3Bass–Serre theory
IV Topics in Geometric Group Theory
3.1 Graphs of spaces
Bass–Serre theory is a way of building spaces by gluing old spaces together, in a
way that allows us to understand the fundamental group of the resulting space.
In this section, we will give a brief (and sketchy) introduction to Bass-Serre
theory, which generalizes some of the ideas we have previously seen, but they
will not be used in the rest of the course.
Suppose we have two spaces
X, Y
, and we want to glue them along some
subspace. For concreteness, suppose we have another space
Z
, and maps
∂
−
:
Z → X
and
∂
+
:
Z → Y
. We want to glue
X
and
Y
by identifying
∂
−
(
z
)
∼ ∂
+
(
z
)
for all z ∈ Z.
If we simply take the disjoint union of
X
and
Y
and then take the quotient,
then this is a pretty poorly-behaved construction. Crucially, if we want to
understand the fundamental group of the resulting space via Seifert–van Kampen,
then the maps
∂
±
must be very well-behaved for the theorem to be applicable.
The homotopy pushout corrects this problem by gluing like this:
∂
−
∂
+
X Y
Z
Definition
(Homotopy pushout)
.
Let
X, Y, Z
be spaces, and
∂
−
:
Z → X
and
∂
+
: Z → Y be maps. We define
X q
Z
Y = (X q Y q Z × [−1, 1])/ ∼,
where we identify ∂
±
(z) ∼ (z, ±1) for all z ∈ Z.
By Seifert–van Kampen, we know π
1
(X q
Z
Y ) is the pushout
π
1
Z π
1
(X)
π
1
Y π
1
(X q
Z
Y )
∂
+
∗
∂
−
∗
In other words, we have
π
1
(X q
Z
Y )
∼
=
π
1
X ∗
π
1
Z
π
1
Y.
In general, this is more well-behaved if the maps
∂
±
∗
are in fact injective, and we
shall focus on this case.
With this construction in mind, we can try to glue together something more
complicated:
Definition
(Graph of spaces)
.
A graph of spaces
X
consists of the following
data
– A connected graph Ξ.
– For each vertex v ∈ V (Ξ), a path-connected space X
v
.
– For each edge e ∈ E(Ξ), a path-connected space X
e
.
–
For each edge
e ∈ E
(Ξ) attached to
v
±
∈ V
(Ξ), we have
π
1
-injective maps
∂
±
e
: X
e
→ X
v
±
.
The realization of X is
|X | = X =
`
v∈V (Ξ)
X
v
q
`
e∈E(Ξ)
(X
e
× [−1, 1])
(∀e ∈ E(Ξ), ∀x ∈ X
e
, (x, ±1) ∼ ∂
±
e
(x))
.
These conditions are not too restrictive. If our vertex or edge space were
not path-connected, then we can just treat each path component as a separate
vertex/edge. If our maps are not
π
1
injective, as long as we are careful enough,
we can attach 2-cells to kill the relevant loops.
Example.
A homotopy pushout is a special case of a realization of a graph of
spaces.
Example. Suppose that the underlying graph looks like this:
v
e
The corresponding gluing diagram looks like
Fix a basepoint
∗ ∈ X
e
. Pick a path from
∂
−
e
(
∗
) to
∂
+
e
(
∗
). This then gives a
loop
t
that “goes around”
X
e
×
[
−
1
,
1] by first starting at
∂
−
e
(
∗
), move along
∗ × [−1, 1], then returning using the path we chose.
Since every loop inside
X
e
can be pushed down along the “tube” to a loop
in X
v
, it should not be surprising that the group π
1
(X) is in fact generated by
π
1
(X
v
) and t.
In fact, we can explicitly write
π
1
X =
π
1
X
v
∗ hti
hh(∂
+
e
)
∗
(g) = t(∂
−
e
)
∗
(g)t
−1
∀g ∈ π
1
X
e
ii
.
This is known as an HNN extension. The way to think about this is as follows
— we have a group
π
1
X
v
, and we have two subgroups that are isomorphic to each
other. Then the HNN extension is the “free-est” way to modify the group so
that these two subgroups are conjugate.
How about for a general graph of spaces? If Ξ is a graph of spaces, then its
fundamental group has the structure of a graph of groups G.
Definition (Graph of groups). A graph of groups G consists of
– A graph Γ
– Groups G
v
for all v ∈ V (Γ)
– Groups G
e
for all e ∈ E(Γ)
– For each edge e with vertices v
±
(e), injective group homomorphisms
∂
±
e
: G
e
→ G
v
±
(e)
.
In the case of a graph of spaces, it was easy to define a realization. One way
to do so is that we have already see how to do so in the two above simple cases,
and we can build the general case up inductively from the simple cases, but that
is not so canonical. However, this has a whole lot of choices involved. Instead,
we are just going to do is to associate to a graph of groups
G
a graph of spaces
X
which “inverts” the natural map from graphs of spaces to graphs of groups,
given by taking π
1
of everything.
This can be done by taking Eilenberg–Maclane spaces.
Definition
(Aspherical space)
.
A space
X
is aspherical if
˜
X
is contractible. By
Whitehead’s theorem and the lifting criterion, this is true iff
π
n
(
X
) = 0 for all
n ≥ 2.
Proposition.
For all groups
G
there exists an aspherical space
BG
=
K
(
G,
1)
such that
π
1
(
K
(
G,
1))
∼
=
G
. Moreover, for any two choices of
K
(
G,
1) and
K
(
H,
1), and for every homomorphism
f
:
G → H
, there is a unique map (up to
homotopy)
¯
f
:
K
(
G,
1)
→ K
(
H,
1) that induces this homomorphism on
π
1
. In
particular, K(G, 1) is well-defined up to homotopy equivalence.
Moreover, we can choose
K
(
G,
1) functorially, namely there are choices of
K
(
G,
1) for each
G
and choices of
¯
f
such that
f
1
◦ f
2
=
f
1
◦
¯
f
2
and
id
G
=
id
K(G,1)
for all f, G, H.
These K(G, 1) are known as Eilenberg–MacLane spaces.
When we talked about presentations, we saw that this is true if we don’t
have the word “aspherical”. But the aspherical requirement makes the space
unique (up to homotopy).
Using Eilenberg–MacLane spaces, given any graph of groups
G
, we can
construct a graph of spaces Ξ such that when we apply
π
1
to all the spaces in
X , we recover G.
We can now set
π
1
G = π
1
|X |.
Note that if Γ is finite, and all the G
v
’s are finitely-generated, then π
1
G is also
finitely-generated, which one can see by looking into the construction of
K
(
G,
1).
If Γ is finite, all
G
v
’s are finitely-presented and all
G
e
’s are finitely-generated,
then π
1
G is finitely-presented.
For more details, read Trees by Serre, or Topological methods in group theory
by Scott and Wall.