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\begin{document}
\parindent=0cm
\section*{I.2 Homology of adjunction space}
Assume $(X,A)$ is a collared pair with a collaring $B$ and given
$f:A\to Y$, obtain $X\cup_f Y$ which is denoted by $Z$ in this
section.
\begin{thm}
Let $p:X\coprod Y \to Z$ be a quotient map and $\overline{f} :=
p|_X:(X,A)\to (Z,Y)$ be its restriction. Then $\overline{f}_* :
\hq (X,A)\to \hq(Z,Y)$ is an isomorphism $\forall q$.
\end{thm}
\begin{pf}
\[
\xymatrix @=2em @C=2em  @*[c] { %
\hq(X,A) \ar[r]^{i_*} \ar[d]^{\overline{f}_*} & \hq (X,B) \ar[d]^{\overline{f}_*} & \hq(X-A,B-A)
\ar[l] \ar[d]^{\overline{f}_*}\\
\hq(Z,Y) \ar[r]^>>>>>{j_*} & \hq(Z,Y\cup \overline{f}(B))
&\hq(Z-Y,\overline{f}(B-A))  \ar[l]
}\]\\[1mm]
���� ������ map���� ��� inclusion�� ���� induce�� map���̹Ƿ� ��
diagram�� commute�Ѵ�. �������� $\overline{f}_*$�� ������ ������
���Ͽ� homeomorphism�� ���Ͽ� induce�Ǿ� isomorphism�̹Ƿ� ����
������ map���� ��� isomorphism����
���̸� ������ ������.\\
3������ 2���� ���� map���� excision theorem�� ����
isomorphism�̰�, 1������ 2���� ���� map���� ���� diagram���� five
lemma�� �����ϸ� isomorphism�̴�.
\[
\xymatrix @=2em @C=2em  @*[c] { %
\hq(A) \ar[r] \ar[d]^{\cong}_{\parbox{1.5cm}{% $\because$
\textrm{\scriptsize deformation retract}}}&
\hq(X)\ar[r]  \ar[d]^{=}& \hq(X,A) \ar[d]\ar[r]& \hq(A) \ar[r]  \ar[d]^{\cong} & \hq(X) \ar[d]^{=} \\
\hq(B) \ar[r] & \hq(X)\ar[r] & \hq(X,B)\ar[r]& \hq(B) \ar[r] &
\hq(X)
}\]\\[1mm]
����, $\overline{f}_*$�� isomorphism�̴�.
\end{pf}
\begin{thm}[MV-sequence] \hfill\\
We have a long exact sequence
\[
\xymatrix @=2em @R=0.5mm @*[c] { %
\cdots \ar[r] & \hq(A) \ar[r] & \hq(X)\bigoplus\hq(Y) \ar[r]
&\hq(Z) \ar[r]^\bd & H_{q-1}(A)\ar[r] & \cdots \\
& \alp \ar@{|->}[r] & (i_*\alp, f_*\alp)&&&\\
&&(x,y)\ar@{|->}[r]&j_*y-\overline{f}_*x&&\\
&&&z\ar@{|->}[r]&\bd \overline{f}_*^{-1}k_*z&
}\]\\
This also holds for reduced homology.
\end{thm}
\begin{pf}
\[
\xymatrix @=2em  @C=1.5em @*[c] { %
\cdots \ar[r] & \hq(A) \ar[r] \ar[d]^{f_*} & \hq(X)\ar[r]
\ar[d]^{\overline{f}_*} & \hq(X,A) \ar[d]^{\overline{f}_*}_\cong
\ar[r]^{\bd_*}
& H_{q-1}(A) \ar[r]  \ar[d]^{f_*} & H_{q-1}(X) \ar[d]^{\overline{f}_*} \ar[r]& \cdots \\
\cdots  \ar[r]  & \hq(Y) \ar[r] & \hq(Z) \ar[r]^{k_*} & \hq(Z,Y)
\ar[r] & H_{q-1}(Y) \ar[r]  & H_{q-1}(Z) \ar[r]& \cdots
}\]\\[1mm]
���� 1�� ���Ͽ� ����� $\overline{f}_*$�� isomorphism�̰� ������
���������� �����ϸ� ���ϴ� ����� ��´�.
\end{pf}
\begin{lem}[Barratt-Whitehead Lemma]\hfill\\
Given long exact sequences,
\[
\xymatrix @=2em   @*[c] { %
\cdots \ar[r] & C_{i+1} \ar[r] \ar[d]^{\gam_{i+1}} & A_i
\ar[r]^{f_i} \ar[d]^{\alp_i} & B_i \ar[d]^{\bet_i} \ar[r]^{g_i}
& C_i \ar[r]^{h_i} \ar[d]^{\gam_i}_\cong & A_{i-1} \ar[d]^{\alp_{i-1}} \ar[r]& \cdots \\
\cdots  \ar[r]  & C_{i+1}' \ar[r] & A_i' \ar[r]^{f_i'} & B_i'
\ar[r]^{g_i'} & C_i' \ar[r]^{h_i'}  & A_{i-1}' \ar[r] & \cdots
}\]\\
If $\gam_i$ are all isomorphisms, then we have a long exact
sequence,
\[
\xymatrix @=1em @R=1mm @*[c] { %
\cdots \ar[r] & A_i \ar[r]^<<<<<{\phi_i} & A_i'\bigoplus B_i
\ar[r]^{\psi_i}
&B_i' \ar[r]^{\bd_i} & A_{i-1} \ar[r] & \cdots \\
& a \ar@{|->}[r] & (\alp(a), f(a))&&&\\
&&(a',b)\ar@{|->}[r]& \bet(b)-f'(a')&&\\
&&&b' \ar@{|->}[r]&h\gam^{-1}g'(b')& }\]
\end{lem}
\begin{pf} Diagram chasing.
\end{pf}


Example 1. Let $(X,A)$ be a collared pair and Y=\{*\}, a point.
Then
$\xf =X/A$ by definition. \\
���� 1 $\Rightarrow \overline{f}_*: \hq(X,A) \overset{\cong}{\to }
\hq(Z,Y)=\hq(X/A,*)=\hhq(X/A)$.\\
���� 2 $\Rightarrow \cdots\to\hhq(A)\to \hhq(X) \to \hhq(X/A) \to
\hh_{q-1}(A)\to \cdots$.\\

2. Wedge (or one point union) of $(X,x)$ and $(Y,y)$, where
$(X,x)$ and $(Y,y)$ are collared pairs.\\
$X\vee Y =X\coprod Y /x\sim y$.\\
$A=\{x\}$, $f:\{x\}\to \{y\}\subset Y$�� �θ�, ���� 2�� ���� exact
sequence,
\[
\xymatrix @=1em   @*[c] { %
\cdots \ar[r] & \hhq(x) \ar[r] & \hhq(X)\bigoplus\hhq(Y) \ar[r] &
\hhq(X\vee Y)\ar[r] & \cdots
}\]\\
�� ��� $\hhq(x)=0$�̹Ƿ� $\hhq(X\vee Y)\cong
\hhq(X)\bigoplus\hhq(Y)$�̴�.\\

3. $(X,A)=(D^n,S^{n-1})$ is a collared pair.
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Then $Z=D^n\cup_f Y$ is a space obtained from $Y $ by attaching an
$n$-cell by $f:S^{n-1}\to Y$. For example, $\snn=D^n \cup_f
\{$point$\}$, $T^2=D^2 \cup_f \{$figure eight$\}$, and
$\mathbb{P}^n=D^n \cup_f \mathbb{P}^{n-1}$.\\

$Z$�� homology�� ���غ���. MV-sequence�κ���
$$ \cdots\to\hhq(S^{n-1}) \to \hhq(Y) \to \hhq(Z) \to \hh_{q-1}(S^{n-1})\to
\cdots$$ �� ��´�. ����, $q\neq n, n-1$�� ��쿡��
$\hhq(Z)\cong\hhq(Y)$���� �� �� �ִ�. ����, $$0\to \hh_n(Y)\to
\hh_n(Z) \to \hh_{n-1}(S^{n-1}) \overset{f_*}{\to} \hh_{n-1}(Y)
\to \hh_{n-1}(Z) \to 0$$���� $\hh_{n-1}(S^{n-1})\cong\zb$�� free
abelian group�̹Ƿ�, $\hh_n(Z)=\hh_n(Y)\bigoplus ker f_*$,
$\hh_{n-1}(Z)=\hh_{n-1}(Y)/imf_*$���� �� �� �ִ�.\\

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Examples.\\
$T^2$=\raisebox{-1.8cm}{
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,        $Y$=\raisebox{-1.8cm}{
\begin{pspicture}(-.1,-.1)(1.1,.5)%
\pscircle(.25,.25){.25} \pscircle(.25,.75){.25}
\rput(.6,.25){$a$}\rput(.6,.75){$b$}
\end{pspicture}}\\[3mm]

$f_*:\hh_1(S^1)\to \hh_1(Y)$�� $\hh_1(S^1)\cong \zb$�� generator
$c$�� $a+b-a-b=0$���� �����Ƿ� zero map�̴�. ����,
$H_2(T^2)=H_2(Y)\bigoplus \zb=\zb$,
$H_1(T^2)=H_1(Y)/0=\zb\bigoplus\zb$�̴�.\\[2mm]


$K$=\raisebox{-1.8cm}{
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    \rput(-.1,.5){$b$}
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\end{pspicture}}
,        $Y$=\raisebox{-1.8cm}{
\begin{pspicture}(-.1,-.1)(1.1,.5)%
\pscircle(.25,.25){.25} \pscircle(.25,.75){.25}
\rput(.6,.25){$a$}\rput(.6,.75){$b$}
\end{pspicture}}\\[3mm]

$f_*(c)=a+b+a-b=2a$�̹Ƿ� $H_2(K)=H_2(Y)\bigoplus ker f_*=0$,
$H_1(K)=H_1(Y)/imf_*=\zb\bigoplus\zb/2$�̴�.\\[2mm]

�������� ������� $\mathbb{P}^2=D^2\cup_f S^1$�� ��쿡��
$f_*(c)=2a$�̹Ƿ�, $H_2(\mathbb{P}^2)=0$,
$H_1(\mathbb{P}^2)=\zb/2$���� �� �� �ִ�.\\

{\bf ���� 6.} Compute the followings.\\
(1) $H_*(\Sigma_g)$\\
(2) $H_*(N_k)$\\
(3) $H_*(D^2\cup_f S^1)$, where $f:\bd D^2=S^1\to S^1$ is given by $f(z)=z^3$\\
(4) $H_*(\mathbb{P}^n)$\\

\end{document}
