\magnification=2000
\overfullrule=0pt
\parindent=0pt

\nopagenumbers

\input amstex

%\voffset=-.6in
%\hoffset=-.5in
%\hsize = 7.5 true in
%\vsize=10.4 true in

\voffset=1.8 true in
\hoffset=-.6 true in
\hsize = 10.2 true in
\vsize=8 true in

\input colordvi

\def\cltr{\Red}		  % Red  VERY-Approx PANTONE RED
\def\cltb{\Blue}		  % Blue  Approximate PANTONE BLUE-072
\def\cltg{\PineGreen}	  % ForestGreen  Approximate PANTONE 349
\def\cltp{\DarkOrchid}	  % DarkOrchid  No PANTONE match
\def\clto{\Orange}	  % Orange  Approximate PANTONE ORANGE-021
\def\cltpk{\CarnationPink}	  % CarnationPink  Approximate PANTONE 218
\def\clts{\Salmon}	  % Salmon  Approximate PANTONE 183
\def\cltbb{\TealBlue}	  % TealBlue  Approximate PANTONE 3145
\def\cltrp{\RoyalPurple}	  % RoyalPurple  Approximate PANTONE 267
\def\cltp{\Purple}	  % Purple  Approximate PANTONE PURPLE

\def\cgy{\GreenYellow}     % GreenYellow  Approximate PANTONE 388
\def\cyy{\Yellow}	  % Yellow  Approximate PANTONE YELLOW
\def\cgo{\Goldenrod}	  % Goldenrod  Approximate PANTONE 109
\def\cda{\Dandelion}	  % Dandelion  Approximate PANTONE 123
\def\capr{\Apricot}	  % Apricot  Approximate PANTONE 1565
\def\cpe{\Peach}		  % Peach  Approximate PANTONE 164
\def\cme{\Melon}		  % Melon  Approximate PANTONE 177
\def\cyo{\YellowOrange}	  % YellowOrange  Approximate PANTONE 130
\def\coo{\Orange}	  % Orange  Approximate PANTONE ORANGE-021
\def\cbo{\BurntOrange}	  % BurntOrange  Approximate PANTONE 388
\def\cbs{\Bittersweet}	  % Bittersweet  Approximate PANTONE 167
%\def\creo{\RedOrange}	  % RedOrange  Approximate PANTONE 179
\def\cma{\Mahogany}	  % Mahogany  Approximate PANTONE 484
\def\cmr{\Maroon}	  % Maroon  Approximate PANTONE 201
\def\cbr{\BrickRed}	  % BrickRed  Approximate PANTONE 1805
\def\crr{\Red}		  % Red  VERY-Approx PANTONE RED
\def\cor{\OrangeRed}	  % OrangeRed  No PANTONE match
\def\paru{\RubineRed}	  % RubineRed  Approximate PANTONE RUBINE-RED
\def\cwi{\WildStrawberry}  % WildStrawberry  Approximate PANTONE 206
\def\csa{\Salmon}	  % Salmon  Approximate PANTONE 183
\def\ccp{\CarnationPink}	  % CarnationPink  Approximate PANTONE 218
\def\cmag{\Magenta}	  % Magenta  Approximate PANTONE PROCESS-MAGENTA
\def\cvr{\VioletRed}	  % VioletRed  Approximate PANTONE 219
\def\parh{\Rhodamine}	  % Rhodamine  Approximate PANTONE RHODAMINE-RED
\def\cmu{\Mulberry}	  % Mulberry  Approximate PANTONE 241
\def\parv{\RedViolet}	  % RedViolet  Approximate PANTONE 234
\def\cfu{\Fuchsia}	  % Fuchsia  Approximate PANTONE 248
\def\cla{\Lavender}	  % Lavender  Approximate PANTONE 223
\def\cth{\Thistle}	  % Thistle  Approximate PANTONE 245
\def\corc{\Orchid}	  % Orchid  Approximate PANTONE 252
\def\cdo{\DarkOrchid}	  % DarkOrchid  No PANTONE match
\def\cpu{\Purple}	  % Purple  Approximate PANTONE PURPLE
\def\cpl{\Plum}		  % Plum  VERY-Approx PANTONE 518
\def\cvi{\Violet}	  % Violet  Approximate PANTONE VIOLET
\def\clrp{\RoyalPurple}	  % RoyalPurple  Approximate PANTONE 267
\def\cbv{\BlueViolet}	  % BlueViolet  Approximate PANTONE 2755
\def\cpe{\Periwinkle}	  % Periwinkle  Approximate PANTONE 2715
\def\ccb{\CadetBlue}	  % CadetBlue  Approximate PANTONE (534+535)/2
\def\cco{\CornflowerBlue}  % CornflowerBlue  Approximate PANTONE 292
\def\cmb{\MidnightBlue}	  % MidnightBlue  Approximate PANTONE 302
\def\cnb{\NavyBlue}	  % NavyBlue  Approximate PANTONE 293
\def\crb{\RoyalBlue}	  % RoyalBlue  No PANTONE match
%\def\cbb{\Blue}		  % Blue  Approximate PANTONE BLUE-072
\def\cce{\Cerulean}	  % Cerulean  Approximate PANTONE 3005
\def\ccy{\Cyan}		  % Cyan  Approximate PANTONE PROCESS-CYAN
\def\cpb{\ProcessBlue}	  % ProcessBlue  Approximate PANTONE PROCESS-BLUE
\def\csb{\SkyBlue}	  % SkyBlue  Approximate PANTONE 2985
\def\ctu{\Turquoise}	  % Turquoise  Approximate PANTONE (312+313)/2
\def\ctb{\TealBlue}	  % TealBlue  Approximate PANTONE 3145
\def\caq{\Aquamarine}	  % Aquamarine  Approximate PANTONE 3135
\def\cbg{\BlueGreen}	  % BlueGreen  Approximate PANTONE 320
\def\cem{\Emerald}	  % Emerald  No PANTONE match
%\def\cjg{\JungleGreen}	  % JungleGreen  Approximate PANTONE 328
\def\csg{\SeaGreen}	  % SeaGreen  Approximate PANTONE 3268
\def\cgg{\Green}	  % Green  VERY-Approx PANTONE GREEN
\def\cfg{\ForestGreen}	  % ForestGreen  Approximate PANTONE 349
\def\cpg{\PineGreen}	  % PineGreen  Approximate PANTONE 323
\def\clg{\LimeGreen}	  % LimeGreen  No PANTONE match
\def\cyg{\YellowGreen}	  % YellowGreen  Approximate PANTONE 375
\def\cspg{\SpringGreen}	  % SpringGreen  Approximate PANTONE 381
\def\cog{\OliveGreen}	  % OliveGreen  Approximate PANTONE 582
\def\pars{\RawSienna}	  % RawSienna  Approximate PANTONE 154
\def\cse{\Sepia}		  % Sepia  Approximate PANTONE 161
\def\cbr{\Brown}		  % Brown  Approximate PANTONE 1615
\def\cta{\Tan}		  % Tan  No PANTONE match
\def\cgr{\Gray}		  % Gray  Approximate PANTONE COOL-GRAY-8
\def\cbl{\Black}		  % Black  Approximate PANTONE PROCESS-BLACK
\def\cwh{\White}		  % White  No PANTONE match


\loadmsbm

\input epsf

\def\ctln{\centerline}
\def\u{\underbar}
\def\ssk{\smallskip}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\hsk{\hskip.1in}
\def\hhsk{\hskip.2in}
\def\dsl{\displaystyle}
\def\hskp{\hskip1.5in}

\def\lra{$\Leftrightarrow$ }
\def\ra{\rightarrow}
\def\mpto{\logmapsto}
\def\pu{\pi_1}
\def\mpu{$\pi_1$}
\def\sig{\Sigma}
\def\msig{$\Sigma$}
\def\ep{\epsilon}
\def\sset{\subseteq}
\def\del{\partial}
\def\inv{^{-1}}
\def\wtl{\widetilde}
%\def\lra{\Leftrightarrow}
\def\del{\partial}
\def\delp{\partial^\prime}
\def\delpp{\partial^{\prime\prime}}
\def\sgn{{\roman{sgn}}}
\def\wtih{\widetilde{H}}
\def\bbz{{\Bbb Z}}
\def\bbr{{\Bbb R}}


{\bf Exact sequences:} Most of the fundamental properties of homology groups
are described in terms of exact sequences. A sequence of homomorphisms
\ssk

\ctln{$\cdots {f_{n+1}\atop\ra} A_n {f_n\atop\ra} A_{n-1} {f_{n-1}\atop\ra} a_{n-2} 
\ra \cdots$}

\ssk

of abelian groups is called {\it exact} if im$(f_n)=\ker(f_{n-1})$ 
for every $n$. In most cases,
we get the most mileage out of an exact sequence when some of the groups
are trivial; 

\ssk

\ctln{$0\ra A {f\atop\ra}B$ is exact \lra\ $f$ is injective, and 
$A {f\atop\ra}B\ra 0$ is exact \lra\ $f$ is surjective.}

\ssk
 
An exact sequence $0\ra A {\ra}B\ra C\ra 0$ is called a {\it short exact sequence}.

\bsk

The main tool we will use turns a family of short exact sequences of chain maps
between three chain complexes into a single {\it long exact homology sequence}.

\ssk

Given chain complexes ${\Cal A}=(A_n,\del)$ , 
${\Cal B}=(B_n,\del^\prime)$ , and ${\Cal C}=(C_n,\del^{\prime\prime})$
and short exact sequences of chain maps (i.e., 
$\del^\prime i_n  = i_n\del $ , $\del^{\prime\prime}j_n = j_n\del^\prime $)

\ssk

\ctln{$0\ra A_n{i_n\atop \ra}B_n{j_n\atop \ra}C_n\ra 0$}

\ssk

there is a general result which provides us with a long exact sequence

\ssk

\ctln{$\cdots {\del \atop \ra} H_n({\Cal A}) {i_{*}\atop \ra} H_n({\Cal B})
{j_{*}\atop \ra} H_n({\Cal C}) {\del\atop\ra} H_{n-1}({\Cal A}) {i_{*}\atop\ra} \cdots$}

\ssk

Most of the work is in defining the ``boundary'' map $\del$. Given an 
element $[z]\in H_n({\Cal C})$, a representative $z\in C_n$ satisfies 
$\del^{\prime\prime}(z)=0$. But $j_n$ is onto, so there is a $b\in B_n$ with
$j_n(b)=z$, Then $ i_{n-1}\del^\prime(b) = \del^{\prime\prime}j_n(b)
=0$, so $\del^\prime(b)\in\ker(j_{n-1})=$im$(a_{n-1})$. So there is an $a\in A_{n-1}$
with $i_{n-1}(a)=\del^\prime(b)$ . But then 
$i_{n-2}\del (a) = \del^\prime i_{n-1}(a)=\del^\prime\del^\prime(b)=0$,
so, since $i_{n-2}$ is injective, $\del a=0$, so $a\in Z_{n-1}({\Cal A})$, and
so represents a homology class $[a]\in H_n({\Cal A})$. We define
$\del([z])=[a]$ . 

\vfill
\eject

Is this well-defined? We need to show that the
class $[a]$ is independent of the choices made along the 
way. The choice of $a$ isn't really one; $i_{n-1}$ is, by assumption, 
injective. For $b$, if $j_n(b)=z=j_n(b^\prime)$, then
$j_n(b-b^\prime)=0$, so $b-b^\prime=i_n(w)$ for some $w\in A_n$. Then 
$\del^\prime b^\prime = \del^\prime b - \del^\prime i_n(w) = 
\del^\prime b - i_{n-1} \del(w)$; choosing $a^\prime = a-\del(w)$ we have
$i_{n-1}(a^\prime)=\del^\prime(b^\prime)$. But then
$[a^\prime]=[a-\del w] =[a] -[\del w] =[a]$. Finally, there is a choice
of $z$ ; if $[z]=[z^\prime]$, then $z^\prime = z+\del^{\prime\prime}w$
for some $w\in C_{n+1}$; but then choosing $b^\prime,w^\prime$ with 
$j_n(b^\prime)=z^\prime$ , $j_{n+1}(w^\prime)=w$ , we have 
$\del^{\prime\prime}w=\del^{\prime\prime}j_{n+1}(w^\prime) = j_n\del^\prime(w^\prime)$ ,
so 
$z^\prime= z+\del^{\prime\prime}w= j_n(b+\del^\prime w^\prime)$, so we may choose
$b^\prime = b+\del^\prime w^\prime$ (since the result is independent of this choice!),
then since $\del^\prime b^\prime = \del^\prime b$ everything continues the same.

\ssk

Now to exactness! We need to show three (types of) equalities, which means six
containments. Three (image $\subseteq$ kernel) 
are shown basically by showing that compositions of
two consecutive homomorphisms are trivial. $j_ni_n=0$ 
immediately implies $j_*i_*=0$ . From the definition of $\del$,
$i_*\del[z] = [i_n(a)] = [\del^\prime(b)] = 0$, and 
$\del j_*[z] = \del[j_n(z)] = [a]$, where $i_{n-1}(a)=\del^\prime(z) = 0$,
so $a=0$ (since $i_{n-1}$ is injective), so $[a]=0$. 

\ssk

For the opposite containments,
if $j_*[z]=[j_n(z)]=0$, then $j_n(z)=\del^{\prime\prime}w$ for some $w$. 
Since $j_{n+1}$ is onto, $w=j_{n+1}(b)$ for some $b$. Then 
$j_n(z-\delp b) = \delpp w-\delpp j_{n+1} b = 0$, so 
$z=\delp b = i_n(a)$ for some $a$, so $i_*[a] = [z-\delp b] = [z]$ . 
So $\ker j_*\subseteq$im$i_*$ . If $i_*[z]=0$, then $i_n(z)=\delp w$ for some $w\in B_{n+1}$.
Setting $c=j_{n+1}(w)$, then $\delpp c = j_n \delp w - i_n i_n(z) = 0$, so 
$[c]\in H_{n+1}({\Cal C})$, and computing $\del [c]$ we can choose $w$ for the 
1st step and $z$ for the 2nd, so $\del [c] = [z]$ . So $\ker j_n\subseteq$im$\del$ .
Finally, if $\del [z] = 0$, then $z=j_n(b)$ for some $b$, and $\delp b = i_{n-1}(a)$ with
$[a]=0$, i.e., $a=\del w$ for some $w$. So $\delp b = i_{n-1}\del w = \delp i_n w$. But
then $\delp (b-i_n w) = 0$, and 
$j_n(b-i_n w) = z-0 = z$, so $z\in$im$(j_n)$, so $[z]\in$im$(j_*)$ . So 
$\ker\del\subseteq$im$(j_n)$ . Which finishes the proof!

\msk

Now all we need are some new chain complexes! 

\bsk




\vfill
\end


















