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\begin{document}
\parindent=0cm
\section*{VII.2 Definition of singular homology}
Let $e_{0}=(0,0,\cdots,0)$ in $\rb^{\infty}$  ( or $\rb^{N})$.\\
\hspace*{1.5em} $e_{1}=(1,0, \cdots,0)$\\
\hspace*{1.5em} $e_{2}=(0,1, \cdots,0)$\\
\hspace*{4.5em} \vdots\\
\hspace*{1.5em} etc.\\

$\bigt^{p}$ = the simplex spanned by $\{e_{0},e_{1}, \cdots ,
e_{p}\}$ = the standard $p$-simplex.\\

The $i$-th face map $f_{p}^{i} : \bigt^{p-1} \rightarrow
\bigt^{p}$ is
an affine map given by\\
\begin{displaymath}
f_{p}^{i}(e_{j}) = \{ \begin{array}{ll} e_{j} & j < i \\
e_{j+1} & j \geq i \end{array}.\\
\end{displaymath}
i.e. "$f_{p}^{i}(\bigt^{p-1}) = f_{p}^{i}(e_{0}, \cdots, e_{p-1})
=
(e_{0}, \cdots, \widehat{e_{i}}, \cdots, e_{p})$" $< \bigt^{p}$.\\

{\bf ��}\\
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\begin{center} $f^{0}(e_{0},e_{1}) = (e_{1}, e_{2})$
\\ $f^{1}(e_{0},e_{1}) = (e_{0}, e_{2})$ \\ $f^{2}(e_{0},e_{1}) =
(e_{0}, e_{1})$ \end{center}


Let $X$ be a topological space.\\
A singular $p$-simplex in $X$ is a map $\sigma : \bigt^{p}
\rightarrow X$.\\
$i$-th face $\sigma^{(i)}$ of $\sigma$ is a map : $\bigt^{p-1}
\overset{f^{i}}{\rightarrow} \bigt^{p}
\overset{\sigma}{\rightarrow} X$,i.e., $\sigma^{(i)} = \sigma
\circ
f_{p}^{i}$\\


$S_{p}(X)$�� singular $p$-simplices in $X$�� ���ؼ� generated�Ǵ�
free abelian group�̶�� ����.(more generally, the free $R$-module
generated by singular $p$-simplices in $X$) �Ǵ�,
the group(module) of singular $p$-chains�̶���� �Ѵ�.\\

�׷���, $c \in S_{p}(X)$�� finite sum, $\sum_{i=1}^{k}
n_{i}\sigma_{i}$�� �����ϰ�
ǥ���ȴ�.\\

���� boundary operator�� ��������.\\
Define a homomorphism $\partial : S_{p}(X) \rightarrow S_{p-1}(X)$
by
\begin{displaymath}
\partial\sigma = \sum^p_{i=0} (-1)^{i}\sigma^{(i)} = \sum^p_{i=0}
(-1)^{i} \sigma \circ f_{p}^{i}
\end{displaymath}
�Ǵ� informally,
\begin{displaymath}
\partial\sigma(e_{0}, \cdots, e_{p-1}) = \sum^{p}_{i=0}
(-1)^{i} \sigma(e_{0}, \cdots, \widehat{e_{i}}, \cdots, e_{p})
\end{displaymath}


{\bf Note}\\
1.$\partial^{2}$=0 : same as before.\\

2. {\bf ���� 21(check)}\hspace{0.5em} In general, $\sigma(e_{0},
\cdots, e_{i}, e_{i+1}, \cdots, e_{p}) \sim -\sigma(e_{0}, \cdots,
e_{i+1},
e_{i}, \cdots, e_{p})$\\


$p$�� 1�� ��쿡 ���ؼ� ���캸��.

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���� ���� �־��� $\tau$�� $\bar{\tau}$�� ���ؼ� $\bar{\tau} + \tau
\sim 0$���� ������. �ٽ� ���� � $c \in S_{2}(X)$�� ���ؼ�
$\bar{\tau} + \tau = \partial c$���� ���̸� �ȴ�. �Ʒ��� �׸���
���� $\sigma, \xi$�� �����ϸ�,

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���⼭ $\xi : \bigt^{2} \rightarrow X$�� $\xi(a)=p , \forall a \in
\bigt^{2}$�� constant map�̴�. ����, $\psi$�� 2-simplex��
1-simplex�� �׸��� ���� collapsing ��Ű�� map�̰� $\sigma$�� $\psi
\circ \tau$�� �����Ѵ�. �׷���, \\

$\partial\sigma = \sigma^{(0)} - \sigma^{(1)} + \sigma^{(2)} =
\sigma(\widehat{0},1,2) - \sigma(0,\widehat{1},2) +
\sigma(0,1,\widehat{2}) = \bar{\tau} - p +\tau$�̰� \\
$\partial\xi = p-p+p = p$�̹Ƿ�,\\
$\bar{\tau} + \tau = \partial\sigma + p = \partial\sigma +
\partial\xi = \partial(\sigma + \xi)$�� �Ǿ $\tau \sim
\bar{\tau}$���� �� �� �ִ�.\\


���� $X$�� singular chain complex�� ��������.\\
$S(X) = \{S_{p}(X),\partial\}$�� singular chain complex of $X$���
�ϰ�, \\
\begin{displaymath}
\cdots \rightarrow S_{p+1}(X) \overset{\partial}{\rightarrow}
S_{p}(X) \overset{\partial}{\rightarrow} S_{p-1}(X) \rightarrow
\cdots\\
\end{displaymath}


$p$-th singular homology group of $X$, $H_{p}(X)$��
\begin{center}
$H_{p}(X) := Z_{p}(X) / B_{p}(X)$ \\
\end{center}
where $Z_{p}(X) = ker \partial_{p}$ (whose element is called a $
cycle$) and $B_{p}(X) = im
\partial_{p+1}$ (whose element is called a $boundary$).\\

��� �����Ѵ�. �̶� $R$-module�� ��� ($H_{p}(X : R)$)�� �����ϱ�
���ؼ� $H_{p}(X : \zb)$��� ���⵵ �Ѵ�.\\



1. {\bf Functorial property for $H_{p}$}\\
Let $f : X \rightarrow Y$ be a map. Then $f$ induces a chain
map.\\


Define $f_{\sharp} : S(X) \rightarrow S(Y)$ by $\sigma \mapsto f
\circ \sigma$.Then\\

$\ulcorner f_{\sharp}(\partial \sigma) = f_{\sharp}(\sum
(-1)^{i}\sigma^{(i)}) = \sum (-1)^{i}f_{\sharp}\sigma^{(i)} = \sum
(-1)^{i}f \circ \sigma^{(i)} = \sum (-1)^{i}f\circ(\sigma\circ
f^{i}) = \sum (-1)^{i}(f \circ \sigma) \circ f^{i} = \partial(f
\circ \sigma) = \partial(f_{\sharp}\sigma)\\
\therefore \hspace{0.2em}f_{\sharp}\partial = \partial f_{\sharp} \lrcorner$\\

Hence $f_{\sharp}$ induces a homomorphism $f_{*}(=H_{p}(f)) :
H_{p}(X) \rightarrow H_{p}(Y)$.

(1) $id : X \rightarrow X \Rightarrow id_{*} = id$ \\
(2) $X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} Z
\Rightarrow (g \circ f)_{\sharp} = g_{\sharp} \circ f_{\sharp}
\Rightarrow (g \circ f)_{*} = g_{*} \circ f_{*}$\\


$\therefore \hspace*{0.2em} H_{p}$ is a covariant functor from the
category of topological spaces to the category of abelian groups
($R$-modules).\\


2. \begin{displaymath}{\bf \textrm{$H_p(point)$}} = \{ \begin{array}{ll} 0 & p \neq 0 \\
\zb & p = 0 \end{array} \end{displaymath}

$X$�� point�̹Ƿ� $S_{p}(X) = <c_p>$�̴�. ���⼭ $c_{p} :
\bigt^{p}
\rightarrow X$�� constant map�̴�. ��\\

\begin{center}
$\cdots \overset{\partial}{\longrightarrow} S_3
\overset{\partial}{\longrightarrow} S_2
\overset{\partial}{\longrightarrow} S_1
\overset{\partial}{\longrightarrow} S_0
\overset{\partial}{\longrightarrow} 0$\\
\end{center}
\hspace*{12.0em} $\Arrowvert \hspace{2.5em} \Arrowvert
\hspace{2.7em}
\Arrowvert$ \\
\hspace*{10.0em} $ <c_2> \hspace{0.5em} <c_1> \hspace{0.5em}
<c_0> $ \\


�̴�. �׷��� ���⼭ $\partial c_{1} = c_0 - c_0$ �̹Ƿ� $S_1
\overset{\partial=0}{\rightarrow} S_0$�� �Ǿ� $H_{0}=\zb$�̴�.
���� $\partial c_{2} = c_1 - c_1 + c_1 = c_1$�̹Ƿ� $S_2
\overset{\partial(\cong)}{\rightarrow} S_1$�� �Ǿ� $H_{1}=0$�̴�.
�̿� ���� ������� $p>2$�� ���ؼ� $H_{p}=0$���� �� �� �ִ�.\\


3. $\{X_{\alp}\}$ : path-components of $X$\\ $\Rightarrow S_{p}(X)
= \underset{\alp}{\bigoplus} S_{p}(X_{\alp})$ since $\bigt^{p}$ is
connected and $\forall c \in S_{p}(X)$ can be written uniquely as
$\sum c_{\alp}$. Moreover,
$\partial : S_{p}(X_{\alp}) \rightarrow S_{p-1}(X_{\alp})$.\\
$\Rightarrow Z_{p}(X)= \underset{\alp}{\bigoplus}Z_{p}(X_{\alp})$
and $B_{p}(X)= \underset{\alp}{\bigoplus}B_{p}(X_{\alp})$\\
$\Rightarrow H_{p}(X)=
\underset{\alp}{\bigoplus}H_{p}(X_{\alp})$\\



4. Let $X$ be path-connected. Then $H_{0}(X) = \zb (or R)$.\\
\begin{proof}
{\bf (Idea)}\\
$x_{0} \in X$�� ������Ű��. �׷��� $x_{0}$�� ������ $x \in X$
���̿��� �� ���� �մ� path�� �����ϰ�, 1-simplex $\rho : \bigt^{1}
\rightarrow X$�� image�� �� path��� �ϸ� $\partial \rho = x -
x_{0}$ �� �ǰ� ���� $x \sim x_{0}$�� �����Ѵ�.\\

�ڼ��� ������ ���캸��.\\
������ $c \in S_{0}(X)$�� $\underset{finite}{\sum} n_{i} x_{i}$��
ǥ���� �� �ִ�. ���⼭ $x_{i}$�� $e_{0}$�� $x_{i} \in X$��
�����ִ� 0-simplex�̴�.\\ �׷��� �켱 $c=\partial c_{1}
\Leftrightarrow \sum n_{i}=0$���� ������.\\
$\because (\Rightarrow) c_{1}=\sum k_{i} \sigma_{i} \Rightarrow
c=\partial c_{1} = \sum k_{i}(\sigma_{i}(1)-\sigma_{i}(0))
\Rightarrow \sum n_{i} = \sum (k_{i}-k_{i})=0$\\
\hspace*{4.0em}���⼭ $\sigma_{i}$�� 1-simplex�̴�.\\
\hspace*{0.8em} $(\Leftarrow) c=\sum n_{i} x_{i} - (\sum
n_{i})x_{0} = \sum n_{i}(x_{i} - x_{0}) = \sum n_{i} \partial \rho_{i}$\\
\hspace*{4.0em}���⼭ $X$�� path-connected��� ������ ������.\\

���� $S_{0}(X) \overset{\eps}{\rightarrow} \zb$ (or $R$) $\sum
n_{i} x_{i} \mapsto \sum n_{i}$�� �����غ���. �׷��� $\eps$��
onto�̰�, $\eps \circ \partial_{1}= 0$�̴�. ���� �տ��� ������
���� �̿��ϸ�, ���� $X$�� path-connected��� $ker \eps = im
\partial_{1}$�� �Ǿ� $S_{1} \overset{\partial_{1}}{\rightarrow}
S_{0} \overset{\eps}{\rightarrow} \zb \rightarrow 0$�� $S_{0}$����
exact�̴�. ���� $H_{0}(X)=S_{0}/ker \eps = \zb$�� �ȴ�.

\end{proof}

\begin{defn}

Given a chain complex $\{C,\partial\}$, an epimorphism $C_{0}
\rightarrow \zb \rightarrow 0$ with $\eps \circ \partial_{1} = 0$
is called an {\bf augmentation}. And the homology of $\cdots
\rightarrow S_{p}(X) \rightarrow \cdots
\overset{\partial}{\rightarrow} S_{0}(X)
\overset{\eps}{\rightarrow} \zb \rightarrow 0$ is called a {\bf
reduced homology of $X$} and denoted by $\widetilde{H_{p}}(X)$.

\end{defn}


{\bf Note}\\
1. $\widetilde{H_{p}}(X) = H_{p}(X)$ if $p \geq 1$\\
2. $H_{0}(X) \cong \widetilde{H_{0}}(X) \bigoplus \zb$, since
$S_{0}(X) \cong ker \eps \bigoplus \zb$ and $im \partial_{1}
\subset ker \eps$.\\
3. $\widetilde{H_{p}}(point)= 0 \hspace{1.0em} \forall p$.


\begin{defn}

A chain complex $\{C_{p},\partial\}$ is called {\bf acyclic} if
$H_{p}(C)=0 \hspace{1.0em} \forall p$.\\
An augmented chain complex $\{C_{p},\partial, \eps\}$ is called
{\bf acyclic} if $\widetilde{H_{p}}(C)=0 \hspace{1.0em} \forall p$. i.e.,\\
\begin{displaymath}
\cdots \rightarrow C_{p+1} \overset{\partial}{\rightarrow} C_{p}
\overset{\partial}{\rightarrow} C_{p-1} \rightarrow \cdots
\end{displaymath}
is acyclic if and only if it is exact.\\

\end{defn}

{\bf Example} $\{S_{p}(pt.), \partial, \eps\}$ is acyclic.

\end{document}
