4The relative homological approach

IV Bounded Cohomology

4.1 Injective modules
When we defined ordinary group cohomology, we essentially defined it as the
right-derived functor of taking invariants. While we do not need the machinery
of homological algebra and derived functors to define group cohomology, having
that available means we can pick different injective resolutions to compute group
cohomology depending on the scenario, and often this can be helpful. It also
allows us to extend group cohomology to allow non-trivial coefficients. Thus, we
would like to develop a similar theory for bounded cohomology.
We begin by specifying the category we are working over.
Definition
(Banach Γ module)
.
A Banach Γ-module is a Banach space
V
together with an action Γ × V V by linear isometries.
Given a Banach Γ-module
V
, we can take the submodule of Γ-invariants
V
Γ
. The relative homological approach tells us we can compute the bounded
cohomology
H
·
b
, R
) by first taking an appropriate exact sequences of Banach
Γ-modules
0 R E
0
E
1
· · · ,
d
(0)
d
(1)
d
(2)
and then take the cohomology of the complex of Γ-invariants
0 E
Γ
0
E
Γ
1
E
Γ
2
· · ·
d
(1)
d
(2)
.
Of course, this works if we take
E
k
=
C
k+1
, A
) and
d
(k)
to be the differentials
we have previously constructed, since this is how we defined bounded cohomology.
The point is that there exists a large class of “appropriate” exact sequences such
that this procedure gives us the bounded cohomology.
We first need the following definition:
Definition
.
An injective morphism
i: A B
of Banach
spaces is admissible if there exists σ : B A with
σi = id
A
; and
kσk 1.
This is a somewhat mysterious definition, but when we have such a situation,
this in particular implies
im A
is closed and
B
=
i
(
A
)
ker σ
. In usual homological
algebra, we don’t meet these kinds of things, because our vector spaces always
have complements. However, here we need them.
Definition
(Injective Banach Γ-module)
.
A Banach Γ-module is injective if for
any diagram
A B
E
i
α
where
i
and
α
are morphisms of Γ-modules, and
i
there exists β : B E a morphism of Γ-modules such that
A B
E
i
α
β
commutes and kβk kαk.
In other words, we can extend any map from a closed complemented subspace
of B to E.
Definition
(Injective resolution)
.
Let
V
be a Banach Γ-module. An injective
resolution of V is an exact sequence
V E
0
E
1
E
2
· · ·
where each E
k
is injective.
Then standard techniques from homological algebra imply the following
theorem:
Theorem. Let E
·
be an injective resolution of R Then
H
·
(E
·
Γ
)
=
H
·
b
, R)
as topological vector spaces.
In case
E
·
admits contracting homotopies, this isomorphism is semi-norm
decreasing.
Unsurprisingly, the defining complex for bounded cohomology were composed
of injective Γ-modules.
Lemma.
`
n
) for n 1 are all injective Banach Γ-modules.
`
alt
n
) for n 1 are injective Banach Γ-modules as well.
This is a verification. More interestingly, we have the following
Proposition. The trivial Γ-module R is injective iff Γ is amenable.
As an immediate corollary, we know that if Γ is amenable, then all the higher
bounded cohomology groups vanish, as 0
R
0
0
· · ·
is an injective
resolution.
Proof.
() Suppose A is injective. Consider the diagram
R `
(Γ)
R
i
,
where
i
(
t
) is the constant function
t
. We need to verify that
i
is an
admissible injection. Then we see that
σ
(
f
) =
f
(
e
) is a left inverse to
i
and
kσk
1. Then there exists a morphism
β : `
(Γ)
R
filling in the
diagram with kβk k id
R
k = 1 and in particular
β(1
Γ
) = 1
Since the action of Γ on
R
is trivial, this
β
is an invariant linear form on
Γ, and we see that this is an invariant mean.
()
Assume Γ is amenable, and let
m: `
(Γ)
R
be an invariant mean.
Consider a diagram
A B
R
i
α
as in the definition of injectivity. Since
i
is an admissible, it has a left
inverse σ : B A. Then we can define
β(v) = m{γ 7→ α(σ(γ
v))}.
Then this is an injective map B R and one can verify this works.
This theory allows us to study bounded cohomology with more general
coefficients. This can also be extended to
G
a locally-compact second-countable
groups with coefficients a
G
-Banach module
E
which is the dual of a continuous
separable Banach module E
b
. This is more technical and subtle, but it works.