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\newcommand{\aff}{\mbox{\it aff}}


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\newcommand{\td}[1]{\widetilde{#1}}
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\begin{document}
\parindent=0cm
\section*{VII.3 Homotopy invariance of homology}

Goal. $f\simeq g : X\to Y \Rightarrow f_* =g_* : H_p(X)\to
H_p(Y)$.
\begin{defn}[Chain homotopy]
Let $f, g : \cc\to\cc{}'$ be chain maps i.e.,\\ $ f=\{f_p : C_p\to
C_p' ~|~ f_{p-1}\bd_p=\bd_p {}'f_p\}$,  $ g=\{g_p : C_p\to
C_p' ~|~ g_{p-1}\bd_p=\bd_p {}'g_p\}$. \\
A \key{Chain homotopy} $D$ between $f$ and $g$ ($D:f\simeq g$) is
a collection of homomorphisms $\{D_p : C_p\to C_{p+1}{}'\}$ such
that $\bd_{p+1}{}'D_p +D_{p-1}\bd_p=f_p-g_p$. (simply denote by
$\bd D +D\bd=f-g$)
\end{defn}

\begin{thm}
If $f\simeq g : \cc\to\cc{}'$, then $f_* =g_* : H_p(\cc)\to
H_p(\cc{}')$.
\end{thm}
\begin{proof}
$z\in Z_p(C) \Rightarrow \bd Dz =f(z)-g(z) \Rightarrow f(z)\sim g
(z)\Rightarrow f_*[z]=g_*[z]$
\end{proof}

Special case : $\cc=\cc{}',\ f=id,\ g=0$\\
Then a chain homotopy $D$ between $id$ and 0 is called a chain
contraction and if $\cc$ has a chain contraction, then $\cc$ is
said to be \key{chain contractible}. In this case, $\bd D+D\bd=1$.

\begin{thm}
If $\cc$ is a free chain complex, i.e., each $C_p$ is a free
abelian group,\\ then $\cc$ is acyclic $\Leftrightarrow$ $\cc$ is
chain contractible.
\end{thm}
\begin{proof}
$\Leftarrow$) by theorem 1, $id_*=0 \Rightarrow H_p(C)=0\ \forall
p$\\
$\Rightarrow$) Define $D_p$ inductively.\\
\[
\xymatrix @=2em @*[r] { %
\cdots \ar[r] & C_2 \ar[d]^{id} \ar[r] & C_1
\ar[d]^{id} \ar[r] \ar@{.>}[dl]^{\exists D_1}&
C_0 \ar[d]^{id} \ar[r] \ar@{.>}[dl]^{\exists D_0}& 0 \ar@{.>}[dl]^{0}&  \textrm{: free}\\
\cdots \ar[r] & C_2 \ar[r] & C_1 \ar[r] & C_0\ar[r] & 0 &
\textrm{: acyclic}}
\]\\[1mm]
�׸������� ���� 0���� $C_0$�� ���� zero homomorphism�� �����ϰ�
�ͳ������� $D$�� �����Ѵ�. \\
$C_p$�� free�̹Ƿ�, $D$�� �������� ���̱� ���ؼ��� $C_p$��
generator�� ���� ������ �����ϸ鼭 $C_{p+1}$�� ���� �Լ���
�������� ���̸� �ȴ�. ��, �� generator $g$�� ���Ͽ� $\bd
D(g)=g-D\bd(g)$�� �����ϴ� $C_{p+1}$�� ���� $D(g)$�� ������Ű��
�ȴ�. (Induction�� ��������
�캯�� $D$�� �̹� ���ǵ� ���̰� �º��� $D$�� �����Ͽ��� �Ѵ�.)\\
�� diagram���� �Ʒ��� sequence�� exact�̹Ƿ�,
$\bd(g-D\bd(g))=0$�̸� $\bd (h)=g-D\bd(g)$�� $h\in C_{p+1}$��
�����ϰ� $D(g)=h$��
�����Ѵ�. \\
����, $\bd(1-D\bd)=0$���� Ȯ���ϸ� �ȴ�. �׷���,
\[\bd(1-D\bd)=\bd-\bd D\bd=
\bd-(1-D\bd)\bd=\bd-\bd+D\bd^2=0\]
�̹Ƿ� ������ ������.
\end{proof}

{\bf Note. Comparison theorem in homological algebra}\\
(1) Existence of chain map lifting a given $k:\zb\to\zb$.
\[
\xymatrix @=2em @*[r] { %
\cc & \cdots \ar[r] & C_2 \ar@{.>}[d]^{\exists k_2} \ar[r]^{\bd} &
C_1 \ar@{.>}[d]^{\exists k_1} \ar[r]^{\bd} & C_0
\ar@{.>}[d]^{\exists k_0} \ar[r]^{\bd} &
\zb \ar[d]^{k} \ar[r]^{\bd} & 0 &  \textrm{: free}\\
\cc{}'& \cdots \ar[r] & C_2{}' \ar[r]^{\bd} & C_1{}' \ar[r]^{\bd}
& C_0{}' \ar[r]^{\bd} & \zb \ar[r]^{\bd} & 0 & \textrm{: acyclic}}
\]

$\bd k_{p+1}=k_p\bd$�� $k_p$���� �������� ���̷��� ���� 2������
���������� $\bd k_p\bd=0$���� Ȯ���ϸ� �ȴ�. $\bd
k_p\bd=k_{p-1}\bd^2=0$�̹Ƿ� �ͳ������� $k_p$�� ������ ���� ��
�ִ�.\\

(2) Existence of chain homotopy between two liftings.
\[
\xymatrix @=2.5em @*[r] { %
\cc & \cdots \ar[r] & C_2 \ar[d]^<<<{k_2,k_2{}'} \ar[r] & C_1
\ar[d]^<<<{k_1,k_1{}'}  \ar[r] \ar@{.>}[dl]^{\exists D_1}&
C_0 \ar[d]^<<<{k_0,k_0{}'}  \ar[r] \ar@{.>}[dl]^{\exists D_0}& \zb \ar[r]\ar@{.>}[dl]^{0}\ar[d]^{k}& 0& \textrm{: free}\\
\cc{}'& \cdots \ar[r] & C_2 \ar[r] & C_1 \ar[r] & C_0\ar[r] & \zb
\ar[r] &0 & \textrm{: acyclic}}
\]
���� $D: \zb\to C_0$�� $D=0$���� �ְ�, $\bd D =k_p-k_p{}'-D\bd$��
�ǵ��� $D$�� �ͳ������� �����س�����. ���� �ռ��� ����������
$\bd(k_p-k_p{}'-D\bd)=0$���� Ȯ���ϸ� �ȴ�. \\
�׷���, $\bd(k_p-k_p{}'-D\bd)=\bd k_p-\bd k_p{}'-\bd
D\bd=k_p\bd-k_p{}'\bd-\bd D\bd=(k_p-k_p{}'-\bd
D)\bd=D\bd\bd=0$�̹Ƿ� ������ ������.

\begin{lem}
$X$ is a star shaped subset of $\rb^n$ $\Rightarrow \td{H}(X)=0$.
\end{lem}
\begin{proof}
We may assume $X$ is star shaped from the origin. Construct a
chain contraction $D$:
\[
\xymatrix @=2.5em @*[r] { %
\cdots \ar[r] & S_2 \ar[d]^{id} \ar[r]^{\bd} & S_1 \ar[d]^{id}
\ar[r]^{\bd} \ar@{.>}[dl]^{D_1}& S_0 \ar[d]^{id} \ar[r]^{\eps}
\ar@{.>}[dl]^{ D_0}
& \zb \ar[r] \ar@{.>}[dl]^{D_{-1}}\ar[d]^{id}& 0 \\
\cdots \ar[r] & S_2 \ar[r] & S_1 \ar[r] & S_0\ar[r] & \zb \ar[r]
&0 }
\]

%���� $D_{-1}$�� $D_{-1}(1)=$constant map to the origin���� �����Ѵ�.
Define $D_{-1}(1)=$constant map to the origin. \\
Given $\sig:\triangle^p\to X$, define $D(\sig)\in S_{p+1}(X)$ by
$D(\sig)(se_0+(1-s)t)=(1-s)"\sig"(t)$, where $s\in I$ and
$t\in\langle e_1,\cdots, e_{p+1}\rangle$. (�ٽø���
$D(\sig)(e_0)=0$ and $D(\sig)\circ f^0_{p+1}=\sig$.

$p=0$�� ��쿡�� $D(\sig)^{(0)}=\sig$, $D(\sig)^{(1)}=0$�̹Ƿ�,
$\bd D(\sig)=\sig-D_{-1}(1)=\sig-D_{-1}\eps(\sig)$�̴�. ����,
$\bd D+D\eps=1$�� �����Ѵ�.\\

$p>0$�� ��쿡�� $D(\sig)^{(0)}=\sig$�̰�,
$D(\sig)^{(i+1)}=D(\sig^{(i)})$�� �����Ѵ�.\\
$(\because D(\sig)^{(i+1)}(e_1,\cdots,e_p)=D\sig
(e_1,\cdots,\hat{e}_{i+1},\cdots, e_{p+1})\\
\hspace*{3ex}
D(\sig^{(i)})(e_1,\cdots,e_p)=\sig^{(i)}(e_0,\cdots,e_{p-1})\\
\hspace*{5em} =\sig(e_0,\cdots,\hat{e}_{i},\cdots,e_p)
=D\sig(e_1,\cdots,\hat{e}_{i+1},\cdots, e_{p+1}))$\\
����
\[
\bd D\sig=\sum_{i=0}^p (-1)^{i+1}(D\sig)^{(i+1)} + D(\sig)^{(0)}
=\sum_{i=0}^p (-1)^{i+1}D(\sig^{(i)}) +\sig =-D(\bd\sig)+\sig
\]
�̹Ƿ�, $\bd D+ D\bd=1$�� �����Ѵ�.
\end{proof}

\begin{lem}
Let \raisebox{-1.4ex}{\parbox{3cm}{$i_0: X\to X\times I $ \\
\hspace*{3ex} $ x\mapsto (x,0)$ }} and
\raisebox{-1.4ex}{\parbox{3cm}{$i_1: X\to X\times I $ \\
\hspace*{3ex} $ x\mapsto (x,1)$ }}. \\

Then for any space $X$, $\forall p, \exists D_X : S_p(X)\to
S_{p+1}(X\times I)$ such that\\
(1) $\bd D +D\bd=i_{1\sharp}-i_{0\sharp}$\\
(2) $D_X$ is natural, i.e., $\forall f: X\to Y$, \\
\hspace*{2em}\parbox{5.5cm}{$
\xymatrix @=1.5em @*[r] { %
S_p(X) \ar[d]^{f_\sharp} \ar[r]^{D_X} & S_{p+1}(X\times I)
\ar[d]^{(f\times id)_\sharp}\\ S_p(Y) \ar[r]^{D_Y} &
S_{p+1}(Y\times I)}$ } commutes.
\end{lem}

\bigskip
���� �� ���������� �����Ͽ��ٸ� ���� ������ �����Ѵ�.
\begin{thm}
$f\simeq g : X\to Y \Rightarrow f_* =g_* : H_*(X)\to H_*(Y)$
\end{thm}
{\bf ����} There exist $F:X\times I \to Y$ such that $f=F\circ
i_0$ and
$g=F\circ i_1$. \\
By the above lemma, there exists $D:i_{0\sharp}\simeq
i_{1\sharp}$. Hence, $i_{0*}= i_{1*}:H(X)\to H(X\times I)$.
Therefore $f_*=(F\circ i_0)_*=F_*\circ i_{0*}=F_*\circ
i_{1*}=(F\circ i_1)_*=g_*$.\hfill\framebox[2mm]{}
\begin{cor}
$X\simeq Y\Rightarrow H_*(X)\cong H_*(Y)$
\end{cor}

\bigskip
���� ���������� �����ϸ� �ȴ�. \\

\newpage
{\bf ���������� ����} (Induction on $p$) \\
idea. ���� $D$�� "model space" $\ssx$�� ���ؼ� �����ϰ�, $D$��
naturality�� �̿��Ͽ� ������ $X$�� ���Ͽ� $D_X$�� �����Ѵ�.\\

���� $p=0$�� ��� $D: S_0(\ssx[0])\to S_1 (\ssx[0]\times I)$��
��������. $\ssx[0]$�� �� point�̹Ƿ�, $S_0(\ssx[0])$�� constant
map $\sig$�� generate�Ǵ� free abelian group�̴�. ���⼭,
$D\sig$�� obvious�� 1-chain���� �����Ѵ�. ��, $D\sig: \ssx[1]\to
\ssx[0]\times I$���� $\ssx[1]$�� $\ssx[0]\times I$�� ����������
���� space�̹Ƿ� $D\sig$�� identity map���� �����Ѵ�. �� ��, $\bd
D\sig=i_1\circ\sig-i_0\circ\sig=i_{1\sharp}(\sig)-i_{0\sharp}(\sig)$�� �ȴ�. \\
$p=0$�� ��쿡�� ���� ������� ������ space $X$�� ���Ͽ� $D_X :
\sg{0}\to\sgi{1}$�� �ٷ� ������ �� �ִµ�, $\sig=x$(constant map),
$x\in X$�� ���Ͽ� $D\sig$�� $\ssx[1]$���� $\{x\}\times I$�� ����
"identity map"���� �����ϸ� ���� ���� (1),(2)�� �������� ���� Ȯ��
�� �� �ִ�.\\

$p>0$�� ���, $p-1$���� (1),(2)�� �����ϴ� $D_X$�� �����Ѵٰ�
��������.\\
$i=id: \ssx\to\ssx$�� ���Ͽ� $\bd Di+D\bd
i=i_{1\sharp}(i)-i_{2\sharp}(i)$�� �����ϴ� $Di\in
\sgi[\ssx]{p+1}$�� ��������.\\
$c=\ish(i)-\iosh(i)-D\bd i\in\sgi[\ssx]{p}$�� �ΰ�, $c=\bd Di$��
�ǵ��� $Di$�� �����ؾ� �Ѵ�. $\ssx\times I$�� star shaped�̹Ƿ�
���� ���������� ���Ͽ� $\td{H}(\ssx\times I)=0$�� �Ǿ�
$\sgi[\ssx]{p}$�� augmented chain�� acyclic�̴�. ����, ���� 2��
���������� ���� $\bd c=0$�� Ȯ���ϸ� �ȴ�.\\
�׷���, induction�� ������ �̿��ϸ�
$$
\bd c=\ish(\bd i)-\iosh(\bd i)%=\bd\ish(i)-\bd\iosh(i)-\bd D\bd i
-(\ish(\bd i)-\iosh(\bd i)- D\bd(\bd i))=0
$$
�̹Ƿ�, $Di$�� �� ���ǵȴ�.\\

���� $Di$�� �̿��Ͽ� ������ space $X$�� ���Ͽ�
$D_X:\sg{p}\to\sgi{p+1}$�� ��������. \\
������ $\sig\in\sgp$�� $\sig:\ssx\to X$�� map�̹Ƿ�,
$\sig_\sharp$��
$\raisebox{-1.4ex}{\parbox{4cm}{$\sig_\sharp: \sgp[\ssx]\to\sgp $ \\
\hspace*{9ex} $ i \mapsto \sig$ }}$ �� map�̴�. ����,
$D_X\sig=(\sig\times id)_\sharp Di$�� �����Ѵ�.\\

$\sg{p}$�� $\sig$��� generate�Ǵ� free abelian group�̹Ƿ�,
$D_X$�� well-defined�̰�, ���� ���� (1),(2)�� Ȯ���ϸ� ������
������. \\

Check (1):\\
\[
\xymatrix @M=1ex @C=-5mm @R=3ex @*[c] { %
%���� ����
& \sg[\ssx]{p+1} \ar[rr]^{\bd} \ar[dl]_{\sig_\sharp}
\ar[dd]_<<<<{\iosh}^<<<<{\ish}
 & &
 \sg[\ssx]{p} \ar[rr]^{\bd}\ar[dl] \ar[dd] \ar@/_3ex/[lldd]_D
& &
\sg[\ssx]{p-1} \ar[dl]\ar[dd]\ar@/_4ex/[lldd]_D\\
%���� ����
\sg{p+1} \ar[rr]\ar[dd]_{\iosh, \ish} & & \sg{p} \ar[rr] \ar[dd]
\ar@/_3ex/[lldd]_{D_X} & &
\sg{p-1}\ar[dd] \ar@/_3ex/[lldd]_{D_X}& \\
%���� �Ʒ���
& \sgi[\ssx]{p+1} \ar[rr] \ar[dl]^<<{(\sig\times id)_\sharp}& &
\sgi[\ssx]{p}  \ar[rr] \ar[dl]& &
 \sgi[\ssx]{p-1} \ar[dl]\\
%���� �Ʒ���
\sgi{p+1} \ar[rr] & & \sgi{p} \ar[rr] & & \sgi{p-1} & }
\]

��,�Ʒ�, ��, ���� �簢������ $\bd$�� $\sharp$�� commute�Ѵٴ�
��Ƿκ��� commute�ϰ�, ���(���η� ���̴�) �簢���� space��
���� map�� commute�ϹǷ� functorial property�κ��� commute�Ѵ�.
($\iosh, \ish$�� naturality.) �������� �񽺵��� �簢����
induction�� �������κ��� commute�Ѵ�. ($D$�� naturality.) ����,
$$
\bd D_X \sig=\bd(\sig\times id)_\sharp Di=(\sig\times
id)_\sharp\bd Di=(\sig\times
id)_\sharp(i_{1\sharp}(i)-i_{0\sharp}(i)-D\bd)
$$

�� ��, $\ish$�� naturality�κ��� ($\iosh$�� ��쵵 ��������)
$$
(\sig\times id)_\sharp i_{1\sharp}(i)
=\ish\sig_\sharp(i)=\ish(\sig)
$$
����, $D$�� naturality�κ���
$$
(\sig\times id)_\sharp D(\bd i)= D_X \sig_\sharp(\bd
i)=D_X\bd\sig_\sharp(i)=D_X\bd\sig
$$
����, $\bd D_X=\ish-\iosh-D_X\bd$���� Ȯ���� �� �ִ�.\\

Check (2):\\
������ map $f:X\to Y$�� ���Ͽ� ���� diagram�� ������ ����.
\[
\xymatrix @M=1ex @C=1ex @R=3ex @*[c] { %
% ����
& \sg{p} \ar[rr]^{D_X} \ar[dd]_<<<<{f_\sharp} & & \sgi{p+1} \ar[dd]^{(f\times id)_\sharp}\\
% ��� ��
\sg[\ssx]{p} \ar[rr]^D \ar[ur]^{\sig_\sharp}
\ar[dr]_{(f\circ\sig)_\sharp} & &
\sgi[\ssx]{p+1} \ar[ur]^{(\sig\times id)_\sharp} \ar[dr]_{(f\circ\sig\times id)_\sharp}& \\
% �Ʒ���
& \sg[Y]{p} \ar[rr] & & \sgi[Y]{p+1} }
\]
������ �ﰢ���� functorial property�� ���Ͽ� commute�ϰ�, $i\in
\sg[\ssx]{p}$�� ���� $D_X$�� ���ǿ� ���Ͽ� ���� �簢����, $D_Y$��
���ǿ� ���Ͽ� �Ʒ��� �簢���� commute�Ѵ�. ����, ���� �簢����
commute�Ѵٴ� ���� Ȯ�� �� �� �ִ�. \hfill\framebox[2mm]{}\\

\textbf{Review of the proof of 2.}\\
Need \\
(i) $\{ S(\bigtriangleup^p \times I),\partial,\epsilon\}$ is
acyclic.(to define $D$ for model space, $\bigtriangleup^p$ and
$i\in
S_p(\bigtriangleup^p))$\\
(ii) $\{ \sigma_{\sharp} i|\sigma : \bigtriangleup^p \rightarrow
X\}$
form a basis for $S_p(X)$.(to define $D_X$ for $\forall X$)\\
(iii) $i_{0\sharp} $ and $i_{1\sharp}$ are natural. (to check
(1))\\
$\Rightarrow$with functorial properties,\\
$\hspace*{1em}\exists$chain homotopy $D:i_0 \simeq i_1$ s.t. naturality of $D$ holds.\\

\textbf{Note.} \\
$\mathcal{T}$= the category of topological spaces.\\
$\mathcal{S}$= the category of chain complexes.(with
augmentation)\\
$ S : \mathcal{T}\longrightarrow\mathcal{S}$ is a
functor.\\
$\hspace*{1.4em} X\mapsto S(X)$\\
$ S' : \mathcal{T}\longrightarrow\mathcal{S}$ is a
functor.\\
$\hspace*{1.4em} X\mapsto S(X\times I)$\\
$i_{0\sharp}$ and $i_{1\sharp}$ are natural transformations : $S\rightarrow S'$, i.e.,\\
$\hspace*{1em} S : S(X)
\hspace{1.5em}\overset{f_{\sharp}}{\longrightarrow}\hspace{1.5em}
S(Y)$\\
$\hspace*{0em} i_{0\sharp}\downarrow \hspace{1.5em}\downarrow
i_{0\sharp}^X\hspace{2em}
\curvearrowright\hspace{2em}\downarrow i_{0\sharp}^Y$\\
$\hspace*{1em} S' : S(X\times I) \overset{(f\times
id)_{\sharp}}{\longrightarrow} S(Y\times I)$

\begin{defn} $\mathcal{T}$= the category of chain complexes , $\mathcal{S}$= the category of
(augmented) chain complexes and chain maps. Let $\mathcal{M}\subset Ob(\mathcal{T})$ (called models).\\
Let $S:\mathcal{T}\rightarrow \mathcal{S}$ be a functor from
$\mathcal{T}$ to $\mathcal{S}$.\\
(1) $S$ is acyclic relative to $\mathcal{M}$ if $S(M)$ is acyclic
$\forall
M\in \mathcal{M}$.\\
(2) $S$ is free relative to $\mathcal{M}$ if $\forall p\geq 0,
\exists J_p$ and $\exists$ an indexed family of
$\{M_{\alp}\}_{\alp \in J_p}, M_{\alp}\in \mathcal{M}$ and
$\exists$ an indexed family of $\{i_{\alp}\}_{\alp \in J_p},
i_{\alp}\in S_p(M_{\alp})$ s.t. $\{S(\sigma)i_{\alp} | \alp\in
J_p, \sigma\in hom(M_{\alp} , X)\}$ is a basis for
$S_p(X)$.\end{defn}

\begin{thm}\textbf{Acyclic Model Theorem}\\
Let $S, S' :\mathcal{T}\rightarrow\mathcal{S}$ be functors and
$\mathcal{M}\subset Ob(\mathcal{T}).\\ S$ : free relative to
$\mathcal{M}$ and $S':$
acyclic relative to $\mathcal{M}$.\\
$\Rightarrow$(a) $\exists $ a natural transformation : $\tau
:S\rightarrow S'$ as a lifting of a given $\tau_0:S_0\rightarrow
S'_0$.(or
$k:\mathbb{Z}\rightarrow\mathbb{Z}$ for augmented case)\\
(b) Given two such natural transformations
$\tau_0,\tau_1:S\rightarrow S', \exists$ a natural chain homotopy
$D:\tau_0\simeq\tau_1$.\end{thm}
\begin{proof}(a) Use induction.\\
1st, define $\tau_p$ for $i_{\alp}\in S_p (M_{\alp})$:\\
\[
\xymatrix @M=1ex @C=2em @R=2em @*[c] { %
\cdots \ar[r] & S_p(M_{\alp}) \ar@{.>}[d]^{\tau_p} \ar[r]^{\bd} &
S_{p-1}(M_{\alp}) \ar[d]^{\tau_{p-1}} \ar[r]^{\partial} &\cdots& \\
\cdots \ar[r] & S'_p(M_{\alp}) \ar[r]^{\partial} &
S'_{p-1}(M_{\alp}) \ar[r]^{\partial} &\cdots& }
\]\\[1mm]
$S'$�� acyclic�̹Ƿ� $\tau_p$�� �������� ���̷��� $\bd\tau_{p-1}
\bd$=0���� ���̸� �ȴ�.
�׷��� induction ������ ���� $\bd\tau_{p-1}\bd = \tau_{p-2}\bd\bd=0$.\\

2nd on X : Use Naturality.
\[
\xymatrix @M=1ex @C=1ex @R=3ex @*[c] { %
% ����
& i_{\alp}\in S_p(M_{\alp}) \ar[dl]^{S_p(\sig)} \ar[rr]^{\bd}
\ar[dd] & & S_{p-1}(M_{\alp})
\ar[dl] \ar[dd]^{\tau_{p-1}}\\
% ��� ��1
S(\sigma)i_{\alp}\in \sg{p} \ar[rr]  \ar@{.>}[dd]_{\tau_p} & & \sg{p-1}  \ar[dd]& \\
% ��� �� 2
& S'_p(M_{\alp}) \ar[dl] \ar[rr] & & S'_{p-1}(M_{\alp})
\ar[dl] \\
% �Ʒ���
S'_{p}(X) \ar[rr] & & S'_{p-1}(X)& }
\]
$S(\sigma)i_{\alp}$���� $S_p(X)$�� basis�� �̷�Ƿ�\\
Define $\tau_p$ for $S(\sigma)i_{\alp}$ : �� diagram box����
$i_{\alp}$�� ���ؼ��� �̹� $\tau_p$�� ���ǵǾ� �����Ƿ� ���� ����
commute�ϵ��� $\tau_p$�� �����ϸ� �ո��� �����ϰ� ��� ����
commute�ϹǷ� �ո鵵 commute.\\
$\therefore \tau_p$ is a chain map, $\forall X$.\\

$\tau_p$ is natural: $f :X\rightarrow Y$�� ��,
\[
\xymatrix @M=1ex @C=1ex @R=3ex @*[c]{ %
%����
& S_p (M_{\alp}) \ar[dl]^{S_p(\sig)} \ar[dd]^>>>>>{\tau} \ar[dr]^{S_p(f\circ\sig)} &\\
S_p(X) \ar[rr]^>>>>{S_p(f)} \ar[dd]^{\tau_X} && S_p(Y) \ar[dd]^{\tau_Y}\\
& S'_p (M_{\alp}) \ar[dl] \ar[dr] &\\
S'_p (X) \ar[rr] && S'_p (Y) }
\]
�� ���� ��տ��� $i_{\alp}$�� ���� ������ $\tau_p$�� ���ǿ�
$S_p$�� functorial property�� ���� �ո鸸 ���� �� commute.\\
$\therefore S(\sigma)i_{\alp}$�� ���� �ո鵵 commute.\\

(b) exactly same as before. (\textbf{���� 22.})\end{proof}

\begin{defn} A chain map $\tau :\mathcal{C}\rightarrow\mathcal{C'}$
is a chain homotopy equivalence if $\exists$ a chain map $\tau'
:\mathcal{C'}\rightarrow\mathcal{C}$ s.t. $\tau\tau' \simeq
id_{\mathcal{C'}}$ and $\tau'\tau\simeq
id_{\mathcal{C}}$.\end{defn}

\textbf{Note.} $\tau :\mathcal{C}\rightarrow\mathcal{C'}:$ chain
homopoty equivalence. $\Rightarrow \tau_* : H_* (\mathcal{C})
\cong H_*(\mathcal{C'}).$

\begin{cor} $S , S':\mathcal{T}\rightarrow\mathcal{S}, \mathcal{T}$
with models $\mathcal{M}$ and $S , S'$: both free and acyclic
relative to $\mathcal{M}$. $\Rightarrow \exists$ a natural chain
map $\tau : S \rightarrow S'$ and any natural chain map is a
natural chain homotopy equivalence.\end{cor}
\begin{proof} AMT. (a) $\Rightarrow$ natural chain map $\tau$
exists and also $\exists\tau' : S'\rightarrow S$. Now $\tau\tau'
:S'\rightarrow S'$\\
AMT (b) $\Rightarrow \tau\tau' \simeq id_{S'},$ similarly
$\tau'\tau \simeq
id_{S}$\end{proof}\\

\textbf{An application. (Equivalence of homologies)}\\
If $K$ is a simplicial complex, then we have\\
$\mathcal{C}(K)\rightsquigarrow H(K) = $ simplicial homology\\
$\bigtriangleup(K)\rightsquigarrow H_{\bigtriangleup}(K) =$
ordered
simplicial homology\\
$S(|K|)\rightsquigarrow H(|K|)$= singular homology for $|K|$

\begin{lem} $K$: a simplicial complex, $w$: a vertex (as a
simplicial complex) not in $K .\Rightarrow
\{C(w*K),\partial,\epsilon\}$ and
$\{\bigtriangleup(w*K),\partial,\epsilon\}$ are acyclic.\end{lem}
\begin{proof} Construct a chain contraction:\\
\[
\xymatrix @=2em @*[r] { %
\cdots \ar[r] & \bigtriangleup_1 \ar[dl]^{D_1} \ar[d]^{id} \ar[r]
& \bigtriangleup_0 \ar[dl]^{D_0} \ar[d]^{id} \ar[r]^{\epsilon} & \mathbb{Z} \ar[dl]^{D_{-1}} \ar[d]^{id}\\
\cdots \ar[r] & \bigtriangleup_1 \ar[r] & \bigtriangleup_0 \ar[r]
& \mathbb{Z}}
\]\\[1mm]

Let $D_{-1}(1) : = w$.Then $\epsilon (w) = 1$�̹Ƿ� OK.\\

Define $D(v_0 , \cdots, v_p ) = (w, v_0 , \cdots, v_p )$ : join
operator.\\
Then $(\bd D + D\bd )(v_0 , \cdots , v_p )\\ \hspace*{2.5em} =
\bd(w, v_0 , \cdots , v_p ) + D (\Sigma (-1)^i (v_0 , \cdots ,
\hat{v_i},
\cdots, v_p))\\
\hspace*{2.5em} = \Sigma (-1)^{j+1} (w , \cdots, \hat{v_j},
\cdots, v_p ) + (v_0 , \cdots , v_p ) + \Sigma (-1)^i (w ,
\cdots, \hat{v_i} ,\cdots, v_p )\\
\hspace*{2.5em} = (v_0 , \cdots, v_p ).$\\

Similarly for $\{\mathcal{C}(w*K),\bd,\epsilon\}$.\end{proof}

\begin{cor}(1) $\forall\sigma\in K ,\hspace{1em} \tilde{H}(\sigma) =
\tilde{H}_{\bigtriangleup}(\sigma) =0 $ \\
(i.e. $\{\mathcal{C},\partial,\epsilon\}$ and
$\{\bigtriangleup,\partial, \epsilon\}$ are acyclic w.r.t.
$\sigma$.)\\
(2) The chain map $\mu
:\bigtriangleup(K)\rightarrow\mathcal{C}(K)$ is a natural homotopy
equivalence.\\
$\hspace*{8em}(v_0,\cdots,v_p )\mapsto [v_0,\cdots,v_p]$\end{cor}
\begin{proof}(1) ��������9�κ��� clear. \\
(2) $\bigtriangleup , \mathcal{C} :\mathcal{C}(K)
\rightarrow\mathcal{S}$ are free and acyclic functors relative to
models $\mathcal{M} = \{\sigma |\sigma \in K\}$, where
$\mathcal{C}(K)$ is a category of subcomplexes of $K$ with
inclusion as morphism. Since $\mu :\bigtriangleup \rightarrow C$
is a natural chain map, it it a
chain homotopy equivalence by the previous cor.(�������� 8)\end{proof}\\



\end{document}
