\chapter{Algebraic Topology}

It would be a mistake to talk about cohomology theory without
introducing homology theory.  We have seen that it is quite challenging
to compute the cohomology module of even some simple manifolds.
Homology theory can greatly simplify the task of computing cohomology
modules of some manifolds.  In this chapter, we explain how the
``homology module'' of a manifold $S$ can give us the
cohomology module of $S$.  We also provide a brief introduction to the
major tolls to find ``homology modules.''

This chapter should not be treated as a full introduction to algebraic
topology.  We present only the important results
about homology and cohomology theory to reach the goal mentioned in the
previous paragraph.  There are fundamentally two approaches to these
two topics: ``simplicial homology and cohomology''\ and
``singular homology and cohomology''.  The first approach is more
geometric and is mostly associated to algebraic geometry while the
second approach is purely algebraic and is associated to algebraic
topology.  We will briefly explore both approaches.

We present a very succinct introduction to simplicial homology and
cohomology in Sections~\ref{sectSimplHomol} and \ref{sectSimplCohom}.
We provide a little more complete introduction to singular homology
and cohomology in Sections~\ref{sectSingHom} and \ref{sectSingCohom}.
The importance of these sections is due to De Rham's theorem,
Theorem~\ref{deRhamThmSing}, in Subsection~\ref{ssectdeRhamSing} that
provides a relation between singular cohomology and de Rham cohomology
as presented in the previous chapter.

There is also a version of de Rham theorem, Theorem~\ref{DeRhamThm},
relating de Rham cohomology and simplicial cohomology.   We only
sketch this proof.  We have chosen to give a full proof of de Rham
theorem in the context of singular cohomology only.  In fact, as we
show in Section~\ref{sectRelSandS}, there is an equivalence between
the simplicial and singular theory.  Therefore, from a practical point
of view, there is no need to study both approaches.  From a
pedagogical point of view, it is nice to study both.
The readers whose main interested is the study of de Rham cohomology
may choose to only browse through Sections~\ref{sectSimplHomol} and
\ref{sectSimplCohom} on simplicial homology and cohomology, and
Section~\ref{sectRelSandS}.  These three sections are more interesting
for those who plan to study algebraic topology later.

Readers interested in learning more about the fascinating
subject of algebraic geometry should consult \cite{ST} for a good
basic introduction.   As for algebraic topology, \cite{GH} is
excellent references.  The content of this chapter is mainly based on
these two references and on \cite{LJM,MUat}.

This chapter requires some basic knowledge of abstract algebra; in
particular, of group theory.  There are many good books in algebra.
Two of them are \cite{He,Hu}.

\input{alg_top_fund_gr}

\input{alg_top_simpl_hom}

\input{alg_top_sing_hom}

\input{alg_top_relation}


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