0Introduction
II Algebraic Topology
0 Introduction
In topology, a typical problem is that we have two spaces
X
and
Y
, and we want
to know if
X
∼
=
Y
, i.e. if
X
and
Y
are homeomorphic. If they are homeomorphic,
we can easily show this by writing down a homeomorphism. But what if they
are not? How can we prove that two spaces are not homeomorphic?
For example, are
R
m
and
R
n
homeomorphic (for
m 6
=
n
)? Intuitively, they
should not be, since they have different dimensions, and in fact they are not.
But how can we actually prove this?
The idea of algebraic topology is to translate these non-existence problems in
topology to non-existence problems in algebra. It turns out we are much better
at algebra than topology. It is much easier to show that two groups are not
isomorphic. For example, we will be able to reduce the problem of whether
R
m
and
R
n
are homeomorphic (for
m 6
=
n
) to the question of whether
Z
and
{e}
are isomorphic, which is very easy.
While the statement that
R
m
6
∼
=
R
n
for
n 6
=
m
is intuitively obvious, algebraic
topology can be used to prove some less obvious results.
Let
D
n
be the
n
dimensional unit disk, and
S
n−1
be the
n−
1 dimensional unit
sphere. We will be able to show that there is no continuous map
F
:
D
n
→ S
n−1
such that the composition
S
n−1
D
n
S
n−1
F
is the identity, where the first arrow is the inclusion map. Alternatively, this
says that we cannot continuously map the disk onto the boundary sphere such
that the boundary sphere is fixed by the map.
Using algebraic topology, we can translate this statement into an algebraic
statement: there is no homomorphism F : {0} → Z such that
Z {0} Z
F
is the identity. This is something we can prove in 5 seconds.
By translating a non-existence problem of a continuous map to a non-existence
problem of a homomorphism, we have made our life much easier.
In algebraic topology, we will be developing a lot of machinery to do this sort
of translation. However, this machinery is not easy. It will take some hard work,
and will be rather tedious and boring at the beginning. So keep in mind that
the point of all that hard work is to prove all these interesting theorems.
In case you are completely uninterested in topology, and don’t care if
R
m
and
R
n
are homeomorphic, further applications of algebraic topology include solving
equations. For example, we will be able to prove the fundamental theorem of
algebra (but we won’t), as well as Brouwer’s fixed point theorem (which says
that every continuous function from
D
2
→ D
2
has a fixed point). If you are not
interested in these either, you may as well drop this course.