4Some group theory
II Algebraic Topology
4.3 Free products with amalgamation
We managed to compute the fundamental group of the circle. But we want
to find the fundamental group of more things. Recall that at the beginning,
we defined cell complexes, and said these are the things we want to work with.
Cell complexes are formed by gluing things together. So we want to know what
happens when we glue things together.
Suppose a space
X
is constructed from two spaces
A, B
by gluing (i.e.
X
=
A ∪ B
). We would like to describe
π
1
(
X
) in terms of
π
1
(
A
) and
π
1
(
B
). To
understand this, we need to understand how to “glue” groups together.
Definition (Free product). Suppose we have groups
G
1
=
hS
1
| R
1
i, G
2
=
hS
2
|
R
2
i, where we assume S
1
∩ S
2
= ∅. The free product G
1
∗ G
2
is defined to be
G
1
∗ G
2
= hS
1
∪ S
2
| R
1
∪ R
2
i.
This is not a really satisfactory definition. A group can have many different
presentations, and it is not clear this is well-defined. However, it is clear that
this group exists. Note that there are natural homomorphisms
j
i
:
G
i
→ G
1
∗ G
2
that send generators to generators. Then we can have the following universal
property of the free product:
Lemma.
G
1
∗ G
2
is the group such that for any group
K
and homomorphisms
φ
i
:
G
i
→ K
, there exists a unique homomorphism
f
:
G
1
∗ G
2
→ K
such that
the following diagram commutes:
G
2
G
1
G
1
∗ G
2
K
φ
2
j
2
φ
1
j
1
f
Proof.
It is immediate from the universal property of the definition of presenta-
tions.
Corollary. The free product is well-defined.
Proof.
The conclusion of the universal property can be seen to characterize
G
1
∗ G
2
up to isomorphism.
Again, we have a definition in terms of a concrete construction of the group,
without making it clear this is well-defined; then we have a universal property
that makes it clear this is well-defined, but not clear that the object actually
exists. Combining the two would give everything we want.
However, this free product is not exactly what we want, since there is little
interaction between
G
1
and
G
2
. In terms of gluing spaces, this corresponds to
gluing
A
and
B
when
A ∩ B
is trivial (i.e. simply connected). What we really
need is the free product with amalgamation, as you might have guessed from
the title.
Definition (Free product with amalgamation). Suppose we have groups
G
1
,
G
2
and H, with the following homomorphisms:
H G
2
G
1
i
2
i
1
The free product with amalgamation is defined to be
G
1
∗
H
G
2
= G
1
∗ G
2
/hh{(j
2
◦ i
2
(h))
−1
(j
1
◦ i
1
)(h) : h ∈ H}ii.
Here we are attempting to identify things “in H” as the same, but we need
to use the maps
j
k
and
i
k
to map the things from
H
to
G
1
∗ G
2
. So we want to
say “for any
h
,
j
1
◦ i
1
(
h
) =
j
2
◦ i
2
(
h
)”, or (
j
2
◦ i
2
(
h
))
−1
(
j
1
◦ i
1
)(
h
) =
e
. So we
quotient by the (normal closure) of these things.
We have the universal property
Lemma.
G
1
∗
H
G
2
is the group such that for any group
K
and homomorphisms
φ
i
:
G
i
→ K
, there exists a unique homomorphism
G
1
∗
H
G
2
→ K
such that the
following diagram commutes:
H G
2
G
1
G
1
∗
H
G
2
K
i
2
i
1
φ
2
j
2
φ
1
j
1
f
This is the language we will need to compute fundamental groups. These
definitions will (hopefully) become more concrete as we see more examples.