

\documentclass[20pt]{extarticle}
\usepackage {tikz}
\include{customization}
\def\Cal{\cal}
\def\R{\mathbb{R}}
\def\X{\wtilde{X}}
\def\x{\wtilde{x_0}}
\def\al{\wtilde{\alpha}}
\def\hb{\hfil \break}
\def\hr{\hrule \vskip 0pt}
\def\t{\vskip 5pt}
\def\s{\vskip -5pt}
\def\u{\vskip -10pt}


%\def\cov{$\commdiag{
 %  \wtilde{X}               \cr 
 % \mapdown^{p}  \cr
%X
%}$}

\def\al{\alpha}

\def\plift{\xymatrix{ & (\wtilde{X}, \wtilde{x}_0)   \ar[d]^p \\ 
I          \ar[r]_-{ \al} \ar@{->}   [ur]^{\exists !\wtilde{\al}}   &   (X, x_0) }}

\def\pdashlift{\xymatrix{ & (\wtilde{X}, \wtilde{x}_0)   \ar[d]^p \\ 
(Y, y_0)          \ar[r]_-{ f_0} \ar@{-->}   [ur]^{\exists !\wtilde{f_0}}   &   (X, x_0) }}


\def\pdashhlift{\xymatrix{ & (\wtilde{X}, \wtilde{x}_0)   \ar[d]^p \\ 
Y \times I           \ar[r]_-{F} \ar@{-->}   [ur]^{\exists !\wtilde{F}}   &   X }}

\def\dashhlift{\xymatrix{ & \wtilde{X}  \ar[d]^p \\ 
Y \times I           \ar[r]_-{F} \ar@{-->}   [ur]^{\exists !\wtilde{F}}   &   X }}

\def\dashlift{\xymatrix{ & \wtilde{X}  \ar[d]^p \\ 
Y          \ar[r]_-{ f_0} \ar@{-->}   [ur]^{\exists !\wtilde{f_0}}   &   X }}

\def\hlift{\xymatrix{ & \wtilde{X}  \ar[d]^p \\ 
Y \times I           \ar[r]_-{F} \ar@{_{(}->}   [ur]^{\wtilde{F}}   &   X }}

\def\lift{\xymatrix{ & \wtilde{X}  \ar[d]^p \\ 
Y          \ar[r]_{f_0} \ar@{_{(}->}  [ur]^{\wtilde{f_0}}   &   X }}



\parskip 20pt
\begin{document}

The following diagram does NOT commute:

%\rightline
{$\commdiag{
\cdots & \rightarrow & C_{n+1}  &         \xrightarrow{~~~~~~~\partial_{n+1}~~~~~~~}                   &    C_n    & \xrightarrow{~~~~~~~\partial_n~~~~~~~}  &   C_{n-1}  & \rightarrow & \cdots \cr 
 &\arrow(-4,-2)& \mapdown^{f_{n+1}}  \mapdown_{g_{n+1}} &\arrow(-3,-2)\lft{K_n} &\mapdown^{f_n}  \mapdown_{g_n}   & \arrow(-3,-2)\lft{K_{n-1}}& \mapdown^{f_{n-1}}  \mapdown_{g_{n-1}}  & \arrow(-4,-2) &\cr
  \cdots & 
\rightarrow       &       D_{n+1}       & \xrightarrow{~~~~\partial'_{n+1}~~~~~} &D_n & \xrightarrow{~~~~\partial'_{n}~~~~~}  &D_{n-1}  & \rightarrow &\cdots
}$}

The chain maps $f_n, g_n:  C_n \rightarrow D_n$ are chain homotopic if \hb
$\exists K_n: C_n \rightarrow D_n$ such that

\centerline{$\partial'_{n+1}K_n + K_{n-1} \partial_n = f_n - g_n$}

Claim:  $f_* = g_*: H_\bullet^C \rightarrow H_\bullet^D$

Proof:  If $\alpha \in Z_n^C$, then $\partial \alpha = 0.  $

$\partial'_{n+1}K_n(\alpha) + K_{n-1} \partial_n(\alpha) = f_n(\alpha) - g_n(\alpha)$


$f_n(\alpha)  = \partial'_{n+1}K_n(\alpha) + g_n(\alpha)$
 
$[f_n(\alpha)]  = [\partial'_{n+1}K_n(\alpha) + g_n(\alpha)] = [g_n(\alpha)]$

Thus $f_*(\alpha) = g_*(\alpha)$


\end{document}
\vfill

\[
\xymatrix{
{0}
 \ar[r]
&
{G}
 \ar@{=}[d]
 \ar@{>->}[r]
&
{R_{0}}
 \ar@{->>}[r]
 \ar@{-->}[d]^-{f_{0}}
&
{C_{1}}
 \ar@{-->}[dr]_-{f_{1}}
 \ar@{>->}[r]
&
{R_{1}}
 \ar@{->>}[r]
 \ar@{-->}[d]^-{f_{1}}
&
{C_{2}}
 \ar@{-->}[dr]_-{f_{2}}
 \ar@{>->}[r]
&
{R_{2}} 
 \ar@{-->}[d]^-{f_{2}}
{}\\
{0}
 \ar[r]
&
{G}
 \ar@{>->}[r]
&
{J_{0}}
 \ar[rr]
&
&
{J_{1}}
  \ar[rr]
&
&
{J_{2}.}
}
\]


\end{document}


