

\magnification=1200
\overfullrule=0pt
\parindent=0pt

\nopagenumbers

\input amstex

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

%\voffset=1.4in
%\hoffset=-.5in
%\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}



\ctln{\bf Math 971 Algebraic Topology}

\ssk

\ctln{March 8, 2005}

\msk

We typically think of building a $\Delta$-complex $X$ inductively. 
The {\it 0-simplices}
(i.e., points), or {\it vertices}, form the 0-skeleton 
$X^{(0)}$. $n$-simplices $\sigma^n = [v_0,\ldots v_n]$ attach 
to the $(n-1)$-skeleton
to form the $n$-skeleton $X^{(n)}$; the restriction
of the attaching map to each face of $\sigma^n$ is, by definition,
an $(n-1)$-simplex in $X$. The attaching map is (by induction)
really determined by a map $\{v_0,\ldots ,v_n\}\ra X^{(0)}$, since this 
determines the attaching maps for the 1-simplices in the boundary of the
$n$-simplex, which 
gives 1-simplices in $X$, which then give the attaching maps for
the 2-simplices in the boundary, etc. Note that the reverse is not true;
the vertices of two different $n$-simplices in $X$ can be the same.
For example, think of the 2-sphere as a pair of 2-simplices whose 
boundaries are glued by the identity. $\Delta$-complexes generalize
{\it simplicial complexes} where the simplices are required to attach
by homeomorphisms to the skeleton, and the intersection of two 
simplices are a (single) sub-simplex of each. This has the advantage
over $\Delta$-complexes that an $n$-simplex is determined uniquely
by the set of vertices in $X^{(0)}$ that it attaches to. This means that,
in principle, a simplicial complex (and everything associated with it, e.g., 
its homology groups!) can be treated purely combinatorially;
the complex is ``really'' a certain collection of subsets of the vertices
(since these determine the simplices), with the property that any subset $B$
of a subset $A$ that has been declared to be a simplex is also a simplex.
But they have the disadvantage that it typically takes far more
simplices to build a simplicial structure on a space $X$ that it does to build a 
$\Delta$-structure. This makes the computations we are about to do take 
far longer.

\msk

The final detail that we need before defining (simplicial) homology
groups is the notion of an {\it orientation} on a simplex of $X$.
Each simplex $\sigma^n$ is determined by a map 
$f:\{v_0,\ldots ,v_n\}\ra X^{(0)}$; an orientation on $\sigma^n$ is an
(equivalence class of) the ordered $(n+1)$-tuple $(f(v_0),\ldots f(v_n)) = (V_0,\ldots ,V_n)$.
Another ordering of the
same vertices represents the same orientation if there is an {\it even} permutation
taking the entries of the first $(n+1)$-tuple to the second. This should be thought 
of as a generalization of the right-hand rule for ${\Bbb R}^3$, interpreted as
orienting the vertices of a 3-simplex. Note that there are precisely two
orientations on a simplex.

\msk

Now to define homology! We start by defining $n$-{\it chains};
these are (finite) formal linear combinations of the (oriented!) $n$-simplices
of $X$, where $-\sigma$ is interpreted as $\sigma$ with the opposite
(i.e., other) orientation. Adding formal linear combinations formally,
we get the $n$-th {\it chain group} 
$C_n(X) = \{\sum n_\alpha \sigma_\alpha$ : $\sigma_\alpha$ an oriented $n$-simplex in $X\}$ .
We next define a {\it boundary operator} $\del:C_n(X)\ra C_{n-1}(X)$, whose image will be 
the $(n-1)$-chains that are the ``boundaries'' of $n$-chains. The idea is that the boundary of a 2-simplex,
for example, should be a ``sum'' of its three faces (since they do make up the boundary of the 
simplex), but we need to take into account their orientations, in order to be getting the correct
sum. Thinking of the orientation on a 1-simplex $[v,w]$ as an arrow pointing from $v$ to $w$, we are lead to 
believe that the boundary of a 2-simplex $[u,v,w]$ should be $[u,v]+[v,w]+[w,u]$. Similarly, the boundary 
of $[u,v]$, on reflection, should be $[v]-[u]$, to distinguish the head of the arrow (the $+$ side) from the
tail (the $-$ side). On the basis of these examples, trying to find a consistent formula, one might eventually
be led to the following formulation.
We define the boundary on the basis
elements $\sigma_\alpha = \sigma$ of $C_n(X)$ as
$\del\sigma = \sum (-1)^i\sigma|_{[v_0,\ldots,\widehat{v_{i}},\ldots ,v_n]}$ , 
where $\sigma:[v_0,\ldots ,v_n]\ra X$ is the characteristic map of $\sigma_\alpha$ .
$\del\sigma$ is therefore an alternating sum of the faces of $\sigma$. 
We then extend the definition by linearity to all of $C_n(X)$. When a notation indicating
dimension is needed, we write $\del=\del_n$ . We define $\del_0=0$. 

\msk

This definition, it utrns out, is cooked up to make the maxim ``boundaries have no boundary'' true;
that is, $\delta_{n-1}\circ \delta_n = 0$, the $0$ map. This is because, for any simplex
$\sigma = [v_0,\ldots v_n]$, 

\msk

$\displaystyle \delta\circ\delta(\sigma) = 
\delta(\sum_{i=0}^n  (-1)^i\sigma|_{[v_0,\ldots,\widehat{v_{i}},\ldots ,v_n]})$

= $\displaystyle (\sum_{j<i}(-1)^j(-1)^i\sigma|_{[v_0,\ldots,\widehat{v_{j}},\ldots,\widehat{v_{i}},\ldots ,v_n]})
+(\sum_{j>i}(-1)^{j-1}(-1)^i\sigma|_{[v_0,\ldots,\widehat{v_{i}},\ldots,\widehat{v_{j}},\ldots ,v_n]})$

\msk

The distinction between the two pieces is that in the second part, $v_j$ is actually the $(j-1)$-st vertex
of the face. Switching the roles of $i$ and $j$ in the second sum, we find that the two are
negatives of one another, so they sum to $0$, as desired.

\msk

And this little calculation is all that it takes to define homology groups! What this tells 
us is that im$(\delta_{n+1})\subseteq\ker(\delta_{n})$ for every $n$. 
$\ker(\delta_{n})=Z_n(X)$ are called the {\it $n$-cycles} of $X$; they are the $n$-chains with
$0$ (i.e., empty) boundary. They form a (free) abelian subgroup of $C_n(X)$. 
im$(\delta_{n+1}) = B_n(X)$ are the {\it $n$-boundaries} of $X$; they are, of course,
the boundaries of $(n+1)$-chains in $X$. The $n$-th homology group of $X$,
$H_n(X)$ is the quotient $Z_n(X)/B_n(X)$ ; it is an abelian group. This consists, essentially, of
the $n$-cycles that are not boundaries; i.e., they represent the ``holes'' in $X$.

\msk

A key observation is that the boundary maps $\delta_n$ are linear, that is,
they are homomorphisms between the free abelian groups 
$\delta_n:C_n(X)\ra C_{n-1}(X)$. Consequently, they can be expressed as
(integer-valued) matrices $\Delta_n$. Row reducing $\Delta_n$ 
(over the integers!) allows us to find a 
basis $v_1,\ldots ,v_k$ for $Z_n(X)$ (clearing denomenators
to get vectors over ${\Bbb Z}$). Then since $\Delta_n\Delta_{n+1}=0$, the
columns of $\Delta_{n+1}$ are in the kernel of $\Delta_n$, so can be
expressed as linear combinations of the $v_i$ . These combinations can be
determined by row reducing the augmented matrix
$( v_1\cdots v_k | \Delta_{n+1} )$ . This will row reduce to 
$\pmatrix I&|&C\\ 0&|&0\\ \endpmatrix$, and $C$ basically describes the boundaries
$B_n(X)$ in terms of the basis $v_1,\ldots ,v_k$ . The homology group
$H_n(X)$ is then the {\it cokernel} of $C$, i.e., ${\Bbb Z}^k/$im$C$ .
Note that $C$ will have integer entries, since we know that 
the columns of $\Delta_{n+1}$ can be expressed as integer linear
combinations of the $v_i$, and, being a basis, there is only
one such expression. 




\vfill
\end