5Seifert-van Kampen theorem

II Algebraic Topology



5.4 The fundamental group of all surfaces
We have found that the torus has fundamental group
Z
2
, but we already knew
this, since the torus is just
S
1
× S
1
, and the fundamental group of a product is
the product of the fundamental groups, as you have shown in the example sheet.
So we want to look at something more interesting. We look at all surfaces.
We start by defining what a surface is. It is surprisingly difficult to get
mathematicians to agree on how we can define a surface. Here we adopt the
following definition:
Definition (Surface). A surface is a Hausdorff topological space such that every
point has a neighbourhood U that is homeomorphic to R
2
.
Some people like
C
more that
R
, and sometimes they put
C
2
instead of
R
2
in
the definition, which is confusing, since that would have two complex dimensions
and hence four real dimensions. Needless to say, the actual surfaces will also
be different and have different homotopy groups. We will just focus on surfaces
with two real dimensions.
To find the fundamental group of all surfaces, we rely on the following
theorem that tells us what surfaces there are.
Theorem (Classification of compact surfaces). If
X
is a compact surface, then
X is homeomorphic to a space in one of the following two families:
(i)
The orientable surface of genus
g
, Σ
g
includes the following (please excuse
my drawing skills):
A more formal definition of this family is the following: we start with the
2-sphere, and remove a few discs from it to get
S
2
\
g
i=1
D
2
. Then we take
g tori with an open disc removed, and attach them to the circles.
(ii)
The non-orientable surface of genus
n
,
E
n
=
{RP
2
, K, · · · }
(where
K
is
the Klein bottle). This has a similar construction as above: we start with
the sphere S
2
, make a few holes, and then glue obius strips to them.
It would be nice to be able to compute fundamental groups of these guys.
To do so, we need to write them as polygons with identification.
Example. To obtain a surface of genus two, written Σ
2
, we start with what we
had for a torus:
a
b
a
b
If we just wanted a torus, we are done (after closing the loop), but now we want
a surface with genus 2, so we add another torus:
a
1
b
1
a
1
b
1
a
2
b
2
a
2
b
2
To visualize how this works, imagine cutting this apart along the dashed line.
This would give two tori with a hole, where the boundary of the holes are just the
dashed line. Then gluing back the dashed lines would give back our orientable
surface with genus 2.
In general, to produce Σ
g
, we produce a polygon with 4g sides. Then we get
π
1
Σ
g
= ha
1
, b
1
, · · · , a
g
, b
g
| a
1
b
1
a
1
1
b
1
1
· · · a
g
b
g
a
1
g
b
1
g
i.
We do we care? The classification theorem tells us that each surface is homeomor-
phic to some of these orientable and non-orientable surfaces, but it doesn’t tell us
there is no overlap. It might be that Σ
6
=
Σ
241
, via some weird homeomorphism
that destroys some holes.
However, this result lets us know that all these orientable surfaces are
genuinely different. While it is difficult to stare at this fundamental group
and say that
π
1
Σ
g
6
=
π
1
Σ
g
0
for
g 6
=
g
0
, we can perform a little trick. We can
take the abelianization of the group
π
1
Σ
g
, where we further quotient by all
commutators. Then the abelianized fundamental group of Σ
g
will simply be
Z
2g
. These are clearly distinct for different values of
g
. So all these surfaces are
distinct. Moreover, they are not even homotopy equivalent.
The fundamental groups of the non-orientable surfaces is left as an exercise
for the reader.