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\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


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\ctln{\bf Math 971 Algebraic Topology}

\ssk

\ctln{March 10, 2005}

\msk

{\bf Some examples:} the Klein bottle $K$ has a $\Delta$-complex structure with 2 2-simplices,
3 1-simplices, and 1 0-simplex; we will call them 
$f_1=[0,1,2],f_2=[1,2,3],
e_1=[0,2]=[1,3],e_2=[1,0]=[2,3],e_3=[1,2]$, 
and $v_1=[0]=[1]=[2]=[3]$.
Computing, we find 
$\del_2 f_1 = \del[0,1,2]=[1,2]-[0,2]+[0,1]=e_3-e_1-e_2$ , $\del_2 f_2 = e_2-e_1+e_3$ , 
$\del_1 e_1 = \del_1 e_2 = \del_1 e_3 = 0$ and $\del_i = 0$ for all other $i$
(as well). So we have the chain complex

$\cdots \ra 0 \ra {\Bbb Z}^2 \ra {\Bbb Z}^3 \ra {\Bbb Z} \ra 0$

and all of the boundary maps are 0, except for $\del_2$, which has the matrix
$\pmatrix
 -1&-1\\ -1&1\\ 1&1\\
\endpmatrix$ . This matrix is injective, so $\ker \del_2 = 0$,
so $H_2(K)=0$, on the other hand, $H_1(K)$ = coker$(\del_2)$, and applying column
operations we can transform the matrix for $\del_2$ to $\pmatrix 1&0\\ 1&2\\ -1&0\\ \endpmatrix$,
which implies that the cokernel is ${\Bbb Z}\oplus{\Bbb Z}_2$, since 
$\pmatrix 1\\ 1\\ -1\\ \endpmatrix ,\pmatrix 0\\ 1\cr 0\\ \endpmatrix , \pmatrix 0\cr 0\\ 1\\ \endpmatrix$
is a basis for ${\Bbb Z}^3$. Finally, $H_0(K)={\Bbb Z}$, since $\del_1,\del_0=0$,
and all higher homology groups are also $0$, for the same reason.

\msk

As another example, the topologist's dunce hat has a $\Delta$-structure with
1 2-simplex, 1 1-simplex, and 1 0-simplex. The boundary maps, we can work out
(starting from $C_2(X)$ ), are $(1),(0)$, and $(0)$, so $H_2(X)=H_1(X)=0$,
and $H_0(X)={\Bbb Z}$. all higher groups are also $0$.

\msk

These homology groups are, in the end, fairly routine to calculate from a 
$\Delta$-complex structure. But there is one very large problem; the calculations
\u{depend} on the $\Delta$ structure! This is not a group defined from the space
$X$; it is defined from the space and a $\Delta$ structure on it. A priori, we don't
know that if we chose a different structure on the same space, that we would get
isomorphic groups! We should really denote our groups by $H_i^\Delta(X)$, to 
acknowledge this dependence on the structure.

\bsk

But we don't {\it want} a group that depends on this structure. We want groups that
just depend on the topological space $X$, i.e., which are topological invariants.
In really turns out that these groups $H_i^\Delta(X)$ \u{are} topological invariants,
but we will need to take a very roundabout route to show this. What we will do
now is to define another sequence $H_i(X)$ of groups, the {\it singular homology
groups}, which their definition makes apparent from the outset 
that they are topological invariants.
But this definition will also make it very unclear how to really compute them! 
Then we will show that for $\Delta$-complexes these two sequences of groups
are really the same. In so doing, we will have built a sequence of topological
invariants that for a large class of spaces are fairly routine to compute. Then
all we will need to show is that they also capture useful information about
a space (i.e., we can prove useful theorems with them!).

\msk

And the basic idea behind defining them is that, with simplicial homology,
we have already done all of the hard work. What we do is, as before, build a 
sequence of (free) abelian groups, the chain groups $C_n(X)$, 
and boundary maps between them,
with consecutive maps composing to 0. Then, as before, the homology groups are
kernels mod images, i.e., cycles mod boundaries. And, as before, the basis
elements for each of our chain groups $C_n(X)$ will be the $n$-simplices
in $X$. But now $X$ is \u{any} topological space. So how do we get $n$-simplices
in such a space? We do the only thing we can; we {\it map} them in. 

\msk

More precisely, we work with {\it singular $n$-chains}, that is,
formal (finite) linear combinations $\sum a_i\sigma_i$, where $a_i\in{Bbb Z}$
and the $\sigma_i$ are {\it singular simplices}, that is, (continuous) 
maps $\sigma_i:\Delta^n\ra X$ from the (standard) $n$-simplex into $X$.
The boundary maps are really exactly as before; they are the alternating sum of
the restrictions of $\sigma_i$ to the $n+1$ faces of $\Delta^n$ . (Formally,
we must precompose these face maps with the (orientation-preserving) linear
isomorphism from the standard $(n-1)$-simplex to each of the faces, preserving
the ordering of their vertices.) The same proof as before (except that we interpret
the faces as restrictions of the map $\sigma_i$, instead of as physical faces)
shows that the composition of two successive boundaries are 0,
and so all of the machinery is in place to define the {\it singular homology
groups} $H_i(X)$ as the kernel of $\del_i$ modulo the image of $\del_{i+1}$ = $Z_i(X)/B_i(X)$ .
They are, by their definition, groups defined using the topological space $X$ as input,
and so are topological invariants of $X$. The elements are equivalence classes of $i$-cycles,
where $z_1\sim z_2$ if $z_1-z_2=\del w$ for some $(i+1)$-chain $w$ . We say that $z_1$ and $z_2$ are
{\it homologous}.


\vfill
\end
