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\begin{document}
\parindent=0cm
\section*{VI Simplicial Homology}
\begin{defn}
(1) An \key{ordered simplex} is a simplex $\sig$ together with a
particular order of vertices of $\sig$ and will be denoted by
$\sig=(v_0,\cdots, v_p)$.\\
(2) An \key{orientation} of a simplex $\sig=(v_0,\cdots, v_p)$ :
Two orderings of vertices of $\sig$ are equivalent if they differ
by an even permutation. A choice of an equivalence class is called
an orientation of $\sig$.\\
(3) An \key{oriented simplex} is a simplex $\sig$ together with an
orientation of $\sig$ and will be denoted by $[v_0,\cdots,v_p]$
representing the equivalence class of $(v_0,\cdots, v_p)$ as an
orientation of $\sig$.
\end{defn}
���� ���, 2-simplex�� ��� $[v_0,v_1,v_2]=[v_1,v_2,v_0]$�̴�.\\

�׸����� ���� ������ ���� �� simplex�� �ΰ��� orientation��
�����Ѵ�.\\

\begin{figure}[h]
\begin{center}
\psset{unit=1cm}
\parbox{2.7cm}{
\begin{pspicture}(1.4,1.4)%
\rput(.2,.2){\psline{->}(0,0)(.55,.55) \psline{*-*}(0,0)(1,1)}
\rput(0,0){$v_0$} \rput(1.4,1.4){$v_1$}
\rput(1.5,0.3){$[v_0,v_1]$}
\end{pspicture}}
or \hspace{1em}
\parbox{2.7cm}{
\begin{pspicture}(1.4,1.4)%
\rput(.2,.2){\psline{-<}(0,0)(.55,.55) \psline{*-*}(0,0)(1,1)}
\rput(0,0){$v_0$} \rput(1.4,1.4){$v_1$}
\rput(1.5,0.3){$[v_1,v_0]$}
\end{pspicture}}\\[6mm]

\psset{unit=1cm} \parbox{2.7cm}{
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\rput(1,0.6){\psarc{->}(0,0){0.3}{100}{440}}
\rput(1,-0.7){$[v_0,v_1,v_2$]}
\end{pspicture}}
or \hspace{1em}
\parbox{2.7cm}{
\begin{pspicture}(0,-.5)(2.5,2)%
\pspolygon[fillstyle=solid,fillcolor=lightgray](0,0)(2,0)(1,1.732)
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\rput(1,0.6){\psarc{<-}(0,0){0.3}{100}{440}}
\rput(1,-0.7){$[v_0,v_2,v_1$]}
\end{pspicture}}
\end{center}
\caption{Orientations}
\end{figure}


{\bf Notation} $\sig$ : an oriented simplex $\Rightarrow$
$\overline{\sig}$ : an oriented simplex with opposite orientation.

\begin{defn}[$p$-th chain group]
$K$ : a simplicial complex\\
$\ch{p}$= the abelian group generated by the oriented
$p$-simplices with the relation $\overline{\sig}=-\sig$.
\end{defn}
��, $\ch{p}$�� $K$�� $p$-simplex��� generate�� free abelian
group���� �� �� �ִ�. �Ϲ������� $p\leq 0$�� ��쿡��
$\ch{p}=0$���� �����Ѵ�. ����, $\ch{p}$�� ���Ҹ� $p$-chain�̶��
�θ��� $\displaystyle{\sum^k_{i=1}}n_i\sig_i,\ n_i\in \zb$��
�����ϰ� ǥ���ȴ�.\\

������ free abelian group�� universal property�κ��� �����ϴ�.\\

{\bf Note.} A function $f$ from the oriented $p$-simplices of $K$
to an abelian group $G$ extends to a homomorphism : $\ch{p} \to G$
uniquely if $f(\overline{\sig})=-f(\sig)$ for all oriented $\sig$.\\

{\bf \underline{Boundary operation}} \\[2mm]
$\partial_p : \ch{p}\to\ch{p-1}$, boundary operator is defined as
follows.
\begin{displaymath}
\sig=[v_0,\cdots, v_p] \Rightarrow
\partial_p\sig=\sum^k_{i=0}(-1)^i[v_0,\cdots,\widehat{v}_i,\cdots,v_p]
\end{displaymath}
$\partial_p$�� orientation�� �����ϴ� equivalence class�� ���Ͽ�
well-defined���� Ȯ���ϱ� ���ؼ���
$\partial_p\sig$�� well-defined�̰� \\
$\partial_p(\overline{\sig})=-\partial_p\sig$���� ���̸� �Ǵµ�,
�̴�
transposition�� ���Ͽ� ��ȣ�� �ݴ밡 ���� ���̸� ����ϴ�.\\
 ��,$\partial
[v_0,\cdots,v_j,v_{j+1},\cdots,v_p]=-\partial[v_0,\cdots,v_{j+1},v_j,\cdots,v_p]$����
Ȯ���ϸ� �ȴ�. �׷��� �̴� $\partial$�� ���ǿ� ���� �ڸ��ϹǷ�,
$\partial_p$�� well-defined homomorphism�� �ȴ�.

\begin{lem}
$\partial_{p-1}\partial_p=0$ ($\partial^2=0$)
\end{lem}
\begin{proof}
$\partial_{p-1}\partial_p[v_0,\cdots,v_p]=\partial_{p-1}\displaystyle{\sum^k_{i=0}}
(-1)^i[v_0,\cdots,\widehat{v}_i,\cdots,v_p]\\
\hspace*{5ex}=\displaystyle{\sum^k_{i=0}}(-1)^i\partial_{p-1}[v_0,\cdots,\widehat{v}_i,\cdots,v_p]\\
\hspace*{5ex}=\displaystyle{\sum^k_{j<i}}(-1)^i(-1)^j[v_0,\cdots,\widehat{v}_j,\cdots,\widehat{v}_i,\cdots,v_p]\\
\hspace*{9ex}+\displaystyle{\sum^k_{i<j}}(-1)^i(-1)^{j-1}[v_0,\cdots,\widehat{v}_i,\cdots,\widehat{v}_j,\cdots,v_p]\\
\hspace*{5ex}=0$
\end{proof}

\newpage
{\bf Remark}\\
For convenience, we can add more generators and relations to
$\ch{p}$. \\
If $v_0,\cdots,v_p$ are vertices not necessarily distinct of some
simplex, we define
\begin{displaymath}
[v_0,\cdots,v_p]=\{\begin{array}{ll} 0 & \textrm{if not
distinct.}\\ \textrm{as before}&\textrm{if distinct}\end{array}.
\end{displaymath}
and define $\partial$ by the same formula
\begin{displaymath}
\partial_p\sig=\sum^k_{i=1}(-1)^i[v_0,\cdots,\widehat{v}_i,\cdots,v_p]
\end{displaymath}
Then this is well defined and $\partial^2=0$.

\begin{defn}
$\cdots \to \ch{p+1} \overset{\partial_{p+1}}{\to}\ch{p}
\overset{\partial_{p}}{\to}\ch{p-1}\to \cdots$ (:=$\{C_p,
\partial\}$) with $\partial^2=0$ \\
\hspace*{7ex}  is called a chain complex.\\
Define $\cy{p}$ = $ker \partial_p$ = the group of $p$-cycles.\\
\hspace*{7ex}$\bd{p}$ = $im \partial_p$ = the group of $p$-boundaries.\\
\hspace*{7ex}$\ho{p}$ = $\cy{p}/\bd{p}$ = the $p$-th homology
group.\\
\end{defn}

��� object�� ���Ͽ� homology group�� ����� ����.\\

1. (0) $K=$ \psset{unit=1cm} \parbox{2.7cm}{
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\pspolygon(0,0)(2,0)(1,1.732) \psline{->}(1,1.732)(0.5,0.866)
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\rput(1,1.94){$v_1$} \rput(-0.2,-0.2){$v_2$}\rput(2.2,-0.2){$v_3$}
\rput(0.1,1){$\sig_1$}
\rput(1,-0.3){$\sig_2$}\rput(1.9,1){$\sig_3$}
\end{pspicture}}\psset{unit=0.5cm}
(=$\partial$\parbox{1cm}{\begin{pspicture}(0,0)(2,1.8)%
\pspolygon[fillstyle=solid,fillcolor=lightgray](0,0)(2,0)(1,1.732)
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 \end{pspicture}})\\[4mm]


���� $\ch{1}=\zb\langle\sig_1,\sig_2,\sig_3\rangle=\zb^3$,
$\ch{0}=\zb\langle v_1,v_2,v_3\rangle=\zb^3$���� �� �� �ְ�, chain
complex
\begin{displaymath}
0\to \ch{1} \overset{\partial}{\to} \ch{0}\to 0
\end{displaymath}
�� ��´�. $\ho{0}$�� ���ϸ�, ���� $\ch{0}$�� $\partial$�� ���Ͽ�
��� 0�� �ǹǷ�, $\cy{0}=\ch{0}$���� �� �� �ִ�. ����, $\partial
\sig_1=v_2-v_1$, $\partial \sig_2=v_3-v_2$, $\partial
\sig_3=v_1-v_3$�̹Ƿ�, $\bd{0}$�� $\{v_2-v_1,v_3-v_2,v_1-v_3\}$��
generate�Ǵ� $\ch{0}$�� subgroup�̴�. �� subgroup�� ������
2�̹Ƿ�, $\ho{0}=\cy{0}/\bd{0}=\zb$���� �� �� �ִ�. \\

���������� $\bd{0}$�� ���ϴ� ���Ҵ� 1-simplex�� boundary�� �ǹǷ�,
$\ho{0}$������ 1-simplex�� ����Ǿ� �ִ� vertex���� ����
equivalent�� ������
���� ������, �Ϲ������� ���� ����� �����Ѵ�.\\
 {\bf Note}.
$\ho{o}\cong\zb^k$, $k$= number of connected
components of $|K|$.\\

�������� $\ho{1}$�� ���غ���. ���� $\bd{1}=0$�̹Ƿ�
$\ho{1}=\cy{1}=ker\partial$�̴�. ������ $c\in \ch{1}$�� ���Ͽ�,
$c=n_1\sig_1+n_2\sig_2+n_3\sig_3$�� �� �� �ִµ�, $\partial c
=0$�� �Ǳ� ���ؼ��� \\
\hspace*{5em} $\partial
c=n_1\partial\sig_1+n_2\partial\sig_2+n_3\partial\sig_3\\
\hspace*{6em}=n_1(v_2-v_1)+n_2(v_3-v_2)+n_3(v_1-v_3)\\
\hspace*{6em}=(n_3-n_1)v_1+(n_1-n_2)v_2+(n_2-n_3)v_3=0$\\
�� �Ǿ�� �ϹǷ�, $n_1=n_2=n_3:=n$���� �� �� �ִ�. ����
$\ho{1}=\cy{1}=\zb\langle\sig_1+\sig_2+\sig_3\rangle\cong\zb$�̴�.\\

(1) $K_1=$ \psset{unit=0.7cm} \parbox{2.7cm}{
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\psline{->}(2,0)(1.5,0.866)}

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$K_1$�� connected�̹Ƿ�, $\ho[K_1]{0}=\zb$�̴�. $\ho[K_1]{1}$��
���ϱ� ���Ͽ� $\cy[K_1]{1}$�� �˾ƺ���. ���� (0)���� ������ �ٸ�
������ 1-chain $c$�� �����ϸ�, $\partial c=0$�̱� ����
�ʿ���������� $c$�� ���ϴ� �� vertex�� ���Ͽ� ������
1-simplex�� ����� ������ 1-simplex�� ����� ��ġ�ؾ� �Ѵٴ�
������ �� �� �ִ�. ��,
\begin{center}
\psset{unit=1cm}
\parbox{2.5cm}{\begin{pspicture}(2.5,2)%
\psline(0,2)(1,1) \psline(0,0)(1,1) \psline(1,1)(2,1)
\psline{->}(0,2)(0.5,1.5) \psline{->}(0,0)(0.5,0.5)
\psline{->}(1,1)(1.5,1) \rput(0.7,0.3){$\sig_1$}
\rput(0.7,1.7){$\sig_2$}\rput(1.5,1.2){$\sig_3$}
\end{pspicture}}
$c=n_1\sig_1+n_2\sig_2+n_3\sig_3+\cdots$
\end{center}
�׸��� ���� $c$�� �־��� ��쿡 $n_1+n_2=n_3$�� �Ǿ�� �Ѵ�.
����, $K_1$�� �׸����� ���� �ﰢ���� ���� cycle�� $c_1$, ������
�ﰢ���� ���� cycle�� $c_2$��� �ϸ� $K_1$�� 1-cycle��
$nc_1+mc_2$�� �÷� ǥ���ȴ�. $\bd[K_1]{1}=0$�̹Ƿ�,
$\ho[K_1]{1}\cong \zb^2$�̴�.\\

(2) $K_2=$ \psset{unit=0.7cm} \parbox{3cm}{
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\rput{30}(2,0){\pspolygon(0,0)(2,0)(1,1.732) \psline{-<}(0,0)(1,0)
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\end{pspicture}}\\

���������� $\ho[K_2]{0}=\zb$�̰�, 1-cycle�� $nc_1+mc_2$�� �÷�
ǥ���ǹǷ� $\ho[K_2]{1}\cong \zb^2$�̴�.\\

(3) $K_3=$ \psset{unit=0.7cm} \parbox{3cm}{
\begin{pspicture}(2.5,2)%
\rput{90}(2,0){\pspolygon(0,0)(2,0)(1,1.732)
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���������� $\ho[K_3]{0}=\zb^2$�̰�, $\ho[K_3]{1}\cong \zb^2$���� Ȯ���� �� �ִ�. \\

�Ϲ�������, $K$�� 1-dimensional ������ ������ $k$��� ���� ��,
$\ho{1}=\zb^k$�� �־����� ���� ���� ������� Ȯ���� �� �ִ�. \\

{\bf ���� 19.} $G$: Graph $\Rightarrow$
$\ho[G]{1}=abel(\pi_1(G))$.\\

���� ������ $K_1$�� $K_2$�� homology group�� ��ġ���� ������ ��
�ִµ�, �Ϲ������� homotopically equivalent�� object���� homology
group�� ������ ��ġ�ϰ� �ȴ�.\\

2. (1) $K=$ \psset{unit=1cm} \parbox{2.7cm}{
\begin{pspicture}(-0.2,-0.2)(2.5,2)%
\pspolygon[fillstyle=solid,fillcolor=lightgray](0,0)(2,0)(1,1.732)
\rput(1,0.6){$\tau$}\rput(1,0.6){\psarc{->}(0,0){0.3}{100}{440}}
\psline{->}(1,1.732)(0.5,0.866) \psline{->}(0,0)(1,0)
\psline{->}(2,0)(1.5,0.866) \rput(1,1.94){$v_1$}
\rput(-0.2,-0.2){$v_2$}\rput(2.2,-0.2){$v_3$}
\rput(0.1,1){$\sig_1$}
\rput(1,-0.3){$\sig_2$}\rput(1.9,1){$\sig_3$}
\end{pspicture}} \\[4mm]

$K$�� connected�̹Ƿ� $\ho{0}=\zb$���� �ٷ� �� �� �ִ�.\\
$K$�� 1-cycle $c$�� �ռ� �˾ƺ� �ٿ� ����
$c=n(\sig_1+\sig_2+\sig_3)=:nc_1$�� ���̴�. �׷��� $c_1=\partial
\tau$�̹Ƿ� $c\sim 0$(homologous to zero)�̴�. ����
$\ho{1}=0$�̴�.\\
���� $\bd{2}=0$�̹Ƿ� $\ho{2}=\cy{2}=ker\partial$���� �� ��
�ִµ�, $\ch{2}$�� generator $\tau$�� ���Ͽ� $\partial
\tau=c_1\neq 0$�̹Ƿ� $ker\partial=0$�� �Ǿ� $\ho{2}=0$�̴�. \\


(2) $K=$ \psset{unit=0.5cm} \parbox{4cm}{
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\pspolygon[fillstyle=solid,fillcolor=lightgray](0,0)(1.5,4)(4.5,5)(6,0)(0,0)
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\psline{->}(0,0)(3,0)
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$\sig_1$}}
\end{pspicture}}
\parbox{8.5cm}
{2���� cell�鿡 �ð�ݴ�������� orientation��
�ش�.\\ $c_1$ : �ٱ��� cycle.\\ $c_2$ : ���� cycle.\\
$\tau = \sig_1+ \sig_2 +\cdots$ where $\sig_i$ are 2-dimensional cells. }\\[4mm]
 % �׿� ���� 1���� simplex�� orientation�� �����ȴ�.

$K$�� connected�̹Ƿ� $\ho{0}=\zb$���� �ڸ��ϴ�.
\begin{center}\psset{unit=0.7cm}
\begin{pspicture}(6,2)%
\pspolygon[fillstyle=solid,fillcolor=lightgray,linestyle=dashed](0,0)(2,0)(1,1.732)
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\end{center}
�� �׸��� ���� ������ �̿��Ͽ� $K$ ������ 1-cycle�� �ٱ�������
homologous�ϰ� �о �� �����Ƿ�, $K$�� essential�� 1-cycle��
$nc_1$�� ���� �ȴ�. ����,
$\ho{1}=\langle c_1\rangle=\zb$.\\
$K$�� ������ 2-chain $c=n_1\sig_1+n_2\sig_2+\cdots$�� ���Ͽ�
$\partial c = n_1\partial\sig_1+n_2\partial\sig_2+\cdots=0$�̸�,
�ռ� ������ �ٿ� ���� (�̹����� �� ���뺯������ incidence��
�����ϸ�) $n_1=n_2=\cdots$�̹Ƿ�, $c$�� cycle�̶��,
$c=n(\sig_1+\sig_2+\cdots)=n\tau$�� ���̴�. �׷���, $\partial
c=n\partial\tau=n(c_1-c_2)=0$�� �Ǿ�� �ϹǷ� $n=0$ ��, $c=0$�̴�.
����, $\ho{2}=0$�̴�.\\

�Ϲ������� boundary�� �ִ� manifold $K$�� ���Ͽ� �̿� ���� ������
���� $\ho{2}=0$�� ���� �� �� �ִ�.
\\

3. Surfaces\\
(0) $K=$ \psset{unit=1cm} \parbox{2.3cm}{
\begin{pspicture}(2,2.5)%
\psline(1,2)(0,0.6)(1,0)(1,2)(2,0.6)(1,0)
\psline[linestyle=dashed](0,0.6)(2,0.6)
\end{pspicture}}
$\approx S^2$ \hspace{0.5cm} \parbox{8cm}{orientable surface�̹Ƿ�
boundary���� ���� ���ǵ��� orientation�� �� �� �ִ�. }\\[4mm]

$\ho{0}=\zb$�̴�.\\
1-cycle�� ���� 2.(2)������ ���� ������ �о�� point��
homologous�ϰ� �ǹǷ� $\ho{1}=0$�̴�. \\
2-cycle�� �ռ���
���������� $c=n\tau$, $\tau=\sig_1+\sig_2+\sig_3+\sig_4$�� ����
�Ǿ�� �ϴµ�, $\partial c=n\partial \tau=0$(orientation��
boundary���� ���� ���ǵ��� �־����Ƿ�)�̹Ƿ�, $c$�� 2-cycle����
Ȯ���� �� �ִ�. ����, $\ho{2}=\langle\tau\rangle=\zb$�̴�. \\

�ڿ��� ���캸������, ���� orientable���� �ʴٸ� boundary����
���ǵ��� orientation�� �� �� �����Ƿ� 2-cycle�� �������� �ʾ�
$\ho{2}=0$�� �ȴ�.\\

(1) Torus\\
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\end{pspicture}}
\parbox{9cm}{Torus�� ������ trangulation�� �ְ�, boundary����
���ǵ��� orientation�� �ش�.}\\[4mm]

$\ho{0}=\zb$�̴�.\\
1-cycle�� �ٱ������� �о �� �����Ƿ�, boundary������ 1-cycle��
����. �ٽø���, $D_1=\cy{1}\cap\{$boundary chains$\}$�� �θ�,
$\cy{1}=D_1+\bd{1}$�̴�. boundary�� figure eight�� �����Ƿ�
$D_1=\cy{1}\cap\{$boundary chains$\} =\langle
a,b\rangle=\zb^2$�̴�. $\bd{1}\cap D_1$�� ���ϸ�, ������ 2-chain
$c$�� ���Ͽ� $\partial c$�� boundary�� ���ԵǱ� ���ؼ���
$c=n\tau$�� ���� �Ǿ�� �ϰ�, �� �� $\partial
c=n\partial\tau=n(b+a-b-a)=0$�̹Ƿ� $\partial$�� image�� �Ǵ�
1-cycle�� 0���̴�. ���� $\ho{1}=\cy{1}/\bd{1}=(D_1+B_1)/B_1=D_1/(B_1\cap D_1) =\zb^2$�̴�.\\
2-cycle�� �ռ��� ���������� $c=n\tau$�� ���� �Ǿ�� �ϴµ�,
$\partial c=n\partial \tau=0$�̹Ƿ� $c$�� 2-cycle����
Ȯ���� �� �ִ�. ����, $\ho{2}=\langle\tau\rangle=\zb$�̴�. \\

(2) $P^2=$ \psset{unit=1.5cm} \parbox{3cm}{
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\rput(0.6,0.8){\pscircle[fillstyle=solid,fillcolor=lightgray](0,0){0.6}
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\end{pspicture}}\\[4mm]

$\ho[P^2]{0}=\zb$�̴�. \\
1-cycle�� boundary������ �����ϸ� �Ǵµ� boundary�� circle��
�����Ƿ� $D_1=\langle a\rangle$�̴�. $\bd[P^2]{1}\cap D_1$��
���ϸ�, ������ 2-chain $c$�� ���Ͽ� $\partial c$�� boundary��
���ԵǱ� ���ؼ��� $c=n\tau$�̰�, �� �� $\partial
c=n\partial\tau=2na$�̹Ƿ� $\bd[P^2]{1}\cap D_1=\langle
2a\rangle$�� �Ǿ� �ᱹ
$\ho[P^2]{1}=\langle a\rangle/\langle 2a\rangle=\zb/2\zb=\zb_2$�̴�.\\
2-cycle�� �ռ��� ���������� $c=n\tau$�� ���� �Ǿ�� �ϴµ�,
$\partial c=n\partial \tau=2na$�̹Ƿ� $\partial c=0$�̸� $n=0$ ��,
 $c=0$�� �Ǿ� $\ho[P^2]{2}=0$�̴�. \\

(3) Klein Bottle\\
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    \psline{->}(0,1)(.55,1)
    \psline{-<<}(1,0)(1,0.55)}
\end{pspicture}}
$\ho{0}=\zb$�̴�. \\[4mm]

�ռ��� ���������� $D_1=\langle a,b\rangle$�̴�.
$\partial\tau=b+a-b+a=2a$�̹Ƿ� $\bd{1}\cap D_1=\langle
2a\rangle$�� �ǰ� �ᱹ
$\ho{1}=\langle a,b\rangle/\langle 2a\rangle=\zb_2\bigoplus\zb$�̴�.\\
���� $\partial \tau=2a\neq 0$�̹Ƿ� $\ho{2}=0$�̴�. \\

{\bf ����. 20} Compute
$\ho[\overset{g}{\underset{1}{\sharp}}T^2]{*}$ and
$\ho[\overset{k}{\underset{1}{\sharp}}P^2]{*}$

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