde{H_{p}}(S^{n}) \cong
\cdots \cong \widetilde{H_{p-n}}(S^{0}) \cong  0\\
\hspace*{2.5em} p<n \Rightarrow  \widetilde{H_{p}}(S^{n}) \cong
\cdots \cong  \widetilde{H_{0}}(S^{n-p}) \cong  0$\\

����
\begin{displaymath}
H_{p}(S^{n}) =\{\begin{array}{ll} \zb & \,\,\, p=0, n\\
0 & \,\,\, \textrm{otherwise}\end{array}.
\end{displaymath}
\begin{displaymath}
H_{p}(S^{0})=\{\begin{array}{ll} \zb \bigoplus \zb & \,\,\, p=0\\
0 & \,\,\, \textrm{otherwise}\end{array}.
\end{displaymath}
�� �����Ѵ�. ���� $H_{1}(S^{1}) \cong \zb$�� generator��
�����غ���. ���� $H_{1}(S^{1}) = \widetilde{H_{1}}(S^{1})
\overset{\bd_{*}}{\cong} \widetilde{H_{0}}(S^{0}) = \zb$�̹Ƿ�
$\widetilde{H_{0}}(S^{0})$�� generator�� ã��.\\

\begin{floatingfigure}[l]{4cm}
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\psarc{*-*}(0,0){0.8}{0}{180} \psarc{|-|}(0,0){0.8}{20}{160}
\psarc{|-|}(0,0){0.8}{200}{340}
\pscurve(-0.85,-0.15)(-0.9,0.3)(-0.85,0.8) \rput(-1.2,0.3){$A$}
\pscurve(0.85,0.15)(0.9,-0.3)(0.85,-0.8) \rput(1.2,-0.3){$B$}
\rput(-0.6,0){$x$} \rput(0.6,0){$y$} }

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\psarc{*->}(0,0){0.8}{180}{275} \rput(0,1.1){$a_{1}$}
\rput(0,-1.1){$a_{2}$} }

\end{pspicture}
\end{floatingfigure}

�켱 $ \cdots \overset{\bd}{\rightarrow} S_{0}(S^{0}) (=\zb
\bigoplus \zb) \overset{\eps}{\rightarrow} \zb \rightarrow 0$��
���캸�� $ker \eps = (a,-a)$�̰� ���� $S_{0}(S^{0})$�� cycle��
$\langle (1,-1)\rangle$ �̴�. �ᱹ $\widetilde{H_{0}}(S^{0})$��
generator�� $x-y$�� �ȴ�.\\

���� ���� �׸��� ���� 1-chain $a_{1}, a_{2}$�� ��������. �׷���
���� ���ǿ� ���ؼ� $\bd_{*}(a_{1}+a_{2}) = \bd a_{1} = x-y$��
�����Ѵ�.
���� $H_{1}(S^{1})$�� generator�� $ \{a_{1}+a_{2}\}$�� �ȴ�.\\

{\bf $H_{2}(S^{2}) = \zb$} generator�� �����ΰ�?(check)\\



(2)Figure eight(=$X$) �� ���� homology�� ����غ���.
\begin{center}
\begin{pspicture}(-0.5,-0.5)(2.5,2.5)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
% S^2%
\rput(1,1){\pscurve(0,0)(-0.7,0.5)(-1.0,0)(-0.7,-0.5)(0,0)
\pscurve(0,0)(0.7,0.5)(1.0,0)(0.7,-0.5)(0,0)
\pscircle[linestyle=dashed](-0.6,0){0.75}
\pscircle[linestyle=dashed](0.6,0){0.75} \rput(-0.6,0.65){$A$}
\rput(0.6,-0.65){$B$} }
\end{pspicture}
\end{center}

�׸��� ���� $A$�� $B$�� �����ϸ�, $A$�� $B$�� $S^{1}$�� $A \cap
B$�� point�� homotopy type�� ����. ���� Mayer-Vietoris sequence��
������Ѻ���,\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & \widetilde{H_{p}}(A \cap B) \ar[r] &
\widetilde{H_{p}}(A) \bigoplus \widetilde{H_{p}}(B) \ar[r] &
\widetilde{H_{p}}(X) \ar[r] & \widetilde{H_{p-1}}(A \cap B) \ar[r]
& \cdots} \]

$\widetilde{H}(pt) = 0 $�̹Ƿ� $\widetilde{H_{p}}(A) \bigoplus
\widetilde{H_{p}}(B) \cong \widetilde{H_{p}}(X)$�� �Ǿ�

\begin{displaymath}
\widetilde{H_{p}}(X)=\{\begin{array}{ll} \zb \bigoplus \zb & , p=1\\
0 & , \textrm{otherwise}\end{array}.
\end{displaymath}

�� �����Ѵ�. �� ����� �̿��ϸ�, �Ϲ������� \\
$X=\underset{r}{S^{1} \vee \cdots \vee S^{1}}$ �� ���

\begin{displaymath}
\widetilde{H_{p}}(X)=\{\begin{array}{ll} \zb^{r} & , p=1\\
0 & , \textrm{otherwise}\end{array}.
\end{displaymath}
�� �����Ѵ�.\\


(3) {\bf Closed surface}
\begin{center}
\begin{pspicture}(-1.0,0.3)(4,1.7)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(-1.0,1){$X$} \rput(-0.5,1){=}%
% Torus%
\rput(1,1){\psellipse(0,0)(1,.7) }
% genus
\rput(1,1){\psarc(0,0.9){1}{240}{300}\psarc(0,-0.95){1}{70}{110} }
 \rput(2.4,1){=}%
\rput(3,0.5){\pspolygon(0,0)(1,0)(1,1)(0,1)
    \psline{->}(0,0)(0,0.55)
    \psline{->}(0,0)(.55,0)
    \psline{->}(0,1)(.55,1)
    \psline{->}(1,0)(1,0.55)
    \rput(1.1,.5){$b$}
    \rput(.5,-.1){$a$}
    \rput(-.1,.5){$b$}
    \rput(.5,1.1){$a$}

    % ����
    \rput(.8,.8){$B$}
    \rput(1.2,.1){$A$}
    \psline{->}(1.1,.1)(.6,.4)
    \rput(.5,.5){\pscircle[fillstyle=solid,fillcolor=lightgray](0,0){0.15}\pscircle(0,0){0.03}}
    }
\end{pspicture}
\end{center}

������ �׸������� ���� Torus���� ������ ���� open set�� $B$, ��
���� �����ϴ� open neighborhood�� $A$��� �ϸ� $A$�� point��
homotopy type�� ����, $B$�� figure eight �׸��� $A \cap B$��
$S^{1}$�� homotopy type�� ����. ���� Mayer-Vietoris sequence��
�����ϸ�,

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & \widetilde{H_{p}}(A \cap B) \ar[r] &
\widetilde{H_{p}}(A) \bigoplus \widetilde{H_{p}}(B) \ar[r] &
\widetilde{H_{p}}(X) \ar[r] & \widetilde{H_{p-1}}(A \cap B) \ar[r]
& \cdots }
\]

�� �ǰ�(���⼭ $\widetilde{H_{p}}(A) = 0$�̴�.), Ư����

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & \widetilde{H_{2}}(X) \ar[r] & \widetilde{H_{1}}(S^{1})
\ar[r] & \widetilde{H_{1}}(B) \ar[r] & \widetilde{H_{1}}(X) \ar[r]
& \widetilde{H_{0}}(S^{1}) \ar[r] & \cdots }
\]

�� ���캸�� $(\widetilde{H_{2}}(B) =0) , \widetilde{H_{1}}(S^{1})
\to \widetilde{H_{1}}(B)$�� zero map �� �ǹǷ�
$\widetilde{H_{2}}(X) \cong \widetilde{H_{1}}(S^{1}) = \zb,
\widetilde{H_{1}}(X) \cong \widetilde{H_{1}}(B) = \zb \bigoplus
\zb$�� �ȴ�. ����, $p \geq 3$�̸�, $\widetilde{H_{p}}(B) = 0,
\widetilde{H_{p-1}}(S^{1})=0$�̹Ƿ�

\[
H_{p}(X)= \begin{cases} \zb & , p=0\\
\zb \bigoplus \zb & ,p=1 \\
\zb & ,p=2 \\
0 & ,\textrm{otherwise} \end{cases}
\]
�� �ȴ�.\\

\begin{center}
\begin{pspicture}(-2.5,-0.5)(10.5,3.5)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
\rput(-1.0,1.5){$X$} \rput(-0.5,1.5){=} \rput(0,1.5){$P^{2}$}
\rput(0.5,1.5){=}


%�׸��ϳ�
\rput(1.0,0){
\pscircle[fillstyle=solid,fillcolor=lightgray](1.5,1.5){1.2}%
\psarc{*-*}(1.5,1.5){1.2}{0}{180} \psarc{*->}(1.5,1.5){1.2}{0}{95}
\psarc{*->}(1.5,1.5){1.2}{180}{275} \rput(1.5,3.1){$a$}
\rput(1.5,-0.1){$a$}}


%�׸���
\rput(4.0,0){ \pscircle(1.5,1.5){1.2}
\psarc{*-*}(1.5,1.5){1.2}{0}{180} \psarc{*->}(1.5,1.5){1.2}{0}{95}
\psarc{*->}(1.5,1.5){1.2}{180}{275} \pscircle(1.5,1.5){0.15}
\pscircle[fillstyle=solid,fillcolor=lightgray](1.5,1.5){0.4}
\rput(2.6,1.5){$B$} \rput(1.3,1.3){$A$} }

%�׸���
\rput(7.0,0){ \pscircle(1.5,1.5){0.8}
\psarc{->}(1.5,1.5){0.8}{0}{135} \rput(1.2,2.3){$a$}
\rput(2.5,1.5){$\cong$} \rput(3.0,1.5){$B$} }

\end{pspicture}
\end{center}

$P^{2}$���� �ι�° �׸��� ���� ������ ���� open set�� $B$, ������
�����ϴ� open neighborhood�� $A$��� �ϸ� $A$�� point, $B$�� $A
\cap B$�� $S^{1}$�� homotopy type�� ����. Mayer-Vietoris
sequence�� �����ϸ�,

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & \widetilde{H_{p+1}}(X) \ar[r] &
\widetilde{H_{p}}(S^{1}) \ar[r] & \widetilde{H_{p}}(B) \ar[r] &
\widetilde{H_{p}}(X) \ar[r] & \widetilde{H_{p-1}}(S^{1}) \ar[r] &
\cdots }
\]
Ư��,

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & \widetilde{H_{2}}(X) \ar[r]^f &
\widetilde{H_{1}}(S^{1}) \ar[r]^g & \widetilde{H_{1}}(B) \ar[r]^h
& \widetilde{H_{1}}(X) \ar[r] & \widetilde{H_{0}}(S^{1}) \ar[r] &
\cdots }
\]

�� ����, $g : \widetilde{H_{1}}(S^{1}) (=\zb)  \to
\widetilde{H_{1}}(B) (=\zb)$�� 1�� $2a$�� ������ map�̹Ƿ� (����°
�׸� ����) injective�̴�. ���� $f$�� zero map �� �Ǿ�,
$\widetilde{H_{2}}(X) = 0$�� �ȴ�. ���� $\widetilde{H_{0}}(S^{1})
= 0$�̹Ƿ� $h$�� surjective �̰�, $\widetilde{H_{1}}(X) \cong
\widetilde{H_{1}}(B) / ker \,\, h = \widetilde{H_{1}}(B) / im \,\,
g \cong \zb / 2\zb$�� �ȴ�. ��,

\[
H_{p}(X)=\begin{cases} \zb & , p=0\\
\zb / 2\zb & , p=1 \\
0 & , \textrm{otherwise}
\end{cases}
\]

�̴�.\\

{\bf Note}
\begin{floatingfigure}[l]{4cm}
\begin{pspicture}(-1.0,0)(3.0,3.0)%
%\psgrid[gridwidth=0.1pt,subgridwidth=0.1pt,gridcolor=red,subgridcolor=green]
% S^2%

\rput(1.5,1.5) {
\psellipse[fillstyle=solid,fillcolor=lightgray](0,0)(1.5,1.2)
\psarc{->}(0,0){1.5}{0}{10} \rput(1.4,0){$c$}
\pscircle[fillstyle=solid,fillcolor=white](-0.5,0){0.3}
\psarc{->}(-0.5,0){0.3}{0}{40} \rput(-0.2,0.3){$a$}
\pscircle[fillstyle=solid,fillcolor=white](0.5,0){0.3}
\psarc{->}(0.5,0){0.3}{0}{50} \rput(0.7,0.3){$b$}
\rput(0,0.8){$D$} \rput(0,-1.0){$\circlearrowleft$} }

\rput(-1.0,1.5){$X$} \rput(-0.5,1.5){=}

\end{pspicture}
\end{floatingfigure}

�׸��� ���� $a,b,c$ �׸��� $D$�� �����ϸ�, $0 = \{\bd D\} = \{c\}
-  \{a\} - \{b\}$�� �����ϹǷ�, $H_{1}(X)$�ȿ��� $\{c\} = \{a\}
+\{b\}$�� �ȴ�.\\

���������� $P^{2}$���� coefficient�� $\zb$��� $\rb$ �Ǵ� $\zb /
2$�� �����غ���. �켱 $\rb$�� ��� $im \,\, g =\rb$�̹Ƿ�
$H_{1}(P^{2},\rb) = 0$�� �ȴ�. ���������� $\zb /2$�� ��쿡��
$g$�� zero map �� �ǹǷ� $H_{2}(P^{2},\zb /2) \cong
H_{1}(S^{1},\zb /2) = \zb /2$,
�׸��� $H_{1}(X,\zb/2) \cong H_{1}(B,\zb/2) = \zb / 2$�� �ȴ�.\\

{\bf ����23} Compute $H_{*}$(closed surface).\\


\end{document}
