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Section 7.7 applies the theory developed in this chapter to our ever-interesting space
P
n
.
The reader is warned that this chapter may be especially hard going if he/she has
not previously studied some abstract algebra. We shall not be using any really
advanced ideas from abstract algebra, but if the reader is new to it and has no one
for a guide, then, as usual, it will take a certain amount of time to get accustomed to
thinking along these lines. Groups and homomorphism are quite a bit different from
topics in calculus and basic linear algebra. The author hopes the reader will perse-
vere because in the end one will be rewarded with some beautiful theories. The next
chapter will make essential use of what is developed here and apply it to the study of
manifolds. Manifolds are the natural spaces for geometric modeling and getting an
understanding of our universe.
Anyone reading this chapter should at least read Sections 7.2.1-7.2.5 as carefully
as possible. They give the reader uninitiated to algebraic topology a taste of what the
subject is about. It is important to pay attention to the details and work through them,
otherwise little will sink in and everything will be just a blur. Two other important
topics are the fundamental group and pseudomanifolds. Although we try to be as clear
as we can be about basic concepts, the proofs in the chapter will tend to get less and
less detailed as we go along because we want to give as much of an overview as pos-
sible. Even if things start getting too abstract, it is recommended that one glance over
all the material (definitions and basic theorems) to at least get an overall picture of
how algebraic topology tries to pin down the structure and classification of topolog-
ical spaces.
7.2
Homology Theory
7.2.1
Homology Groups
The motivation for the homology groups (and especially the homotopy groups defined
later in the chapter) is based on the intuitive idea that topological spaces can be char-
acterized in terms of the number and type of “holes” that they have. There is no precise
definition of a hole. The simplest examples of spaces without holes are the Euclidean
spaces
R
n
. The prototype of a space that has an “n-dimensional hole” is
S
n
. The hole
exists because
S
n
cannot be contracted to a point
within S
n
without tearing it. In our
discussion we shall use surfaces and their one-dimensional holes as examples because
one can draw nice simple pictures in this case to illustrate what we are talking
about.
Consider the infinite cylinder
C
in Figure 7.1. In terms of holes, we would say that
the cylinder has a “vertical hole.” The existence of this hole is demonstrated by the
fact that we have closed curves (circles), such as a and b, that cannot be contracted
to a point
in C
. The curves a and b actually determine the same hole in
C
. One reason
for this is that one can deform one curve into the other. On the other hand, the closed
curves a and b in the torus
T
in Figure 7.1 correspond to the presence of two distinct
holes, an “inside” and “outside” hole. Of course, some closed curves, such as g in the
cylinder
C
, can be contracted to a point and do not correspond to any hole. It follows
that if we are going to study (one-dimensional) holes by means of closed curves then
