6Simplicial complexes
II Algebraic Topology
6 Simplicial complexes
So far, we have taken a space
X
, and assigned some things to it. The first was
easy —
π
0
(
X
). It was easy to calculate and understand. We then spent a lot of
time talking about
π
1
(
X
). What are they good for? A question we motivated
ourselves with was to prove
R
m
∼
=
R
n
implies
n
=
m
.
π
0
was pretty good for the
case when
n
= 1. If
R
m
∼
=
R
, then we would have
R
m
\ {
0
}
∼
=
R \ {
0
} ' S
0
. We
know that
|π
0
(
S
0
)
|
= 2, while
|π
0
(
R
m
\ {
0
}
)
|
= 1 for
m 6
= 1. This is just a fancy
way of saying that R \ {0} is disconnected while R
m
\ {0} is not for m 6= 1.
We can just add 1 to
n
, and add 1 to our subscript. If
R
m
∼
=
R
2
, then we
have
R
m
\{
0
}
∼
=
R
2
\{
0
} ' S
1
. We know that
π
1
(
S
1
)
∼
=
Z
, while
π
1
(
R
m
\{
0
}
)
∼
=
π
1
(S
m−1
)
∼
=
1 unless m = 2.
The obvious thing to do is to create some
π
n
(
X
). But as you noticed,
π
1
took us quite a long time to define, and was really hard to compute. As we
would expect, this only gets harder as we get to higher dimensions. This is
indeed possible, but will only be done in Part III courses.
The problem is that π
n
works with groups, and groups are hard. There are
too many groups out there. We want to do some easier algebra, and a good
choice is linear algebra. Linear algebra is easy. In the rest of the course, we
will have things like
H
0
(
X
) and
H
1
(
X
), instead of
π
n
, which are more closely
related to linear algebra.
Another way to motivate the abandoning of groups is as follows: recall last
time we saw that if
X
is a finite cell, reasonably explicitly defined, then we can
write down a presentation
hS | Ri
for
π
1
(
X
). This sounds easy, except that we
don’t understand presentations. In fact there is a theorem that says there is no
algorithm that decides if a particular group presentation is actually the trivial
group.
So even though we can compute the fundamental group in terms of presenta-
tions, this is not necessarily helpful. On the other hand, linear algebra is easy.
Computers can do it. So we will move on to working with linear algebra instead.
This is where homology theory comes in. It takes a while for us to define it,
but after we finish developing the machinery, things are easy.