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\begin{document}
\parindent=0cm
\section*{VI.2 Properties of Simplicial Homology}
\begin{defn} $f : |K| \rightarrow |L|$ , a simplicial map. \\
Define $f_{\sharp,p} :C_p (K)\rightarrow C_p (L)$ by $f_{\sharp
,p}[v_0 , \cdots,v_p ] = [f(v_0 ), \cdots , f(v_p )].\\ f_\sharp $
is
well-defined, i.e., $f(\overline{\sigma}) = - f(\sigma).\\
\{f_{\sharp ,p}\}$ is a "chain map" induced by
$f$, i.e., $f_\sharp \circ \partial = \partial \circ f_\sharp .$\\

$\cdots\hspace{0.3em} \rightarrow C_{p+1}\hspace{0.3em}
\overset{\partial}{\rightarrow}\hspace{0.3em} C_p\hspace{0.3em}
\overset{\partial}{\rightarrow}
\hspace{0.3em}C_{p-1}\hspace{0.3em}
\overset{\partial}{\rightarrow}\hspace{0.3em} \cdots\\
\hspace*{2.4em}\downarrow f_{p+1}
\curvearrowright\hspace{0.8em}\downarrow f_{p}
\curvearrowright\hspace{0.8em}\downarrow f_{p-1}\hspace{1em}
\{f_p\} $
is called a chain map if $\partial _p \circ f_p = f_{p-1}\circ\partial_p\\
\cdots \hspace{0.3em}\rightarrow D_{p+1}\hspace{0.3em}
\overset{\partial}{\rightarrow}\hspace{0.3em} D_p\hspace{0.3em}
\overset{\partial}{\rightarrow}\hspace{0.3em}
D_{p-1}\hspace{0.3em} \overset{\partial}{\rightarrow}
\hspace{0.3em}\cdots$
\end{defn}

1. $f_\sharp$ commutes with $\partial$ and $f_\sharp$ induces a
homomorphism $f_* :H_p (K) \rightarrow H_p (L)$.\\
\begin{proof}
$\partial \circ f_\sharp [v_0 ,\cdots , v_p ] = \partial [f(v_0 )
, \cdots , f(v_p )]\\ \hspace*{9em} = \Sigma (-1)^i [f(v_0
),\cdots,\hat{f(v_i )},\cdots,f(v_p )]\\ \hspace*{9em} = f_\sharp
\partial [v_0 , \cdots , v_p ] \\ \partial \circ f_\sharp =
f_\sharp \circ \partial \Rightarrow f_\sharp :Z_p \rightarrow Z_p
$ and $ B_p \rightarrow B_p \Rightarrow f_* : H_p \rightarrow
H_p$.
\end{proof}

2. (i) $f : K\rightarrow L$ and $g: L\rightarrow M$ : simplicial
maps $\Rightarrow (g \circ f)_* = g_* \circ f_*.$\\
(ii) id : $K \rightarrow K \Rightarrow$ id$_*$ = id.\\
\begin{proof}
clear from definition since $(g\circ f)_\sharp = g_\sharp \circ
f_\sharp$ and id$_\sharp$ = id.
\end{proof}

\textbf{ Topological invariance of Simplicial Homology}(key idea)\\
1. Let $K'$ be a subdivision of $K$ and let $\lam : C_p(K)
\rightarrow C_p(K')$ be an obvious subdivision operator. Then it
can be shown that $ \lam_*
: H_p (K) \rightarrow H_p(K')$ is an isomorphism.\\

2. $K $ and $L$ are simplicial complex structure for $X$. Choose a
common subdivision $M$ for $K$ and $L$. Then from 1, $H_p (K)
\cong H_p
(M) \cong H_p(L).$\\

\textbf{ Ordered Simplicial Homology}\\
Ordered chain complex :\\
$\bigtriangleup_p(K) :=$ the free abelian group generated by the
ordered p-simplices in $K$\\
and let $\partial (v_0, \cdots , v_p ) = \Sigma (-1)^i (v_0,
\cdots,
\hat{v_i }, \cdots ,v_p )$.\\

Then $\partial ^2 = 0 :$ same as before. \\ $\Rightarrow
$We have a chain complex $\{\bigtriangleup_p (K) ,\partial\}$ called the ordered chain complex.\\
$\cdots\overset{\partial}{\rightarrow}\bigtriangleup_{p+1}(K)
\overset{\partial}{\rightarrow}\bigtriangleup_{p}(K)
\overset{\partial}{\rightarrow}\bigtriangleup_{p-1}(K)\overset{\partial}{\rightarrow}\cdots$\\

$\Rightarrow H_p^{\bigtriangleup}(K) = $ker $\partial_p /$ im$
\partial_{p+1}$ : ordered simplicial homology.
\end{document}