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\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


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\ctln{\bf Math 971 Algebraic Topology}

\ssk

\ctln{April 12, 2005}

\msk

There 
is another piece of homological algebra
that we will find useful ; the Five Lemma. It allows us to 
\u{compare} the information contained in two long exact sequences.

\msk

{\bf Five Lemma:} If we have abelian groups and maps

\ssk

\ctln{$\displaystyle 
\matrix 
A_n&{\buildrel f_n\over\ra}&B_n&{\buildrel g_n\over\ra}&C_n & {\buildrel h_n\over\ra} & D_n & {\buildrel i_n\over\ra} & E_n\cr
\alpha\downarrow & & \beta\downarrow & & \gamma\downarrow & & \delta\downarrow & & \epsilon\downarrow & \cr
A_{n-1} & {\buildrel f_{n-1}\over\ra} & B_{n-1} & {\buildrel g_{n-1}\over\ra} & C_{n-1} & {\buildrel h_{n-1}\over\ra} & D_{n-1} & {\buildrel i_{n-1}\over\ra} & E_{n-1}\cr
\endmatrix$}

\ssk

where the rows are exact, and the maps $\alpha,\beta,\delta,\epsilon$ are all isomorphisms, then $\gamma$ is an isomorphism.

\msk

The proof is in some sense literally a matter of doing the only thing you can. To show injectivity, suppose
$x\in C_n$ and $\gamma x = 0$, 
then $h_{n-1}\gamma x = \delta h_n x = 0$, so, 
since $\delta$ is injective,
$h_n x = 0$. So 
by the exactness at $C_n$, $x=g_n y$ for some $y\in B_n$. 
Then $g_{n-1} \beta y = \gamma g_n y = \gamma x = 0$, so 
by exactness at $B_{n-1}$, $\beta y = f_{n-1} z$ for some $z\in A_{n-1}$. Then 
since $\alpha$ is surjective,
$f_{n-1}z = \alpha w$ for some $w$. Then 
$0=g_n f_n w$ . But 
$\beta f_n w = f_{n-1} \alpha w \ f_{n-1} z = \beta y$, so since
$\beta$ is injective, 
$y= f_n w$ . So $0=g_n f_n w = g_n y = x$. So $x=0$.

\ssk

For surjectivity, suppose $x\in C_{n-1}$.
Then $h_{n-1} x \in D_{n-1}$, so
since $\delta$ is surjective, 
$h_{n-1} x = \delta y$ for some $y\in D_n$.
Then $\epsilon i_n y = i_{n-1}\delta y = i_{n-1} h_{n-1} x = 0$, so 
since $\epsilon$ is injective, $i_n y= 0$.
So by exactness at $D_n$, 
$y=h_n z$ for some $z\in C_n$. Then 
$h_{n-1}\gamma z = \delta h_n z = \delta y = h_{n-1} x$,
so $h_{n-1} (\gamma z-x) = 0$, so 
by exactness at $C_{n-1}$, 
$\gamma z-x = g_{n-1}w$ for some $w\in B_{n-1}$. Then
since $\beta$ is surjective, 
$w= \beta u$ for some $u\in B_n$. Then 
$\gamma g_n u = g_{n-1} \beta u = g_{n-1}w = \gamma z-x$,
so $x=\gamma z - \gamma g_n u = \gamma (z-g_n u)$ . 
So $\gamma$ is onto.

\bsk

The second result that this machinery gives us is what is properly known as {\it excision}:

\msk

If $B\sset A\sset X$ and cl$_X(B)\sset$ int$_X(A)$, then for every $k$ the inclusion-induced map 
$H_k(X\setminus B,A\setminus B)\ra H_k(X,A)$ is an isomorphism. 

\msk

An equivalent formulation of this is that if $A,B\sset X$ and int$_X(A)\cup$ int$_X(B) = X$, then the
inclusion-induced map $H_k(B,A\cap B)\ra H_k(X,A)$ is an isomorphism. [From first to second
statement, set $B^\prime = X\setminus B$ .] To prove the second statement, we know that
the inclusion $C_n^{\{A,B\}}(X) \ra C_n(X)$ induce isomorphisms on homology, as does 
$C_n(A) \ra C_n(A)$, so, by the five lemma, the induced map
$C_n^{\{A,B\}}(X)/C_n(A) \ra C_n(X)/C_n(A) = C_n(X,A)$ induces isomorphisms on homology. 
But the inclusion $C_n(B) \ra C_n^{\{A,B\}}(X)$ induces a map 
$C_n(B,A\cap B) = C_n(B)/C_n(A\cap B) \ra C_n^{\{A,B\}}(X)/C_n(A)$ which is an isomorphism of chain groups;
a basis for $C_n^{\{A,B\}}(X)/C_n(A)$ consists of singular simplices which map into $A$ or $B$, but don't map into $A$,
i.e., of simplices mapping into $B$ but not $A$, i.e., of simplices mapping into $B$ but not $A\cap B$.
But this is the \u{same} as the basis for $C_n(B,A\cap B)$ !

\msk

With these tools, we can start making some \u{real} homology computations. First, we show that 
if $\emptyset\neq A\sset X$ is ``nice enough'', then $H_n(X,A)\cong \widetilde{H}_n(X/A)$ .
The definition of nice enough, like Seifert - van Kampen, is that $A$ is closed and has an open neighborhood
${\Cal U}$ that deformation retracts to $A$ (think: $A$ is the subcomplex of a CW-complex $X$).
Then using ${\Cal U}.X\setminus A$ as a cover of $X$, and ${\Cal U}/A,(X\setminus A)/A$ as a cover of $X/A$,
 we have

\ssk

$\widetilde{H}_n(X/A) {\buildrel {(1)}\over \cong} H_n(X/A,A/A){\buildrel {(2)}\over \cong} 
H_n(X/A,{\Cal U}/A) {\buildrel {(3)}\over \cong} H_n(X/A\setminus A/A,{\Cal U}/A\setminus A/A) {\buildrel {(4)}\over \cong}
H_n(X\setminus A,{\Cal U}\setminus A){\buildrel {(5)}\over \cong} H_n(X,A)$

\ssk

Where (1),(2) follow from the LES for a pair, (3),(5) by excision, and (4) because the restriction of the quotient
map $X\ra X/A$ gives a homeomorphism of pairs.

\msk

Second, if $X,Y$ are $T_1$, $x\in X$ and $y\in Y$ each have neighborhoods 
${\Cal U},{\Cal V}$ which deformation retract to each point, then the 
one-point union $Z=X\vee Y = (X\coprod Y)/(x=y)$ has $\widetilde{H}_n(Z) \cong \widetilde{H}_n(X)\oplus \widetilde{H}_n(Y)$;
this follows from a similar sequence of isomorphisms. Setting $z$ = the image of $\{x,y\}$ in $Z$, we have

\ssk

$\widetilde{H}_n(Z) \cong H_n(Z,z) \cong H_n(Z,{\Cal U}\vee{\Cal V}) \cong H_n(Z\setminus z,{\Cal U}\vee{\Cal V}\setminus z)
\cong H_n([X\setminus x]\coprod[Y\setminus y],[{\Cal U}\setminus x]\coprod [{\Cal V}\setminus y])
\cong H_n(X\setminus x,{\Cal U}\setminus x)\oplus H_n(Y\setminus y,{\Cal V}\setminus y) 
\cong H_n(X,x)\oplus H_n(Y,y)\cong \widetilde{H}_n(X)\oplus \widetilde{H}_n(Y)$

\ssk

By induction, we then have $\displaystyle \widetilde{H}_n(\vee_{i=1}^k X_i) \cong \oplus_{i=1}^k \widetilde{H}_n(X_i)$

\msk

As an application, we can compute the homology groups of the closed orientable surfaces of genus $g$, $F_g$.  We will argue by induction on $g$ that 
$\wtih_i(F_g) = $ $\bbz$ for $i=2$, $\bbz^{2g}$ for $i=1$, and $0$ for all other $i$. 
For the base case $g$=1 we have the 2-torus, whose homology we have computed previously. For the inductive step, we look at the homology of the pair
$(F_{g+1},A)$ depicted here; with what we have learned above, in the LES for the pair, we will have
$H_i(F_{g+1},A)\cong \wtih_i(F_{g+1}/A) \cong  \wtih_i(F_{g}\vee T^2) \cong  \wtih_i(F_{g})\oplus\wtih_i(T^2)$ , which we can compute,
by our inductive hypothesis. And so we find that we have

\ssk

$\wtih_2(A) \ra \wtih_2(F_{g+1}) \ra \wtih_2(F_{g}\oplus\wtih_2(T^2) \ra \wtih_1(A) \ra \wtih_1(F_{g+1}) \ra \wtih_1(F_{g})\oplus\wtih_2(T^2) 
\ra \wtih_0(A)$

\ssk

which renders as
\hhsk
$0\ra \wtih_2(F_{g+1}) \ra \bbz^2 \ra \bbz \ra \wtih_1(F_{g+1}) \ra \bbz^{2g+2}\ra 0$ 
\hhsk . The point, though is that
the map $\wtih_1(A) \ra \wtih_1(F_{g+1})$ is the $0$ map, since the circle $A$ is null-homologous in $F_{g+1}$;
it bounds the once-punctured torus $X$ on the right of the figure. [Formally, what we mean is that the generator of 
$\wtih_1(A)$, a singular 1-simplex wrapping once around $A$, is a boundary in $C_1(F_{g+1})$; writing $X$ as
a sum of singular 2-simplices provides a demonstration.] So we can cut out exact sequence into two pieces

$0\ra \wtih_2(F_{g+1}) \ra \bbz^2 \ra \bbz \ra 0$ \hhsk and \hhsk $0 \ra \wtih_1(F_{g+1}) \ra \bbz^{2g+2}\ra 0$

The first implies that $\wtih_2(F_{g+1})$ is a direct summand, with $\bbz$, for $\bbz^2$, and so is $\bbz$; the second
asserts that $\wtih_1(F_{g+1}) \cong \bbz^{2g+2}$, as desired. So our inductive step is proved, and our computation
follows by induction.

\leavevmode

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\ctln{{\epsfbox{0412f1.ai}}}

\vfill
\end