1. Cohomology of groups
Let G be a group (which has to be thought as endowed with the discretetopology). We begin by recalling the usual definition of the cohomologyof G with coefficients in a G-module V . All the results described here areclassical, and date back to the works of Eilenberg and Mac Lane, Hopf,Eckmann, and Freudenthal in the 1940s, and to the contributions of Cartanand Eilenberg to the birth of the theory of homological algebra in the early1950s.
For the whole course, unless otherwise stated, groups actions will alwaysbe on the left, and modules over (possibly non-commutative) rings will al-ways be left modules. If R is any commutative ring (in fact, we will beinterested only in the cases R = Z, R = R), we denote by R[G] the groupring associated to R and G, i.e. the set of finite linear combinations of ele-ments of G with coefficients in R, endowed with the operations
∑
g∈G
agg
+
∑
g∈G
bgg
=
∑
g∈G
(ag + bg)g ,
∑
g∈G
agg
·
∑
g∈G
bgg
=
∑
g∈G
∑
h∈G
aghbh−1g
(when the sums on the left-hand sides of these equalities are finite, then thesame is true for the corresponding right-hand sides). Observe that an R[G]-module V is just an R-module V endowed with an action of G by R-linearmaps, and that an R[G]-map between R[G]-modules is just an R-linear mapwhich commutes with the actions of G. When R is understood, we will oftenrefer to R[G]-maps as to G-maps. If V is an R[G]-module, then we denoteby V G the subspace of G-invariants of V , i.e. the set
V G = {v ∈ V | g · v = v for every g ∈ G} .
We are now ready to describe the complex of cochains which defines thecohomology of G with coefficients in V . For every n ∈ N we set
Cn(G,V ) = {f : Gn+1 → V } ,
and we define δn : Cn(G,V )→ Cn+1(G,V ) as follows:
δnf(g0, . . . , gn+1) =
n+1∑
i=0
(−1)if(g0, . . . , gi, . . . , gn+1) .
It is immediate to check that δn+1 ◦ δn = 0 for every n ∈ N, so the pair(C•(G,V ), δ•) is indeed a complex, which is usually known as the homoge-neous complex associated to the pair (G,V ). The formula
(g · f)(g0, . . . , gn) = g(f(g−1g0, . . . , g−1gn))
endows Cn(G,V ) with an action of G, whence with the structure of an R[G]-module. It is immediate to check that δn is a G-map, so the G-invariants
1
2
C•(G,V )G provide a subcomplex of C•(G,V ), whose homology is by defi-nition the cohomology of G with coefficients in V . More precisely, if
Zn(G,V ) = Cn(G,V )G ∩ ker δn , Bn(G,V ) = δn−1(Cn−1(G,V )G)
(where we understand that B0(G,V ) = 0), then Bn(G,V ) ⊆ Zn(G,V ), and
Hn(G,V ) = Zn(G,V )/Bn(G,V ) .
Definition 1.1. The R-module Hn(G,V ) is the n-th cohomology moduleof G with coefficients in V .
By definition, we have H0(G,V ) = V G for every R[G]-module V . In thiscourse we will be mainly concerned with the case when R = V = R (or, lessfrequently, when R = V = Z), and the action of G on V is trivial.
1.1. Functoriality. Group cohomology provides a binary functor. If G isa group and α : V1 → V2 is an R[G]-map, then f induces an obvious changeof coefficients map α• : C•(G,V1) → C•(G,V2) obtained by composing anycochain with values in V1 with α. It is not difficult to show that, if
0 −→ V1α−→ V2
β−→ V3 −→ 0
is an exact sequence of R[G]-modules, then the induced sequence of com-plexes
0 −→ C•(G,V1)G α•
−→ C•(G,V2)G β•
−→ C•(G,V3)G −→ 0
is also exact, so there is a long exact sequence
0 −→ H0(V1, G) −→ H0(V2, G) −→ H0(V3, G) −→ H1(G,V1) −→ H1(G,V2) −→ . . .
in cohomology.Let us now consider the functioriality of cohomology with respect to the
first variable. Let ψ : G1 → G2 be a group homomorphism, and let V be anR[G2]-module. Then G1 acts on V via ψ, so V is endowed with a naturalstructure of R[G2]-module. If we denote this structure by ψ−1V , then themaps
ψn : Cn(G2, V )→ Cn(G1, ψ−1V ) , ψn(f)(g0, . . . , gn) = f(ψ(g0), . . . , ψ(gn))
provide a chain map such that ψn(Cn(G2, V )G2) ⊆ Cn(G1, ψ−1V )G1 . As a
consequence, we get a well-defined map
Hn(ψn) : Hn(G2, V )→ Hn(G1, ψ−1V )
in cohomology. We will consider this map mainly in the case when V is thetrivial module R. In that context, the discussion above shows that everyhomomorphism ψ : G1 → G2 induces a map
Hn(ψn) : Hn(G2, R)→ Hn(G1, R)
in cohomology.
3
1.2. The topological interpretation of group cohomology. Let us re-call the well-known topological interpretation of group cohomology. Werestrict our attention to the case when V = R is a trivial G-module. If Xis any topological space, then we denote by C•(X,R) (resp. C•(X,R)) thecomplex of singular chains (resp. cochains) with coefficients in R, and byH•(X,R) (resp. H•(X,R)) the corresponding singular homology module.
Suppose now that X is any path-connected topological space satisfyingthe following properties:
(1) the fundamental group of X is isomorphic to G,
(2) the space X admits a universal covering X, and
(3) X is R-acyclic, i.e. Hn(X,R) = 0 for every n ≥ 1.
Then H•(G,R) is canonically isomorphic to H•(X,R) (see Subsection 7.4).If X is a CW-complex, then condition (2) is automatically satisfied, andWhitehead Theorem implies that condition (3) may be replaced by one ofthe following equivalent conditions:
(3’) X is contractible, or(3”) πn(X) = 0 for every n ≥ 2.
If a CW-complex satisfies conditions (1), (2) and (3) (or (3’), or (3”)),then one usually says that X is a K(G, 1), or an Eilenberg-MacLane space.Whitehead Theorem implies that the homotopy type of a K(G, 1) only de-pends on G, so if X is any K(G, 1), then it makes sense to define H i(G,R)by setting H i(G,R) = H i(X,R). It is not difficult to show that this defi-nition agrees with the definition of group cohomology given above. In fact,associated to G there is the ∆-complex BG, having one n-simplex for ev-ery ordered (n + 1)-tuple of elements of G, and obvious face operators (seee.g. [Hat02]). When endowed with the weak topology, BG is clearly con-tractible. Moreover, the group G acts freely and simplicially on BG, so thequotient XG of BG by the action of G inherits the structure of a ∆-complex,and is such that π1(XG) = G. In other words, XG is a K(G, 1). By defini-tion, the space of the simplicial n-cochains on BG coincides with the moduleCn(G,R) introduced above, and the simplicial homology ofXG is isomorphicto the homology of the G-invariant simplicial cochains on BG. This allowsus to conclude that H•(XG, R) is canonically isomorphic to H•(G,R).
2. Bounded cohomology of groups
Let us now shift our attention to bounded cohomology of groups. In orderto do so we first need to define the notion of normed G-module. Let Rand G be as above. For the sake of simplicity, we assume that R = Z orR = R, and we denote by | · | the usual absolute value on R. A normedR[G]-module V is an R[G]-module endowed with an invariant norm, i.e. amap ‖ · ‖ : V → R such that:
• ‖v‖ = 0 if and only if v = 0,• ‖r · v‖ = |r| · ‖v‖ for every r ∈ R, v ∈ V ,
4
• ‖v + w‖ ≤ ‖v‖ + ‖w‖ for every v,w ∈ V ,• ‖g · v‖ = ‖v‖ for every g ∈ G, v ∈ V .
A G-map between normed R[G]-modules is a G-map between the underlyingR[G]-modules, which is bounded with respect to the norms.
Let V be a normed R[G]-module, and recall that Cn(G,V ) is endowedwith the structure of an R[G]-module. For every f ∈ Cn(G,V ) one mayconsider the ℓ∞-norm
‖f‖∞ = sup{‖f(g0, . . . , gn)‖ | (g0, . . . , gn) ∈ Gn+1} ∈ [0,+∞] .
We set
Cnb (G,V ) = {f ∈ Cn(G,V ) | ‖f‖∞ <∞}
and we observe that Cnb (G,V ) is an R[G]-submodule of Cn(G,V ). There-fore, Cnb (G,V ) is a normed R[G]-module. The differential δn : Cn(G,V )→Cn+1(G,V ) restricts to a G-map of normed R[G]-modules δn : Cnb (G,V )→
Cn+1b (G,V ), so one may define as usual
Znb (G,V ) = ker δn ∩ Cnb (G,V )G , Bnb (G,V ) = δn−1(Cn−1
b (G,V )G)
(where we understand that B0b (G,V ) = {0}), and set
Hnb (G,V ) = Znb (G,V )/Bn
b (G,V ) .
The ℓ∞-norm on Cnb (G,V ) restricts to a norm on Znb (G,V ), which descendsto a seminorm on Hn
b (G,V ) by taking the infimum over all the representa-tives of a coclass: namely, for every α ∈ Hn
b (G,V ) one sets
‖α‖∞ = inf{‖f‖∞ | f ∈ Znb (G,V ), [f ] = α} .
Definition 2.1. The R-module Hnb (G,V ) is the n-th bounded cohomology
module of G with coefficients in V . The seminorm ‖ · ‖∞ : Hnb (G,V )→ R is
called the canonical seminorm of Hnb (G,V ).
The canonical seminorm on Hnb (G,V ) is a norm if and only if the sub-
space Bnb (G,V ) is closed in Znb (G,V ) (whence in Cnb (G,V )). However, this
is not always the case: for example, if G is the fundamental group of a closedhyperbolic surface, then it is proved in [Som97, Som98] that H3
b (G,R) con-tains non-trivial elements with null seminorm. On the other hand, it wasproved independently by Ivanov [Iva90] and Matsumoto and Morita [MM85]that H2
b (G,R) is a Banach space for every group G (see Theorem 11.1).
2.1. Functoriality. Just as in the case of ordinary cohomology, also boundedcohomology provides a binary functor. The discussion carried out in Sub-section 1.1 applies word by word in the bounded case. Namely, every exactsequence of normed R[G]-modules
0 −→ V1α−→ V2
β−→ V3 −→ 0
induces a long exact sequence
0 −→ H0b (V1, G) −→ H0
b (V2, G) −→ H0b (V3, G) −→ H1
b (G,V1) −→ H1b (G,V2) −→ . . .
5
in cohomology. Moreover, if ψ : G1 → G2 is a group homomorphism, andV is a normed R[G2]-module, then V admits a natural structure of normedR[G1]-module, which is denoted by ψ−1V . Then, the homomorphism ψinduces a well-defined map
Hnb (ψn) : Hn
b (G2, V )→ Hnb (G1, ψ
−1V )
in bounded cohomology. In particular, in the case of trivial coefficients weget a map
Hn(ψn) : Hnb (G2, R)→ Hn
b (G1, R) .
2.2. The comparison map. The inclusion C•b (G,V ) → C•(G,V ) induces
a map
c : H•b (G,V )→ H•(G,V )
called the comparison map. We will see soon that the comparison map isneither injective nor surjective in general. One could approach the studyof Hn
b (G,V ) by looking separately at the kernel and at the image of thecomparison map. This strategy is described in Section 4 for bounded coho-mology groups in low degree.
2.3. Looking for a topological interpretation of bounded cohomol-ogy of groups. Let us now restrict to the case when V = R is a trivialnormed G-module. We would like to compare the bounded cohomology ofG with a suitably defined singular bounded cohomology of a suitable topo-logical model for G. There is a straightforward notion of boundedness forsingular cochains, so the notion of bounded singular cohomology of a topo-logical space is easily defined (see Section 10). Then, it is still true that thebounded cohomology of a group G is canonically isomorphic to the boundedsingular cohomology of a K(G, 1) (in fact, even more is true: in the caseof real coefficients, the bounded cohomology of G is isometrically isomor-phic to the bounded singular cohomology of any countable CW-complex Xsuch that π1(X) = G – see Theorem 10.8). However, unfortunately theproof described in Subsection 1.2 for classical cohomology does not carryover to the bounded context. It is still true that homotopically equivalentspaces have isometrically isomorphic bounded cohomology, but, if XG is theK(G, 1) defined in Subsection 1.2, then there is no clear reason why thebounded simplicial cohomology of XG (which coincides with the boundedcohomology of G) should be isomorphic to the bounded singular cohomologyof XG.
For example, if X is any finite ∆-complex, then every simplicial cochainon X is obviously bounded, so the cohomology of the bounded simplicialcochains on X coincides with the classical singular cohomology of X, whichin general is very different from the bounded singular cohomology of X.
6
3. The bar resolution
We have defined the cohomology (resp. the bounded cohomology) of Gas the cohomology of the complex C•(G,V )G (resp. C•
b (G,V )G). Of course,
an element f ∈ C•(G,V )G is completely determined by the values it takeson (n+ 1)-tuples having 1 as first entry. More precisely, if we denote by
C0(G,V ) = V , C
n(G,V ) = Cn−1(G,V ) = {f : Gn → V } ,
and we consider Cn(G,V ) simply as an R-module for every n ∈ N, then we
have R-isomorphisms
Cn(G,V )G → Cn(G,V )
ϕ 7→ ((g1, . . . , gn) 7→ ϕ(1, g1, g1g2, . . . , g1 · gn)) .
Under these isomorphisms, the diffential δ• : C•(G,V )G → C•+1(G,V )G
translates into the differential δ•: C
•(G,V )→ C
•+1(G,V ) such that
δ0(v)(g) = g · v − v v ∈ V = C
0(G,V ), g ∈ G ,
and
δn(f)(g1, . . . , gn+1) = g1 · f(g2, . . . , gn+1)
+
n∑
i=1
(−1)if(g1, . . . , gigi+1, . . . , gn+1)
+ (−1)n+1f(g1, . . . , gn) ,
for n ≥ 1. The complex
0 −→ C0(G,V )
δ0
−→ C1(G,V )
δ2
−→ . . .δ
n−1
−→ Cn(G,V )
δn
−→ . . .
is usually known as the inhomogeneous complex associated to the pair(G,V ). By construction, the cohomology of this complex is canonicallyisomorphic to H•(G,V ).
Just as we did for the homogeneous complex, if V is a normed R[G]-module, then we may define the submodule C
nb (G,V ) of bounded elements
of Cn(G,V ). For every n ∈ N, the isomorphism Cn(G,V )G ∼= C
n(G,V )
restricts to an isometric isomorphism Cnb (G,V )G ∼= Cnb (G,V ), so C
•
b(G,V )
is a subcomplex of C•(G,V ), whose cohomology is canonically isometrically
isomorphic to H•b (G,V ).
4. (Bounded) cohomology of groups in low degree
In this section we analyze the (bounded) cohomology modules of a groupG in degree 0, 1, 2. We restrict our attention to the case when V = R isequal either to Z, or to R, both endowed with the structure of trivial R[G]-module. In order to simplify the computations, it will be convenient to workwith the inhomogeneous complexes of (bounded) cochains.
7
By the very definitions, we have C0(G,R) = C
0b(G,R) = R, and δ
0= 0,
soH0(G,R) = H0
b (G,R) = R .
4.1. (Bounded) group cohomology in degree one. Let us now describe
what happens in degree one. By definition, for every ϕ ∈ C1(G,R) we have
δ(f)(g1, g2) = f(g1) + f(g2)− f(g1g2)
(recall that we are assuming that the action of G on R is trivial). In other
words, if we denote by Z•(G,R) and B
•(G,R) the spaces of cocycles and
coboundaries of the inhomogeneous complex C•(G,R), then we have
H1(G,R) = Z1(G,R) = Hom(G,R) .
Moreover, every bounded homomorphism with values in Z or in R is obvi-ously trivial, so
H1b (G,R) = Z
1b(G,R) = 0
(here and henceforth, we denote by Z•
b(G,R) and B•
b(G,R) the spaces of co-
cycles and coboundaries of the bounded inhomogeneous complex C•
b(G,R)).
4.2. Group cohomology in degree two. The situation in degree two ismore interesting. A central extension of G by R is an exact sequence
1 −→ Rι−→ G′ π
−→ G→ 1
such that ι(R) is contained in the center of G. In what follows, in thissituation we will identify R with ι(R) ∈ G via ι. Two such extensions areequivalent if they may be put into a commutative diagram as follows:
1 // Rι1
//
Id��
G1π1
//
f
��
G
Id��
// 1
1 // Rι2
// G2π2
// G // 1
(by the commutativity of the diagram, the map f is necessarily an isomor-phism). Associated to an exact sequence
1 −→ Rι−→ G′ π
−→ G→ 1
there is a cocyle ϕ ∈ C2(G,R) which is defined as follows: let s : G→ G′ be
any map such that π ◦ s = IdG. Then we set
ϕ(g1, g2) = s(g1g2)−1s(g1)s(g2) .
By construction we have ϕ(g1, g2) ∈ ker π = R, so ϕ is indeed an element
in C2(G,R). It is easy to check that δ(ϕ) = 0, so ϕ ∈ Z
2(G,R). Moreover,
different choices for the section s give rise to cocycles which differ one fromthe other by a coboundary, so any central extension of G by R defines anelement in H2(G,R). It is not difficult to reverse this construction to showthat every element in H2(G,R) is represented by a central extension, and
8
two central extensions are equivalent if and only if they define the sameelement in H2(G,R) (see e.g. [Bro82] for full details). Therefore:
Proposition 4.1. The group H2(G,R) is in natural bijection with the setof equivalence classes of central extensions of G by R.
Remark 4.2. Proposition 4.1 may be easily generalized to the case of ar-bitrary extensions of G by R. Once such an extension is given, if G′ isthe middle term of the extension, then the action of G′ on R by conjugacydescends to a well-defined action of G on R, thus defining on R the struc-ture of a (possibly non-trivial) G-module (this structure is trivial preciselywhen R is central in G′). Then, one may associate to the extension an el-ement of the cohomology group H2(G,R) with coefficients in the (possiblynon-trivial) R[G]-module R.
4.3. Bounded group cohomology in degree two: quasimorphisms.As mentioned above, the study of H2
b (G,R) may be reduced to the study ofthe kernel and of the image of the comparison map
c : H2b (G,R)→ H2(G,R) .
We will describe in Section 11 a characterization of groups with injectivecomparison map due to Matsumoto and Morita [MM85]. In this subsectionwe describe the relationship between the kernel of the comparison map andthe space of quasimorphisms on G.
Definition 4.3. A map f : G → R is a quasimorphism if there exists aconstant D ≥ 0 such that
|f(g1) + f(g2)− f(g1g2)| ≤ D
for every g1, g2 ∈ G. The least D ≥ 0 for which the above inequalityis satisfied is the defect of f , and it is denoted by D(f). The space ofquasimorphisms is an R-module, and it is denoted by Q(G,R), or simply byQ(G) when R is understood.
By the very definition, a quasimorphism is an element of C1(G,R) having
bounded differential. Of course, both bounded functions (i.e. elements in
C1b(G,R)) and homomorphisms (i.e. elements in Z
1(G,R) = Hom(G,R))
are quasimorphisms.If every quasimorphism could be obtained just by adding a bounded func-
tion to a homomorphism, the notion of quasimorphism would not really in-troduce something new. Observe that every bounded homomorphism with
values in R is necessarily trivial, so C1b(G,R) ∩ Hom(G,R) = {0}, and we
may think of C1b(G,R) ⊕ Hom(G,R) as of the space of “trivial” quasimor-
phisms on G.The following result is an immediate consequence of the definitions, and
shows that the existence of “non-trivial” quasimorphisms is equivalent tothe non-injectivity of the comparison map c : H2
b (G,R)→ H2(G,R).
9
Proposition 4.4. The differential δ : Q(G,R)→ C2b(G,R) induces an iso-
morphism
Q(G,R)/(
C1b(G,R)⊕HomZ(G,R)
)∼= ker c .
Therefore, in order to show that H2b (G,R) is non-trivial it is sufficient
to construct quasimorphisms which do not stay at finite distance from ahomomorphism.
4.4. Homogeneous quasimorphisms. Let us introduce the following:
Definition 4.5. A quasimorphism f : G → R is homogeneous if f(gn) =n · f(g) for every g ∈ G, n ∈ Z. The space of homogeneous quasimorphismsis a submodule of Q(G,R), and it is denoted by Qh(G,R).
Of course, there are no non-trivial bounded homogeneous quasimorphisms.In particular, for every quasimorphism f there exists at most one homoge-neous quasimorphism f such that ‖f−f‖∞ < +∞, so a homogeneous quasi-morphism which is not a homomorphism cannot stay at finite distance froma homomorphism. When R = Z, it may happen that homogeneous quasi-morphisms are quite sparse in Q(G,Z): for example, for every α ∈ R \ Z,if we denote by ⌊x⌋ the largest integer which is not bigger than x, then thequasimorphism fα : Z→ Z defined by
(1) fα(n) = ⌊αn⌋
is not at finite distance from any element in Qh(Z,Z) = Hom(Z,Z). WhenR = R homogeneous quasimorphisms play a much more important role, dueto the following:
Proposition 4.6. Let f ∈ Q(G,R) be a quasimorphism. Then, there ex-ists a unique element f ∈ Qh(G,R) that stays at a finite distance from f .Moreover, we have
‖f − f‖∞ ≤ D(f) , D(f) ≤ 4D(f) .
Proof. For every g ∈ G, m,n ∈ N we have
|f(gmn − nf(gm)| ≤ (n− 1)D(f) ,
so∣∣∣∣f(gn)
n−f(gm)
m
∣∣∣∣ ≤∣∣∣∣f(gn)
n−f(gmn)
mn
∣∣∣∣+∣∣∣∣f(gmn)
mn−f(gm)
m
∣∣∣∣ ≤(
1
n+
1
m
)D(f) .
Therefore, the sequence f(gn)/n is a Cauchy sequence, and the same is truefor the sequence f(g−n)/(−n). Since f(gn)+f(g−n) ≤ f(0)+D(f) for everyn, we may conclude that the limit
f(g) = limn→∞
f(gn)
n
10
exists for every g ∈ G. Moreover, the inequality |f(gn) − nf(g)| ≤ (n −1)D(f) implies that ∣∣∣∣f(g)−
f(gn)
n
∣∣∣∣ ≤ D(f)
so by passing to the limit we obtain that ‖f−f‖∞ ≤ D(f). This immediatelyimplies that f is a quasimorphism such that D(f) ≤ 4D(f). Finally, thefact that f is homogeneous is obvious. �
In fact, the stronger inequality D(f) ≤ 2D(f) holds (see e.g. [Cal09] fora proof). Propositions 4.4 and 4.6 imply the following:
Corollary 4.7. The space Q(G,R) decomposes as a direct sum
Q(G,R) = Qh(G,R)⊕ C1b(G,R) .
Moreover, the restriction of δ to Qh(G,R) induces an isomorphism
Qh(G,R)/Hom(G,R) ∼= ker c .
4.5. Quasimorphisms on abelian groups. Suppose now thatG is abelian.Then, for every g1, g2 ∈ G, every element f ∈ Qh(G,R), and every n ∈ N
we have
|nf(g1g2)− nf(g1)− nf(g2)| = |f((g1g2)n)− f(gn1 )− f(g2)
n|
= |f(gn1 gn2 )− f(gn1 )− f(gn2 )| ≤ D(f) .
Dividing by n this inequality and passing to the limit for n → ∞ we getthat f(g1g2) = f(g1) + f(g2), i.e. f is a homomorphism. Therefore, everyhomogeneous quasimorphism on G is a homomorphism. Putting togetherProposition 4.4 and Corollary 4.7 we obtain the following:
Corollary 4.8. If G is abelian, then the comparison map
c : H2b (G,R)→ H2(G,R)
is injective.
In fact, a much stronger result holds: if G is abelian, then it is amenable(see Definition 6.1), so Hn
b (G,R) = 0 for every n ≥ 1 (see Corollary 6.7). Westress that Corollary 4.8 does not hold in the case with integer coefficients.For example, we have the following:
Proposition 4.9. We have H2b (Z,Z) = R/Z.
Proof. The short exact sequence
0 −→ Z −→ R −→ R/Z −→ 0
induces a short exact sequence of complexes
0 −→ C•
b(Z,Z) −→ C•
b(Z,R) −→ C•(Z,R/Z) −→ 0 .
Let us consider the following portion of the long exact sequence induced incohomology:
. . . −→ H1b (Z,R) −→ H1(Z,R/Z) −→ H2
b (Z,Z) −→ H2b (Z,R) −→ H2(Z,R/Z) −→ . . .
11
Recall from Subsection 4.1 thatH1b (Z,R) = 0. Moreover, we haveH2(Z,R) =
H2(S1,R) = 0, so H2b (Z,R) is equal to the kernel of the comparison map
c : H2b (Z,R) → H2(Z,R), which vanishes by Corollary 4.8. Therefore, we
have
H2b (Z,Z) ∼= H1(Z,R/Z) = Hom(Z,R/Z) ∼= R/Z .
�
It follows from the proof of the previous proposition that the isomorphismR/Z→ H2
b (Z,Z) is given by the map
R/Z ∋ [α] 7→ [δfα] ∈ H2b (Z,Z) ,
where fα is the quasimorphism defined in (1). It is interesting to notice thatthe moduleH2
b (Z,Z) is not finitely generated over Z. This fact already showsthat bounded cohomology may be very different from classical cohomology.The same phenomenon may occur in the case of real coefficients: in thefollowing section we show that, if F2 is the free group on 2 elements, thenH2b (F2,R) is an infinite dimensional vector space.
4.6. The image of the comparison map. In Subsection 4.2 we have in-terpreted elements in H2(G,R) as equivalence classes of central extensionsof G by R. Of course, one may wonder whether elements of H2(G,R) admit-ting bounded representatives represent peculiar central extensions. It turnsout that this is indeed the case: bounded classes represent central exten-sions which are quasi-isometrically trivial (i.e. quasi-isometrically equivalentto product extension). Of course, in order to give a sense to this statementwe need to restrict our attention to the case when G is finitely generated,and R = Z (or, more in general, R is a finitely generated abelian group).Under these assumptions, let us consider the central extension
1 // Z // G′π
// G // 1 ,
and the following condition:
(*) there exists a quasi-isometry q : G′ → G × Z which makes the fol-lowing diagram commute:
G′π
//
q
��
G
Id��
G× Z // G
where the horizontal arrow on the bottom represents the obviousprojection.
Condition (*) is equivalent to the existence of a Lipschitz section s : G→ G′
such that π◦s = IdG (see e.g. [KL01, Proposition 8.2]). Gersten proved thata sufficient condition for a central extension to satisfy condition (*) is that itscoclass in H2(G,Z) admits a bounded representative (see [Ger92, Theorem3.1]). Therefore, the image of the comparison map H2
b (G,Z) → H2(G,Z)
12
determines extensions which satisfy condition (*). As far as the authorknows, it is not known whether the converse implication holds, i.e. if everycentral extension satisfying (*) is represented by a cohomology class lyingin the image of the comparison map.
Let us now consider the case with real coefficients, and list some resultsabout the surjectivity of the comparison map:
(1) If G is the fundamental group of a closed locally symmetric spaceof non-compact type, then the comparison map c : Hn
b (G,R) →Hn(G,R) ∼= R is surjective [LS06].
(2) If G is word hyperbolic, then the comparison map c : Hnb (G,V ) →
Hn(G,V ) is surjective for every n ≥ 2 when V is any normed R[G]module or any finitely generated group (considered as a trivial Z[G]-module) [Min01].
(3) If G is finitely presented and the comparison map c : H2b (G,V ) →
H2(G,V ) is surjective for every normed R[G]-module, thenG is wordhyperbolic [Min02].
(4) The set of closed 3-manifolds M for which the comparison mapH•b (π1(M),R)→ H•(π1(M),R) is surjective in every degree is com-
pletely characterized in [FS02].
Observe that points (2) and (3) provide a characterization of word hyperbol-icity in terms of bounded cohomology. Moreover, putting together point (2)with the discussion above, we see that every extension of a word hyperbolicgroup by Z is quasi-isometrically trivial.
5. The second bounded cohomology group of free groups
Let F2 be the free group on two generators. Since H2(F2,R) = H2(S1 ∨S1,R) = 0, the computation of H2
b (F2,R) is reduced to the analysis of thespace Q(G,R) of real quasimorphisms on F2. There are several constructionsof elements in Q(G,R) available in the literature. The first such exampleis probably due to Johnson [Joh72], who proved that H2
b (F2,R) 6= 0. Af-terwords, Brooks produced an infinite family of quasimorphisms inducinglineraly independent elements of H2
b (F2,R) [Bro81]. Since then, other con-structions have been provided by many authors for more general classes ofgroups (see Remark 5.3 below). We describe here a family of quasimorphismwhich is due to Rolli [Rol].
Let s1, s2 be the generators of F2, and let
ℓ∞odd(Z) = {α : Z→ R |α(n) = −α(−n) for every n ∈ Z} .
For every α ∈ ℓ∞odd(Z) we consider the map
fα : F2 → R
defined by
fα(sn1
i1· · · snk
ik) =
k∑
j=1
α(nk) ,
13
where we are identifying every element of F2 with the unique reduced wordrepresenting it. It is elementary to check that fα is a quasimorphism (see [Rol,Proposition 2.1]). Moreover, we have the following:
Proposition 5.1 ([Rol]). The map
ℓ∞odd(Z)→ H2b (F2,R) α 7→ [δ(fα)]
is injective.
Proof. Suppose that [δ(fα)] = 0. By Proposition 4.4, this implies that fα =h + b, where h is a homomorphism and b is bounded. Then, for i = 1, 2,k ∈ N, we have k ·h(si) = h(ski ) = fα(s
ki )−b(s
ki ) = α(k)−b(ski ). By dividing
by k and letting k going to ∞ we get h(si) = 0, so h = 0, and fα = b isbounded.
Observe now that for every k, l ∈ Z we have fα((sl1sl2)k) = 2k ·α(l). Since
fα is bounded, this implies that α = 0, whence the conclusion. �
Corollary 5.2. Let G be a group admitting an epimorphism
ϕ : G→ F2 .
Then the vector space H2b (G,R) is infinite-dimensional.
Proof. Since F2 is free, the epimorphism admits a right inverse ψ. Thecomposition
H2b (F2,R)
ϕ∗
−→ H2b (G,R)
ψ∗
−→ H2b (F2,R)
of the induced maps in bounded cohomology is the identity, and Propo-sition 5.1 ensures that H2
b (F2,R) is infinite-dimensional. The conclusionfollows. �
Remark 5.3. The previous corollary implies that a large class of groupshas infinite-dimensional second bounded cohomology module. In fact, thisholds true for every group belonging to one of the following families (groupsas in (1) also satisfy condition (2), and groups as in (2) also satisfy (3): thecited papers generalized one the result of the other):
(1) non-elementary Gromov hyperbolic groups [EF97];(2) groups acting geometrically on Gromov hyperbolic spaces with limit
set consisting of at least three points [Fuj98];(3) groups admitting a non-elementary weakly proper discontinuous ac-
tion on a Gromov hyperbolic space [BF02];(4) groups having infinitely many ends [Fuj00].
An important application of point (3) is that every subgroup of a the map-ping class group of a compact surface either is virtually abelian, or hasinfinite-dimensional second bounded cohomology. A nice geometric inter-pretation of (homogeneous) quasimorphisms is given in [Man05].
14
6. Amenability
The original definition of amenable group is due to von Neumann [vN29],who introduced the class of amenable groups while studying the Banach-Tarski paradox. As usual, we will restrict our attention to the defini-tion of amenability in the context of discrete groups, referring the readere.g. to [Pie84, Pat88] for a thorough account on amenability in the widercontext of locally compact groups.
Let G be a group. We denote by ℓ∞(G) = C0b (G,R) the space of bounded
real functions on G, endowed with the usual structure of normed R[G]-module (in fact, of Banach G-module). A mean on G is a map m : ℓ∞(G)→R satisfying the following properties:
(1) m is linear;(2) if 1G denotes the map taking every element of G to 1 ∈ R, then
m(1G) = 1;(3) m(f) ≥ 0 for every non-negative f ∈ ℓ∞(G).
If conditions (1) and (2) are satisfied, then condition (3) is equivalent to
(3’) infg∈G f(g) ≤ m(f) ≤ supg∈G f(g) for every f ∈ ℓ∞(G).
In particular, the dual norm of a mean on G, when considered as a functionalon ℓ∞(G), is equal to 1. A mean m is left invariant (or simply invariant) ifit satisfies the following additional condition:
(4) m(g · f) = m(f) for every g ∈ G, f ∈ ℓ∞(G).
Definition 6.1. A group G is amenable if it admits an invariant mean.
The following lemma describes some equivalent definitions of amenabilitythat will prove useful later.
Lemma 6.2. Let G be a group. Then, the following conditions are equiva-lent:
(1) G is amenable;(2) there exists a non-trivial left invariant continuous functional ϕ ∈
ℓ∞(G)′;(3) G admits a left invariant finitely additive probability measure.
Proof. If A is a subset of G, we denote by χA the characteristic function ofA.
(1) ⇒ (2): If m is an invariant mean on ℓ∞(G), then the map f 7→ m(f)defines the desired functional on ℓ∞(G).
(2)⇒ (3): Let ϕ : ℓ∞(G)→ R be a non-trivial continuous functional. Forevery A ⊆ G we define a non-negative real number µ(A) as follows. Forevery partition P of A into a finite number of subsets A1, . . . , An, we set
µP(A) = |ϕ(χA1)|+ . . .+ |ϕ(χAn
)| .
Observe that, if εi is the sign of ϕ(χAi), then µP(A) = ϕ(
∑i εiχAi
) ≤ ‖ϕ‖,so
µ(A) = supP
µP(A)
15
is a finite non-negative number for every A ⊆ G. By the linearity of ϕ, if P ′
is a refinement of P, then µP(A) ≤ µP ′(A), and this easily implies that µ isa finitely additive measure on G. Recall now that the set of characteristicfunctions generates a dense subspace of ℓ∞(G). As a consequence, since ϕis non-trivial, there exists a subset A ⊆ G such that µ(A) ≥ |ϕ(χA)| > 0.In particular, we have µ(G) > 0, so after rescaling we may assume thatµ is a probability measure on G. The fact that µ is G-invariant is now aconsequence of the G-invariance of ϕ.
(3)⇒ (1): Let µ be a left invariant finitely additive measure on G, and letZ ⊆ ℓ∞(G) be the subspace generated by the functions χA, A ⊆ G. Usingthe finite additivity of µ, it is not difficult to show that there exists a linearfunctionalmZ : Z → R such that m(χA) = µ(A) for every A ⊆ G. Moreover,mZ has operator norm equal to one, so it is continuous. Since Z is dense inℓ∞(G), the functional mZ uniquely extends to a continuous functional m ∈ℓ∞(G)′. The fact that m is a mean is an immediate consequence of the factthat every non-negative function in ℓ∞(G) may be approximated by linearcombinations of characteristic functions with positive coefficients. Finally,the G-invariance of µ implies that mZ , whence m, is also invariant. �
The action of G on itself by right translations endows ℓ∞(G) also with thestructure of a right Banach G-module, and it is well-known that the existenceof a left-invariant mean on G implies the existence of a bi-invariant meanon G: in other words, G is amenable if and only if it admits a bi-invariantmean. However, we won’t use this fact in the sequel.
Finite groups are obviously amenable: an invariant mean m is obtainedby setting m(f) = (1/|G|)
∑g∈G f(g) for every f ∈ ℓ∞(G). Of course, we
are interested in finding less obvious examples of amenable groups.
6.1. Abelian groups are amenable. A key result in the theory of amenablegroups is the following theorem, which is originally due to von Neumann:
Theorem 6.3 ([vN29]). Every abelian group is amenable.
Proof. We follow here the proof given in [Pat88], which is based on theMarkov-Kakutani Fixed Point Theorem.
Let G be an abelian group, and let ℓ∞(G)′ be the topological dual ofℓ∞(G), endowed with the weak∗ topology. We consider the subset K ⊆ℓ∞(G)′ given by (not necessarily invariant) means on G, i.e. we set
K = {ϕ ∈ ℓ∞(G)′ |ϕ(1G) = 1 , ϕ(f) ≥ 0 for every f ≥ 0} .
The set K is non-empty (it contains the map ϕ sending every element ofℓ∞(G) to the value it takes on the identity of G). Moreover, it is closed,convex, and contained in the closed ball of radius one in ℓ∞(G)′. Therefore,K is compact by the Banach-Alaouglu Theorem.
For every g ∈ G let us now consider the map Lg : ℓ∞(G)′ → ℓ∞(G)′
defined by
Lg(ϕ)(f) = ϕ(g · f) .
16
We want to show that the Lg admit a common fixed point in K. To thisaim, one may apply the Markov-Kakutani Theorem cited above, which weprove here (in the case we are interested in) for completeness.
We first show that, if H is any non-empty compact convex subset of Kwhich is left invariant by a single Lg, then Lg admits a fixed point in H.Let us fix ϕ ∈ H. For every n ∈ N we set
ϕn =1
n+ 1
n∑
i=0
Lig(ϕ) .
By convexity, ϕn ∈ H for every n, so there exists a subsequence ϕnitending
to ϕ ∈ H. Recall that ‖y‖ = 1 for every y ∈ K, so for every f ∈ ℓ∞(G) wehave
|ϕni(g · f)−ϕni
(f)| = |(Lg(ϕni)−ϕni
)(f)| =|(Ln+1
g (ϕ) − ϕ)(f)|
ni + 1≤
2‖f‖
ni + 1.
By passing to the limit for ni tending to ∞, we obtain that ϕ(g · f) = ϕ(f)for every f ∈ ℓ∞(G), so Lg has a fixed point in H.
Let us now observe that K is Lg-invariant for every g ∈ G. Therefore,the set Kg of points of K which are fixed by Lg is non-empty. Moreover, itis easily seen that Kg is closed (whence compact) and convex. Since G isabelian, if g1, g2 are elements of G, then the maps Lg1 and Lg2 commute witheach other, so Kg1 is Lg2-invariant. The argument above shows that Lg2 hasa fixed point in Kg1 , so Kg1 ∩ Kg2 6= ∅. One may iterate this argumentto show that every finite intersection Kg1 ∩ . . . ∩ Kgn
is non-empty. Inother words, the family Kg, g ∈ G satisfies the finite intersection property,so KG =
⋂g∈GKg is non-empty. But KG is precisely the set of invariant
means on G. �
6.2. Other amenable groups. Starting from abelian groups, it is possibleto construct larger classes of amenable groups:
Proposition 6.4. Let G,H be amenable groups. Then:
(1) Every subgroup of G is amenable.(2) If the sequence
1 −→ H −→ G′ −→ G −→ 1
is exact, then G′ is amenable (in particular, the direct product of afinite number of amenable groups is amenable).
(3) Every direct union of amenable groups is amenable.
Proof. (1): Let K be a subgroup of G. Let S be a set of representatives inG for the set K\G of right lateral classes of K in G. For every f ∈ ℓ∞(K)
we define f ∈ ℓ∞(G) by setting f(g) = f(k), where g = ks, s ∈ S, k ∈ K.
If m is an invariant mean on G, then we set mH(f) = m(f). It is easy tocheck that mH is an invariant mean on K, so K is amenable.
(2): Let mH ,mG be invariant means on H,G respectively. For everyf ∈ ℓ∞(G′) we construct a map fG ∈ ℓ∞(G) as follows. For g ∈ G, we
17
take g′ in the preimage of g in G′, and we define fg′ ∈ ℓ∞(H) by settingfg′(h) = f(g′h). Then, we set fG(g) = mH(fg′). Using that mH is H-invariant one may show that fG(g) does not depend on the choice of g′, sofG is well-defined. We obtain the desired mean on G′ by setting m(f) =mG(fG).
(3): Let G be the direct union of the amenable groups Gi, i ∈ I. Forevery i we consider the set
Ki = {ϕ ∈ ℓ∞(G)′ |ϕ(1G) = 1 , ϕ(f) ≥ 0 for every f ≥ 0, ϕ isGi−invariant} .
If mi is an invariant mean on Gi, then the map f 7→ mi(f |Gi) defines an
element of Ki. Therefore, each Ki is non-empty. Moreover, the fact that theunion of the Gi is direct implies that the family Ki, i ∈ I satisfies the finiteintersection property. Finally, each Ki is closed and compact in the weak∗
topology on ℓ∞(G)′, so the intersection K =⋂i∈I Ki is non-empty. But K
is precisely the set of G-invariants means on G, whence the conclusion. �
The class of elementary amenable groups is the smallest class of groupswhich contains all finite and all abelian groups, and is closed under theoperations of taking subgroups, forming quotients, forming extensions, andtaking direct unions [Day57]. For example, virtually solvable groups areelementary amenable. Proposition 6.4 (together with Theorem 6.3) showsthat every elementary amenable group is indeed amenable. However, anygroup of intermediate growth is amenable [Pat88] but is not elementaryamenable [Cho80] In particular, there exist amenable groups which are notvirtually solvable.
Remark 6.5. We have already noticed that, if G1, G2 are amenable groups,then so is G1 × G2. More precisely, in the proof of Proposition 6.4-(2) wehave shown how to construct an invariant mean m on G1 × G2 startingfrom invariant means mi on Gi, i = 1, 2. Under the identification of meanswith finitely additive probability measures, the mean m corresponds to theproduct measure m1 ×m2, so it makes sense to say that m is the productof m1 and m2. If G1, . . . , Gn are amenable groups with invariant meansm1, . . . ,mn, then we denote by m1 × . . . × mn the invariant mean on Ginductively defined by the formula m1 × . . .×mn = m1 × (m2 × . . .×mn).
6.3. Amenability and bounded cohomology. In this subsection we showthat amenable groups are somewhat invisible to bounded cohomology withcoefficients in normed R[G]-modules (things are quite different in the caseof Z[G]-modules). We say that an R[G]-module V is a dual normed R[G]-module if V is isomorphic (as a normed R[G]-module) to the topologicaldual of some normed R[G]-module W . In other words, V is the space ofbounded linear functionals on the normed R[G]-module W , endowed withthe action defined by g · f(w) = f(g−1w), g ∈ G, w ∈W .
Theorem 6.6. Let G be an amenable group, and let V be a dual normedR[G]-module. Then Hn
b (G,V ) = 0 for every n ≥ 1.
18
Proof. Recall that the bounded cohomology H•b (G,V ) is defined as the co-
homology of the G-invariants of the homogeneous complex
0 −→ C0(G,V )δ0−→ C1(G,V )→ . . .→ Cn(G,V )→ . . .
Of course, the cohomology of the complex C•b (G,V ) of possibly non-invariant
cochains vanishes in positive degree, since the maps
kn+1 : Cn+1b (G,V )→ Cnb (G,V ) , kn+1(f)(g0, . . . , gn) = f(1, g0, . . . , gn) , n ≥ 0
provide a (partial) homotopy between the identity and the zero map ofC•b (G,V ) in positive degree. In order to prove the theorem, we are going to
show that, under the assumption thatG is amenable, a similar homotopy canbe defined on the complex of invariant cochains. Roughly speaking, while k•
is obtained by coning over the identity of G, we will define a G-invariant ho-motopy j• by averaging the cone operator over all the elements of G. Let usfix an invariant mean m on G, and let f be an element of Cn+1
b (G,V ). Recallthat V = W ′ for some R[G]-module W , so f(g, g0, . . . , gn) is a bounded func-tional on W for every (g, g0, . . . , gn) ∈ G
n+2. For every (g0, . . . , gn) ∈ Gn+1,
w ∈W , we consider the function
fw : G→ R , fw(g) = f(g, g0, . . . , gn)(w) .
It follows from the definitions that fw is an element of ℓ∞(G), so we mayset (jn+1(f))(g0, . . . , gn)(w) = m(fw). It is immediate to check that thisformula defines a continuous functional on W , whose norm is bounded interms of the norm of f . In other words, jn+1(f)(g0, . . . , gn) is indeed anelement of V , and the map jn+1 : Cn+1
b (G,V ) → Cnb (G,V ) is bounded.The G-invariance of the mean m implies that jn+1 is G-equivariant. Thecollection of maps jn+1, n ∈ N provides the required partial G-equivarianthomotopy between the identity and the zero map of Cnb (G,V ), n ≥ 1. �
Corollary 6.7. Let G be an amenable group. Then Hnb (G,R) = 0 for every
n ≥ 1.
Recall from Section 4.3 that the second bounded cohomology group withtrivial real coefficients is strictly related to the space of real quasimorphisms.Putting together Corollary 6.7 with Proposition 4.4 and Corollary 4.7 we getthe following result, which extends to amenable groups the characterizationof real quasimorphisms on abelian groups we described in Subsection 4.5:
Corollary 6.8. Let G be an amenable group. Then every real quasimor-phism on G is at bounded distance from a homomorphism. Equivalently,every homogeneous real quasimorphism on G is a homomorphism.
From Corollary 5.2 and Corollary 6.7 we deduce that there exist manygroups which are not amenable:
Corollary 6.9. Suppose that the group G contains a non-abelian free sub-group. Then G is not amenable.
19
Proof. We know from Corollary 5.2 that non-abelian free groups have non-trivial second bounded cohomology group with real coefficients, so they can-not be amenable. The conclusion follows from the fact that the subgroup ofan amenable group is amenable. �
It was a long-standing problem to understand whether the previous corol-lary could be sharpened into a characterization of amenable groups as thosegroups which do not contain any non-abelian free subgroup. This ques-tion is usually attributed to von Neumann, and appeared first in [Day57].Ol’shanskii answered von Neumann’s question in the negative in [Ol’80]. Thefirst examples of finitely presented groups which are not amenable but do notcontain any non-abelian free group are due to Ol’shanskii and Sapir [OS02].
6.4. Johnson’s characterization of amenability. We have just seen thatthe bounded cohomology of any amenable group with values in any dual co-efficient module vanishes. In fact, also the converse implication holds. Moreprecisely, amenability of G is ensured by the vanishing of a specific class(of degree one) in a specific bounded cohomology module. We denote byℓ∞(G)/R the quotient of ℓ∞(G) by the (trivial) R[G]-submodule of constantfunctions. Such a quotient is itself endowed with the structure of a normedR[G]-module. Note that the topological dual (ℓ∞(G)/R)′ may be identifiedwith the subspace of elements of ℓ∞(G)′ that vanish on constant functions.For example, if δg is the element of ℓ∞(G)′ such that δg(f) = f(g), then forevery pair (g0, g1) ∈ G
2 the map δg0 − δg1 defines an element in (ℓ∞(G)/R)′.For later purposes, we observe that the action of G on ℓ∞(G)′ is such that
g · δg0(f) = δg0(g−1 · f) = f(gg0) = δgg0(f) .
Let us consider the element
J ∈ C1b (G, (ℓ
∞(G)/R)′) , J(g0, g1) = δg1 − δg0 .
Of course we have δJ = 0. Moreover, for every g, g0, g1 ∈ G we have
(g·J)(g0, g1) = g(J(g−1g0, g−1g1)) = g(δg−1g0−δg−1g1) = δg0−δg1 = J(g0, g1)
so J is G-invariant, and defines an element [J ] ∈ H1b (G, (ℓ
∞(G)/R)′), calledthe Johnson class of G.
Theorem 6.10. Suppose that the Johnson class of G vanishes. Then G isamenable.
Proof. By Lemma 6.2, it is sufficient to show that the topological dualℓ∞(G)′ contains a non-trivial G-invariant element.
We keep notation from the preceding paragraph. If [J ] = 0, then J = δψ
for some ψ ∈ C0b (G, (ℓ
∞(G)/R)′)G. For every g ∈ G we denote by ψ(g) ∈ℓ∞(G)′ the pullback of ψ(g) via the projection map ℓ∞(G) → ℓ∞(G)/R.
We consider the element ϕ ∈ ℓ∞(G)′ defined by ϕ = δ1 − ψ(1). Since ψ(1)vanishes on constant functions, we have ϕ(1G) = 1, so ϕ is non-trivial, andwe are left to show that ϕ is G-invariant.
20
The G-invariance of ψ implies the G-invariance of ψ, so
(2) ψ(g) = (g · ψ)(g) = g · (ψ(1)) for every g ∈ G .
From J = δψ we deduce that
δg1 − δg0 = ψ(g1)− ψ(g0) for every g0, g1 ∈ G .
In particular, we have
(3) δg − ψ(g) = δ1 − ψ(1) for every g ∈ G .
Therefore, using Equations (2), (3), for every g ∈ G we get
g · ϕ = g · (δ1 − ψ(1)) = (g · δ1)− g · (ψ(1)) = δg − ψ(g) = δ1 − ψ(1) = ϕ ,
and this concludes the proof. �
Putting together Theorems 6.6 and 6.10 we get the following result, whichcharacterizes amenability in terms of bounded cohomology:
Corollary 6.11. Let G be a group. Then G is amenable if and only ifHnb (G,V ) = 0 for every dual normed R[G]-module V and every n ≥ 1.
�
7. Group cohomology via resolutions
Computing group cohomology by means of its very definition is usuallyvery hard. The topological interpretation of group cohomology already pro-vides a powerful tool for computations: for example, one may estimate thecohomological dimension of a group G (i.e. the maximal n ∈ N such thatHn(G,R) 6= 0) in terms of the dimension of a K(G, 1) as a CW-complex. Wehave already mentioned that a topological interpretation for the boundedcohomology of a group is still available, but in order to prove this fact moremachinery has to be introduced. Before going into the bounded case, wedescribe some well-known results which hold in the classical case: namely,we will show that the cohomology of G may be computed by looking atseveral complexes of cochains. This crucial fact was already observed in thepioneering works on group cohomology of the 1940s and the 1950s. Thereare several ways to define group cohomology in terms of resolutions. Weprivilege here an approach that better extends to the case of bounded co-homology. We briefly compare our approach to more traditional ones inSubsection 7.3.
7.1. Relatively injective modules. We begin by introducing the notionsof relatively injective R[G]-module and of strong G-resolution of an R[G]-module. The counterpart of these notions in the context of normed R[G]-modules will play an important role in the theory of bounded cohomologyof groups. The importance of these notions is due to the fact that thecohomology of G may be computed by looking at any strong G-resolutionof the coefficient module by relatively injective modules (see Corollary 7.5below).
21
A G-map ι : A → B between R[G]-modules is strongly injective if thereis an R-linear map σ : B → A such that σ ◦ ι = IdA (in particular, ι isinjective). We emphasize that, even if A and B are G-modules, the map σis not required to be G-equivariant.
Definition 7.1. An R[G]-module U is relatively injective (over R[G]) ifthe following holds: whenever A,B are G-modules, ι : A → B is a stronglyinjective G-map and α : A→ U is a G-map, there exists a G-map β : B → Usuch that β ◦ ι = α.
0 // A ι//
α
��
B
β~~~
~
~
~
σuu
U
In the case when R = R, a map is strongly injective if and only if it isinjective, so in the context of R[G]-modules the notions of relative injectivityand the traditional notion of injectivity coincide. However, relative injec-tivity is rather weaker than injectivity in general. For example, if R = Z
and G = {1}, then Ci(G,R) is isomorphic to the trivial G-module Z, whichof course is not injective over Z[G] = Z. On the other hand, we have thefollowing:
Lemma 7.2. For every R[G]-module V and every n ∈ N, the R[G]-moduleCn(G,V ) is relatively injective.
Proof. Let us consider the extension problem described in Definition 7.1,with U = Cn(G,V ). Then we define β as follows:
β(b)(g0, . . . , gn) = α(g0σ(g−10 b))(g0, . . . , gn) .
The fact that β is R-linear and that β ◦ ι = α is straightforward, so wejust need to show that β is G-equivariant. But for every g ∈ G, b ∈ B and(g0, . . . , gn) ∈ G
n+1 we have
g(β(b))(g0, . . . , gn) = g(β(b)(g−1g0, . . . , g−1gn))
= g(α(g−1g0σ(g−10 gb))(g−1g0, . . . , g
−1gn))
= g((g−1(α(g0σ(g−10 gb)))(g−1g0, . . . , g
−1gn))
= α(g0σ(g−10 gb))(g0, . . . , gn)
= β(gb)(g0, . . . , gn)
whence the conclusion. �
7.2. Complexes and resolutions. AnR[G]-complex (or simply aG-complexor a complex ) is a sequence of R[G]-modules Ei and G-maps δi : Ei → Ei+1,i ∈ N, such that δi+1 ◦ δi = 0 for every i:
0 −→ E0 δ0−→ E1 δ1
−→ . . .δn
−→ En+1 δn+1
−→ . . .
22
Such a sequence will be denoted by (E•, δ•). Moreover, we set Zn(E•) =ker δn ∩ (En)G, Bn(E•) = δn−1((En−1)G) (where again we understand thatB0(E•) = 0), and we define the cohomology of the complex E• by setting
Hn(E•) = Zn(E•)/Bn(E•) .
A chain map between G-complexes (E•, δ•E) and (F •, δ•F ) is a sequenceof G-maps {αi : Ei → F i | i ∈ N} such that δiF ◦ α
i = αi+1 ◦ δiE for everyi ∈ N. If α•, β• are chain maps between (E•, δ•E) and (F •, δ•F ), a G-homotopybetween α• and β• is a sequence of G-maps {T i : Ei → F i−1 | i ≥ 0} suchthat T 1 ◦ δ0E = α0 − β0 and δi−1
F ◦ T i + T i+1 ◦ δiE = αi − βi for everyi ≥ 1. Every chain map induces a well-defined map in cohomology, andG-homotopic chain maps induce the same map in cohomology.
If E is an R[G]-module, an augmented G-complex (E,E•, δ•) with aug-mentation map ε : E → E0 is a complex
0 −→ Eε−→ E0 δ0
−→ E1 δ1−→ . . .
δn
−→ En+1 δn+1
−→ . . .
A resolution of E (over G) is an exact augmented complex (E,E•, δ•) (overG). A resolution (E,E•, δ•) is relatively injective if En is relatively injectivefor every n ≥ 0. It is well-known that any map between modules extendsto a chain map between injective resolutions of the modules. Unfortunately,the same result for relatively injective resolutions does not hold. The pointis that relative injectivity guarantees the needed extension property onlyfor strongly injective maps. Therefore, we need to introduce the notion ofstrong resolution.
A contracting homotopy for a resolution (E,E•, δ•) is a sequence of R-linear maps ki : Ei → Ei−1 such that δi−1 ◦ ki + ki+1 ◦ δi = IdEi if i ≥ 0,and k0 ◦ δ−1 = IdE:
0 // E ε// E0
δ0//
k0
uu
E1δ1
//
k1
tt . . .k2
tt
δn−1
// Enδn
//
kn
tt . . .kn+1
tt
Note however that it is not required that ki be G-equivariant. A resolutionis strong if it admits a contracting homotopy.
The following proposition shows that the chain complex C•(G,V ) pro-vides a relatively injective strong resolution of V :
Proposition 7.3. Suppose that R = R. Let ε : V → C0(G,V ) be defined byε(v)(g) = v for every v ∈ V , g ∈ G. Then the augmented complex
0 −→ Vε−→ C0(G,V )
δ0−→ C1(G,V )→ . . .→ Cn(G,V )→ . . .
provides a relatively injective strong resolution of V .
Proof. We already know that each Ci(G,V ) is relatively injective. In orderto show that the augmented complex (V,C•(G,V ), δ•) is a strong resolutionit is sufficient to observe that the maps
kn+1 : Cn+1(G,V )→ Cn(G,V ) kn+1(f)(g0, . . . , gn) = f(1, g0, . . . , gn)
23
provide the required contracting homotopy. �
The resolution of V described in the previous proposition is obtained justby augmenting the homogeneous complex associated to (G,V ), and it isusually called the standard resolution of V (over G).
The following result may be proved by means of standard homologicalalgebra arguments (see e.g. [Iva87], [Mon01, Lemmas 7.2.4 and 7.2.6] fora detailed proof in the bounded case – the argument there extends to thiscontext just by forgetting any reference to the norms). It implies that anyrelatively injective strong resolution of a G-module V may be used to com-pute the cohomology modules H•(G,V ) (see Corollary 7.5).
Theorem 7.4. Let α : E → F be a G-map between R[G]-modules, let(E,E•, δ•E) be a strong resolution of E, and suppose that (F,F •, δ•F ) is anaugmented complex such that F i is relatively injective for every i ≥ 0. Thenα extends to a chain map α•, and any two extensions of α to chain mapsare G-homotopic.
Corollary 7.5. Let (V, V •, δ•V ) be a relatively injective strong resolution ofV . Then for every n ∈ N there is a canonical isomorphism
Hn(G,V ) ∼= Hn(V •) .
Proof. By Proposition 7.3, both (V, V •, δ•V ) and the standard resolution ofV are relatively injective strong resolutions of V over G. Therefore, The-orem 7.4 provides chain maps between C•(G,V ) and V •, which are onethe G-homotopy inverse of the other. Therefore, these chain maps induceisomorphisms in cohomology. �
7.3. The classical approach to group cohomology via resolutions. Inthis subsection we describe the relationship between the description of groupcohomology via resolutions given above and more traditional approaches tothe subject (see e.g. [Bro82]). The reader who is not interested can safelyskip the section, since the results cited below will not be used elsewhere inthe course.
The category of abelian groups endowed with a left action by G obviouslycoincides with the category of left Z[G]-modules. If V,W are two such mod-ules, then the space HomZ(V,W ) is endowed with the structure of a Z[G]-module by setting (g ·f)(v) = g(f(g−1v)) for every g ∈ G, f ∈ HomZ(V,W ),v ∈ V . Then, the cohomology H•(G,V ) is often defined in the followingequivalent ways (we refer e.g. to [Bro82] for the definition of projectivemodule (over a ring R); for our discussion, it is sufficient to know that anyfree R-module is projective over R, so any free resolution is projective):
(1) (Via injective resolutions of V ): Let (V, V •, δ•) be an injectiveresolution of V over Z[G], and take the complex W • = HomZ(V •,Z),endowed with the action g·f(v) = f(g−1v). Then (W •)G = HomZ[G](V
•,Z)(where Z is endowed with the structure of trivial G-module), and onemay define H•(G,V ) as the homology of the G-invariants of W •.
24
(2) (Via projective resolutions of Z): Let (Z, P•, d•) be a projectiveresolution of Z over Z[G], and take the complex Z• = HomZ(P•, V ).We have again that (Z•)G = HomZ[G](P•, V ), and again we maydefine H•(G,V ) as the homology of the G-invariants of the complexZ•.
The fact that these two definitions are indeed equivalent is proved e.g. in [Bro82].We have already observed that, if V is a Z[G]-module, the module Cn(G,V )
is not injective in general. However, this is not really a problem, since thecomplex C•(G,V ) may be recovered from a projective resolution of Z overZ[G]. Namely, let Cn(G,Z) be the free Z-module admitting the set Gn+1
as a basis. The diagonal action of G onto Gn+1 endows Cn(G,Z) with thestructure of a Z[G]-module. The modules Cn(G,Z) may be arranged into aresolution
0←− Zε←− C0(G,Z)
d1←− C1(G,Z)d2←− . . .
dn←− Cn(G,Z)dn+1
←− . . .
of the trivial Z[G]-module Z over Z[G] (the homology of the G-coinvariantsof this resolution is by definition the homology of G, we refer to Section 11for more details). Now, it is easy to check that Cn(G,Z) is free, whenceprojective, as a Z[G]-module. Moreover, the module HomZ(Cn(G,Z), V ) isZ[G]-isomorphic to Cn(G,V ). This shows that, in the context of the tradi-tional definition of group cohomology via resolutions, the complex C•(G,V )arises from a projective resolution of Z, rather than from an injective reso-lution of V .
7.4. The topological interpretation of group cohomology revisited.Let us show how Corollary 7.5 may be exploited to prove that the cohomol-ogy of G is isomorphic to the singular cohomology of any path-connectedtopological space X satisfying conditions (1), (2) and (3) described in Sub-section 1.2. We fix an identification between G and the group of the cov-
ering automorphisms of the universal covering X of X. The action of
G on X induces an action of G on C•(X,R), whence an action of G on
C•(X,R), which is defined by (g · ϕ)(c) = ϕ(g−1c) for every c ∈ C•(X,R),
ϕ ∈ C•(X,R), g ∈ G. Therefore, for n ∈ N both Cn(X,R) and Cn(X,R)are endowed with the structure of R[G]-modules. We have a natural identi-
fication C•(X,R)G ∼= C•(X,R).
Lemma 7.6. For every n ∈ N, the singular cochain module Cn(X,R) isrelatively injective.
Proof. For every topological space Y , let us denote by Sn(Y ) the set ofsingular simplices with values in Y .
We denote by Ln(X) a set of representatives for the action of G on Sn(X)
(for example, if F is a set of representatives for the action of G on X , we
may define Ln(X) as the set of singular n-simplices whose first vertex lies in
F ). Then, for every n-simplex s ∈ Sn(X), there exist a unique gs ∈ G, and
25
a unique s ∈ Ln(X) such that gs · s = s. Let us now consider the extensionproblem:
0 // A ι//
�
B
βzzvv
vv
v
σtt
Cn(X,R)
We define the desired extension β by setting
β(b)(s) = α(σ(g−1s · b))(s)
for every s ∈ Sn(X). It is easy to verify that the map β is an R[G]-map,and that α = β ◦ ι.
An alternative proof (providing the same solution to the extension prob-
lem) is the following. Using that the action of G on X is free, it is easy to
show that Ln(X), when considered as a subset of Cn(X,R), is a free basis of
Cn(X,R) over R[G]. In other words, every c ∈ Cn(X,R) may be expressed
uniquely as a sum of the form c =∑k
i=1 aigisi, ai ∈ R, gi ∈ G, si ∈ Ln(X).Therefore, the map
ψ : C0(G,C0(Ln(X), R))→ Cn(X,R) , ψ(f)
(k∑
i=1
aigisi
)=
k∑
i=1
aif(gi)(si)
is well-defined. If we endow C0(Ln(X), R) with the structure of trivialG-module, then a straightforward computation shows that ψ is in fact aG-isomorphism, whence the conclusion by Lemma 7.2. �
Proposition 7.7. Let ε : R→ C0(X,R) be defined by ε(t)(s) = t for every
singular 0-simplex s in X. Suppose that Hi(X,R) = 0 for every i ≥ 1. Thenthe augmented complex
0 −→ Rε−→ C0(X,R)
δ1−→ C1(X,R)→ . . .
δn−1
−→ Cn(X,R)δn
−→ Cn+1(X,R)
is a relative injective strong resolution of the trivial R[G]-module R.
Proof. Since X is path-connected, we have that Im ε = ker δ0. Observe now
that Cn(X,R) is R-free for every n ∈ N. As a consequence, the obvious
augmented complex associated to C•(X,R), being acyclic, is homotopically
trivial over R. Since Cn(X,R) ∼= HomR(Cn(X,R), R), we may concludethat the augmented complex described in the statement is a strong resolutionof R over R[G]. Then the conclusion follows from Lemma 7.6. �
Putting together Propositions 7.3, 7.7 and Corollary 7.5 we may providethe following topological characterization of Hn(G,R):
Corollary 7.8. Let X be a path-connected space admitting a universal cov-
ering X, and suppose that Hi(X,R) = 0 for every i ≥ 1. Then H i(X,R) iscanonically isomorphic to H i(π1(X), R) for every i ∈ N.
26
8. Bounded cohomology via resolutions
Just as in the case of classical cohomology, it is often useful to have alter-native ways for computing the bounded cohomology of a group. This sectionis devoted to an approach to bounded cohomology which closely follows thetraditional approach to classical cohomology via homological algebra. Thecircle of ideas we are going to describe first appeared (in the case with trivialreal coefficients) in a paper by Brooks [Bro81], where it was exploited to givean independent proof of Gromov’s result that the isomorphism type of thebounded cohomology of a space (with real coefficients) only depends on itsfundamental group [Gro82]. Brooks’ theory was then developed by Ivanovin his foundational paper [Iva87]. Ivanov gave a new proof of the vanish-ing of the bounded cohomology (with real coefficients) of simply connectedspace (this result played an important role in Brooks’ argument, and wasoriginally due to Gromov [Gro82]), and managed to incorporate the semi-norm into the homological algebra approach to bounded cohomology, thusproving that the bounded cohomology of a space is isometrically isomor-phic to the bounded cohomology of its fundamental group (in the case withreal coefficients). Ivanov’s theory was further developed by Monod [Mon01],who paid a particular attention to the continuous bounded cohomology oftopological groups.
Both Ivanov’s and Monod’s theory are concerned with Banach G-modules,which are in particular R[G]-modules. For the moment, we prefer to con-sider also the (quite different) case with integral coefficients. Therefore, welet R be etiher Z or R, and we concentrate our attention on the category ofnormed R[G]-modules introduced in Section 2. In the next subsections wewill see that relative injective modules and strong resolutions may be de-fined in this context just by adapting to normed R[G]-modules the analogousdefinitions for generic R[G]-modules.
8.1. Relative injectivity. Throughout the whole section, unless other-wise stated, we will deal only with normed R[G]-modules. Therefore, G-morphisms will be always assumed to be bounded.
The following definitions are taken from [Iva87] (where only the case whenR = R and V is Banach is dealt with). A bounded linear map ι : A→ B ofnormed R-modules is strongly injective if there is a linear map σ : B → Awith ‖σ‖ ≤ 1 and σ ◦ ι = IdA (in particular, ι is injective). We emphasizethat, even when A and B are R[G]-modules, the map σ is not required tobe G-equivariant.
Definition 8.1. A normed R[G]-module E is relatively injective if for everystrongly injective G-morphism ι : A→ B of normed R[G]-modules and everyG-morphism α : A→ E there is a G-morphism β : B → E satisfying β◦ι = α
27
and ‖β‖ ≤ ‖α‖.
0 // A ι//
α
��
B
β~~~
~
~
~
σuu
E
Remark 8.2. Let E be a normed R[G]-module, and let E be the underlyingR[G]-module. Then no obvious implication exists between the fact that E isrelatively injective (in the category of normed R[G]-modules, i.e. according
to Definition 8.1), and the fact that E is (in the category of R[G]-modules,i.e. according to Definition 7.1). This could suggest that the use of the samename for these different notions could indeed be an abuse. However, unlessotherwise stated, henceforth we will deal with relatively injective modulesonly in the context of normed R[G]-modules, so the reader may safely takeDefinition 8.1 as the only definition of relative injectivity.
The following result is due to Ivanov [Iva87] (see also Remark 8.5), andshows that the modules involved in the definition of bounded cohomologyare relatively injective.
Lemma 8.3. Let V be a normed R[G]-module. Then the normed R[G]-module Cnb (G,V ) is relatively injective.
Proof. Let us consider the extension problem described in Definition 8.1,with E = Cnb (G,V ). Then we define β as follows:
β(b)(g0, . . . , gn) = α(g0σ(g−10 b))(g0, . . . , gn) .
It is immediate to check that β ◦ ι = α. Moreover, since ‖σ‖ ≤ 1, we have‖β‖ ≤ ‖α‖. Finally, the fact that β commutes with the actions of G may beproved by the very same computation given in the proof of Lemma 7.2. �
8.2. Resolutions of normed R[G]-modules. A normed R[G]-complex isan R[G]-complex whose modules are normed R[G]-spaces, and whose dif-ferential is a bounded R[G]-map in every degree. A chain map between(E•, δ•E) (F •, δ•F ) is a chain map between the underlying R[G]-complexeswhich is bounded in every degree, and a G-homotopy between two suchchain maps is just a G-homotopy between the underlying maps of R[G]-modules, which is bounded in every degree. The cohomology H•
b (E•) of the
normed G-complex (E•, δ•E) is defined as usual by taking the cohomologyof the subcomplex of G-invariants. The norm on En restricts to a normon G-invariant cocycles, which induces in turn a seminorm on Hn(E∗) forevery n ∈ N.
An augmented normed G-complex (E,E•, δ•) with augmentation mapε : E → E0 is a G-complex
0 −→ Eε−→ E0 δ0
−→ E1 δ1−→ . . .
δn
−→ En+1 δn+1
−→ . . .
28
(in particular, ε is bounded). A resolution of E (as a normed R[G]-complex)is an exact augmented normed complex (E,E•, δ•). It is relatively injectiveif En is relatively injective for every n ≥ 0. From now on, we will call simplycomplex any normed complex.
Let (E,E•, δ•E) be an augmented complex, and suppose that (F,F •, δ•F )is a relatively injective resolution of F . We would like to be able to extendany G-map E → F to a chain map between E• and F •. As observed in thepreceding section, the assumption that (F,F •, δ•F ) is relatively injective isnot sufficient to ensure that any G-map between E and F extends to a chainmap between (E,E•, δ•E) and (F,F •, δ•F ). The point is that relative injec-tivity guarantees the needed extension property only for strongly injectivemaps, so a corresponding notion of strong resolution is needed.
A contracting homotopy for a resolution (E,E•, δ•) is a sequence of linearmaps ki : Ei → Ei−1 such that ‖ki‖ ≤ 1 for every i ∈ N, δi−1◦ki+ki+1◦δi =IdEi if i ≥ 0, and k0 ◦ δ−1 = IdE :
0 // E ε// E0
δ0//
k0
uu
E1δ1
//
k1
tt . . .k2
tt
δn−1
// Enδn
//
kn
tt . . .kn+1
tt
Note however that it is not required that ki be G-equivariant. A resolutionis strong if it admits a contracting homotopy.
Proposition 8.4. Let V be a normed R[G]-space, and let ε : V → C0b (G,V )
be defined by ε(v)(g) = v for every v ∈ V , g ∈ G. Then the augmentedcomplex
0 −→ Vε−→ C0
b (G,V )δ0−→ C1
b (G,V )→ . . .→ Cnb (G,V )→ . . .
provides a relatively injective strong resolution of V .
Proof. We already know that each Cib(G,V ) is relatively injective, so in orderto conclude it is sufficient to observe that the map
kn+1 : Cn+1b (G,V )→ Cnb (G,V ) kn+1(f)(g0, . . . , gn) = f(1, g0, . . . , gn)
provides a contracting homotopy for the resolution (V,C•b (G,V ), δ•). �
The resolution described in Proposition 8.4 is the standard resolution ofV as a normed R[G]-module.
Remark 8.5. Let us briefly compare our notion of standard resolution withIvanov’s and Monod’s ones. In [Iva87], for every n ∈ N the set Cnb (G,R) isdenoted by B(Gn+1), and it is endowed with the structure of a right BanachG-module by the action g · f(g0, . . . , gn) = f(g0, . . . , gn · g). Moreover, thesequence of modules B(Gn), n ∈ N, is equipped with a structure of G-complex
0 −→ Rd−1
−→ B(G)d0−→ B(G2)
d1−→ . . .dn−→ B(Gn+2)
dn+1
−→ . . . ,
29
where d−1(t)(g) = t and
dn(f)(g0, . . . , gn+1) = (−1)n+1f(g1, . . . , gn+1)+n∑
i=0
(−1)n−if(g0, . . . , gigi+1, . . . , gn+1)
for every n ≥ 0 (here we are using Ivanov’s notation also for the differential).Now, it is readily seen that Ivanov’s resolution is isomorphic to our standardresolution via the isometric G-chain isomorphism ϕ• : B•(G) → B(G•+1)defined by
ϕn(f)(g0, . . . , gn) = f(g−1n , g−1
n g−1n−1, . . . , g
−1n g−1
n−1 · · · g−11 g−1
0 )
(its inverse is given by (ϕn)−1(f)(g0, . . . , gn) = f(g−1n gn−1, g
−1n−1gn−2, . . . , g
−11 g0, g
−10 )).
We also observe that the contracting homotopy described in Proposition 8.4is conjugated by ϕ• into Ivanov’s contracting homotopy for the complex(B(G•), d•) (which is defined in [Iva87]).
Our notation is much closer to Monod’s one. In fact, in [Mon01] the moregeneral case of a topological group G is addressed, and the n-th module ofthe standard G-resolution of R is inductively defined by setting
C0b (G,R) = Cb(G,R), Cnb (G,R) = Cb(G,C
n−1b (G,R)) ,
where Cb(G,E) denotes the space of continuous bounded maps fromG to theBanach space E. However, as observed in [Mon01, Remarks 6.1.2 and 6.1.3],the case when G is an abstract group may be recovered from the general casejust by equipping G with the discrete topology. In that case, our notion ofstandard resolution coincides with Monod’s one (see also [Mon01, Remark7.4.9]).
The following result can be proved by means of standard homologicalalgebra arguments (see [Iva87], [Mon01, Lemmas 7.2.4 and 7.2.6] for fulldetails):
Theorem 8.6. Let α : E → F be a G-map between normed R[G]-modules,let (E,E•, δ•E) be a strong resolution of E, and suppose (F,F •, δ•F ) is anaugmented complex such that F i is relatively injective for every i ≥ 0. Thenα extends to a chain map α•, and any two extensions of α to chain mapsare G-homotopic.
Corollary 8.7. Let V be a normed R[G]-modules, and let (V, V •, δ•V ) be arelatively injective strong resolution of V . Then for every n ∈ N there is acanonical isomorphism
Hnb (G,V ) ∼= Hn
b (V •) .
Moreover, this isomorphism is bi-Lipschitz with respect to the seminorms ofHnb (G,V ) and Hn(V •).
Proof. By Proposition 8.4, the standard resolution of V is also a relativelyinjective strong resolution of V over G. Therefore, Theorem 7.4 provideschain maps between C•
b (G,V ) and V •, which are one the G-homotopy in-verse of the other. Therefore, these chain maps induce isomorphisms in
30
cohomology. The conclusion follows from the fact that bounded chain mapsinduce bounded maps in cohomology. �
By Corollary 8.7, every relatively injective strong resolution of V inducesa seminorm on H•
b (G,V ). Moreover, the seminorms defined in this wayare pairwise equivalent. However, in many applications, it is important tobe able to compute the exact canonical seminorm of elements in Hn
b (G,V ),i.e. the seminorm induced on Hn
b (G,V ) by the standard resolution C•b (G,V ).
Unfortunately, it is not possible to capture the isometry type of Hnb (G,V )
via arbitrary relatively injective strong resolutions. Therefore, a specialrole is played by those resolutions which compute the canonical seminorm.The following fundamental result is due to Ivanov, and implies that thesedistinguished resolutions are in some sense extremal:
Theorem 8.8. Let V be a normed R[G]-module, and let (V, V •, δ•) beany strong resolution of V . Then the identity of V can be extended to achain map α• between V • and the standard resolution of V , in such a waythat ‖αn‖ ≤ 1 for every n ≥ 0. In particular, the canonical seminormof H•
b (G,V ) is not bigger than the seminorm induced on H•b (G,V ) by any
relatively injective strong resolution.
Proof. One can define αn by induction setting, for every v ∈ En and gj ∈ G:
αn(v)(g0, . . . , gn) = αn−1(g0
(kn(g−10 (v)
)))(g1, . . . , gn),
where k• is a contracting homotopy for the given resolution (V, V •, δ•). Itis not difficult to prove by induction that α• is indeed a norm-decreasingchain G-map (see [Iva87], [Mon01, Theorem 7.3.1] for the details). �
Corollary 8.9. Let V be a normed R[G]-module, let (V, V •, δ•) be a rela-tively injective strong resolution of V , and suppose that the identity of V maybe extended to a chain map α• : C•
b (G,V )→ V • such that ‖αn‖ ≤ 1 for everyn ∈ N. Then α• induces an isometric isomorphism between H•
b (G,V ) andH•(V •). In particular, the seminorm induced by the resolution (V, V •, δ•)coincides with the canonical seminorm on H•
b (G,V ).
9. More on amenability
The following result establishes an interesting relationship between theamenability of G and the relative injectivity of normed R[G]-modules.
Proposition 9.1. The following facts are equivalent:
(1) The group G is amenable.(2) Every dual normed R[G]-module is relatively injective.(3) The trivial R[G]-module R is relatively injective.
Proof. (1) ⇒ (2): Let W be a normed R[G]-module, and let V = W ′ be thedual normed R[G]-module. We first construct a left inverse (over R[G]) of
31
the augmentation map ε : V → C0b (G,V ). We fix an invariant mean m on
G. For f ∈ C0b (G,V ) and w ∈W we consider the function
fw : G→ R , fw(g) = f(g)(w) .
It follows from the definitions that fw is an element of ℓ∞(G), so we maydefine a map r : C0
b (G,V )→ V by setting r(f)(w) = m(fw). It is immediateto check that r(f) is indeed a bounded functional on W , whose norm isbounded by ‖f‖∞. In other words, the map r is well-defined and norm non-increasing. The G-invariance of the mean m implies that r is G-equivariant,and an easy computation shows that r ◦ ε = IdV .
Let us now consider the diagram
0 // A ι//
α
��
B
β′
~~
�
�
�
�
�
βzzt
t
t
tt
σtt
V
�
C0b (G,V )
r
UU
By Lemma 8.3, C0b (G,V ) is relatively injective, so there exists a bounded
R[G]-map β′ such that ‖β′‖ ≤ ‖ε◦α‖ ≤ ‖α‖ and β′◦ι = ε◦α. The R[G]-mapβ := r ◦β′ satisfies ‖β‖ ≤ ‖α‖ and β ◦ ι = α. This shows that V is relativelyinjective.
(2) ⇒ (3) is obvious, so we are left to show that (3) implies (1). If R isrelatively injective and σ : ℓ∞(G) → R is the map defined by σ(f) = f(1),then there exists an R[G]-map β : ℓ∞(G) → R such that ‖β‖ ≤ 1 and thefollowing diagram commutes:
R ε//
Id
��
ℓ∞(R)
βwwooooooooooooo
σss
R
Using that β(1G) = 1 and ‖β‖ ≤ 1 it is easy to show that β is a mean.Being an R[G]-map, β is G-invariant, whence the conclusion. �
The previous proposition allows us to provide an alternative proof ofTheorem 6.6, which we recall here for the convenience of the reader:
Theorem 9.2. Let G be an amenable group, and let V be a dual normedR[G]-module. Then Hn
b (G,V ) = 0 for every n ≥ 1.
Proof. The complex
0 −→ VId−→ V −→ 0
provides a relatively injective strong resolution of V , so the conclusion fol-lows from Corollary 8.7. �
32
9.1. Amenable spaces. The notion of amenable space was introduced byZimmer [Zim78] in the context of actions of topological groups on standardmeasure spaces (see e.g. [Mon01, Section 5.3] for several equivalent defini-tions). In our case of interest, i.e. when G is a discrete group acting on aset S (which may be thought as endowed with the discrete topology), theamenability of S as a G-space amounts to the amenability of the stabilizersin G of elements of S [AEG94, Theorem 5.1]. Therefore, we may take thischaracterization as a definition:
Definition 9.3. A left action G×S → S of a group G on a set S is amenableif the stabilizer of every s ∈ S is an amenable subgroup of G. In this case,we equivalently say that S is an amenable G-set.
The importance of amenable G-sets is due to the fact that they may beexploited to isometrically compute the bounded cohomology of G. If S is anyG-set and V is any normed R[G]-module, then we denote by ℓ∞(Sn+1, V )the space of bounded functions from Sn+1 to V . This space may be endowedwith the a structure of a normed R[G]-module via the action
(g · f)(s0, . . . , sn) = g · (f(g−1s0, . . . , g−1sn)) .
The differential δn : ℓ∞(Sn+1)→ ℓ∞(Sn+1) defined by
δn(f)(s0, . . . , sn+1) =n∑
i=0
(−1)if(s0, . . . , si, . . . , sn)
endows the pair (ℓ∞(S•+1, V ), δ•) with the structure of a normed R[G]-complex. Togehter with the augmentation ε : V → ℓ∞(S, V ) given byε(v)(s) = v for every s ∈ S, such a complex provides a strong resolution ofV :
Lemma 9.4. The augmented complex
0 −→ V −→ ℓ∞(S, V )δ0−→ ℓ∞(S2, V )
δ1−→ ℓ∞(S3, V )
δ2−→ . . .
provides a strong resolution of V .
Proof. Let s be a fixed element of S. Then the maps
kn : ℓ∞(Sn+1, V )→ ℓ∞(Sn, V ) , kn(f)(s0, . . . , sn−1) = f(s, s0, . . . , sn−1)
provide the required contracting homotopy. �
Lemma 9.5. Suppose that S is an amenable G-space, and that V is a dualnormed R[G]-module. Then ℓ∞(Sn+1, V ) is relatively injective for everyn ≥ 0.
Proof. Of course, if S is amenable, then the same is true for Sn, n ≥ 1, sowe may assume n = 0. Moreover, let W be the normed R[G]-module suchthat V = W ′.
33
Let us consider the extension problem described in Definition 8.1, withℓ∞(S, V ):
0 // A ι//
α
��
B
βzzvv
vv
v
σtt
ℓ∞(S, V )
We denote by R ⊆ S a set of representatives for the action of G on S, and forevery r ∈ R we denote by Gr the stabilizer of r, endowed with the invariantmean µr. Moreover, for every s ∈ S we choose an element gs ∈ G such thatg−1s (s) = rs ∈ R. Then gs is uniquely determined up to right multiplication
by elements in Grs .Let us fix an element b ∈ B. In order to define β(b), for every s ∈ S we
need to know the value taken by β(b)(s) on every w ∈W . Therefore, we fixs ∈ S, w ∈W , and we set
(β(b)(s))(w) = µrs(fb) ,
where fb ∈ ℓ∞(Grs ,R) is defined by
fb(g) =((gsg) · α(σ(g−1g−1
s b)))(s) (w)
Since ‖σ‖ ≤ 1 we have that ‖β‖ ≤ ‖α‖, and the behaviour of means onconstant functions implies that β ◦ ι = α.
Observe that the element β(b)(s) does not depend on the choice of gs ∈ G.In fact, if we replace gs by gsg
′ for some g′ ∈ Grs , then the function f definedabove is replaced by the function
f ′b(g) =((gsg
′g) · α(σ(g−1g−1g−1s b))
)(s) (w) = fb(g
′g) ,
and µrs(f′b) = µrs(fb) by the invariance of the mean µrs . This fact allows
us to prove that β is a G-map. In fact, let us fix g ∈ G and let s = g−1(s).Then we may assume that gs = g−1gs, so
(g(β(b)))(s)(w) = β(b)(s)(g−1w) = µrs(f) ,
where f b ∈ ℓ∞(Grs ,R) is given by
f b(g) =((g−1gsg) · α(σ(g−1g−1
s gb)))(s) (g−1w)
=((gsg) · α(σ(g−1g−1
s gb)))(s) (w)
= fgb(g) .
This concludes the proof. �
As anticipated above, we are now able to show that amenable spaces maybe exploited to compute bounded cohomology:
Theorem 9.6. Let S be an amenable G-set and let V be a dual normedR[G]-module. Then the homology of the complex
0 −→ ℓ∞(S, V )Gδ0−→ ℓ∞(S2, V )G
δ1−→ ℓ∞(S3, V )G
δ2−→ . . .
34
is canonically isometrically isomorphic to H•(G,V ).
Proof. The previous lemmas imply that the augmented complex (V, ℓ∞(S•+1), δ•)provides a realtively injective strong resolution of V , so Corollary 8.7 impliesthat the homology of the G-invariants of (V, ℓ∞(S•+1), δ•) is isomorphic tothe bounded cohomology of G with coefficients in V . By Corollary 8.9, inorder to prove that such an isomorphism is isometric we are left to constructa norm non-increasing chain map
α• : C•b (G,V )→ ℓ∞(S•+1, V ) .
We keep notation from the proof of the previous lemma, i.e. we fix a setof representatives R for the action of G on S, and for every s ∈ S we choosean element gs ∈ G such that g−1
s s = rs ∈ R. For every r ∈ R we also fix aninvariant mean µr on the stabilizer Gr.
Let us fix an element f ∈ Cnb (G,V ), and take (s0, . . . , sn) ∈ Sn+1. IfV = W ′, we also fix an element w ∈ W . For every i we denote by ri ∈ Rthe representative of the orbit of si, and we consider the invariant meanµr1 × . . .× µrn on Gr1 × . . .×Grn (see Remark 6.5). Then we consider thefunction fs0,...,sn
∈ ℓ∞(Gr1 × . . .×Grn ,R) defined by
fs0,...,sn(g0, . . . , gn) = f(gs0g0, . . . , gsn
gn)(w) .
By construction we have ‖fs0,...,sn‖∞ ≤ ‖f‖∞ · ‖w‖W , so we may set
(αn(f)(s0, . . . , sn))(w) = (µr1 . . . × . . . µrn)(fs0,...,sn) ,
thus defining an element αn(f)(s0, . . . , sn) ∈W′ = V such that ‖αn(f)‖V ≤
‖f‖∞. We have thus shown that αn : Cnb (G,V ) → ℓ∞(Sn+1, V ) is a well-defined norm non-increasing linear map. The fact that αn commutes withthe action of G follows from the invariance of the means µr, r ∈ R, and thefact that α• is a chain map is obvious. �
Let us prove some direct corollaries of the previous results. Observe that,if W is a dual normed R[H]-module and ψ : G → H is a homomorphism,then the induced module ψ−1(W ) is a dual normed R[G]-module.
Theorem 9.7. Let ψ : G→ H be a surjective homomorphism with amenablekernel, and let W be a dual normed R[H]-module. Then the induced map
ψ• : H•b (H,W )→ H•
b (G,ψ−1W )
is an isometric isomorphism.
Proof. The action G × H → H defined by (g, h) 7→ ψ(g)h endows H withthe structure of an amenable G-set. Therefore, Theorem 9.6 implies thatthe bounded cohomology of G with coefficients in ψ−1W is isometricallyisomorphic to the cohomology of the complex ℓ∞(H•+1, ψ−1W )G. However,since ψ is surjective, we have a (tautological) isometric identification betweenℓ∞(H•+1, ψ−1W )G and C•
b (G,W )G, whence the conclusion. �
35
Corollary 9.8. Let ψ : G→ H be a surjective homomorphism with amenablekernel. Then Hn
b (G,R) is isometrically isomorphic to Hnb (H,R) for every
n ∈ N.
9.2. Alternating cochains.
10. Bounded cohomology of topological spaces
Let X be a topological space, and let R = Z,R. Recall that C•(X,R)(resp. C•(X,R)) denotes the usual complex of singular chains (resp. cochains)on X with coefficients in R. For i ∈ N, we let Si(X) be the set of singulari–simplices in X. We also regard Si(X) as a subset of Ci(X,R), so that forany cochain ϕ ∈ Ci(X,R) it makes sense to consider its restriction ϕ|Si(X).
For every ϕ ∈ Ci(X,R), we set
‖ϕ‖ = ‖ϕ‖∞ = sup {|ϕ(s)| | s ∈ Si(X)} ∈ [0,∞].
We denote by C•b (X,R) the submodule of bounded cochains, i.e. we set
C•b (X,R) = {ϕ ∈ C•(X,R) | ‖ϕ‖ <∞} .
Since the differential takes bounded cochains into bounded cochains, C•b (X,R)
is a subcomplex of C•(X,R). We denote by H•(X,R) and H•b (X,R) respec-
tively the homology of the complexes C•(X,R) and C•b (X,R). Of course,
H•(X,R) is the usual singular cohomology module of X with coefficients inR, while H•
b (X,R) is the bounded cohomology module of X with coefficientsin R. Just as in the case of groups, the norm on Ci(X,R) descends (afterthe suitable restrictions) to a seminorm on each of the modules H•(X,R),H•b (X,R). More precisely, if ϕ ∈ H is a class in one of these modules, which
is obtained as a quotient of the corresponding module of cocycles Z, thenwe set
‖ϕ‖ = inf {‖ψ‖ | ψ ∈ Z, [ψ] = ϕ in H} .
This seminorm may take infinite values on elements in H•(X,R) and maybe null on non-zero elements in H•
b (X,R) (but not on non-zero elementsin H•(X,R): this is clear in the case with integer coefficients, and it isa consequence of the Universal Coefficient Theorem in the case with realcoefficients, since a real cohomology class with vanishing seminorm has tobe null on any cycle, whence null in H•(X,R)).
The inclusion of bounded cochains into possibly unbounded cochains in-duce a map in cohomology
c• : H•b (X,R)→ H•(X,R),
which is called comparison map.
10.1. Basic properties of bounded cohomology of spaces. Boundedcohomology enjoys some of the fundamental properties of classical singularcohomology: for example, H i
b({pt.}, R) = 0 if i > 0, and H0b ({pt.}, R) = R
(more in general, H0b (X,R) is canonically isomorphic to ℓ∞(S), where S
is the set of the path connected components of X). The usual proof of
36
the homotopy invariance of singular homology is based on the constructionof an algebraic homotopy which takes every n-simplex onto the sum of atmost n+ 1 (n+ 1)-dimensional simplices. As a consequence, the homotopyoperator induced in cohomology preserves bounded cochains. This impliesthat bounded cohomology is a homotopy invariant of topological spaces.Moreover, if (X,Y ) is a topological pair, then there is an obvious definitionof H•
b (X,Y ), and it is immediate to check that the analogous of the longexact sequence of the pair in classical singular cohomology also holds in thebounded case.
Perhaps the most important peculiarity of bounded cohomology with re-spect to classical singular cohomology is the lacking of any Mayer-Vietorissequence (or, equivalently, of any excision theorem). This is also at thebasis of the phenomenon for which spaces with finite-dimensional boundedcohomology may be tamely glued to each other to get spaces with infinite-dimensional bounded cohomology (see Remark 10.5 below).
Recall from Subsection 1.2 that Hn(X,R) ∼= Hn(π1(X), R) for everyaspherical CW-complex X. We are now going to prove an analogous re-sult in the context of bounded cohomology. As anticipated in Subsec-tion 2.3, a fundamental result by Gromov provides an isometric isomorphismHnb (X,R) ∼= Hn
b (π1(X),R) even without any assumption on the aspheric-ity of X. This section is devoted to a proof of Gromov’s result. We willclosely follows Ivanov’s argument [Iva87], which deals with the case whenX is (homotopically equivalent to) a countable CW-complex. Before goinginto the general case, we will concentrate our attention on the easier case ofaspherical spaces.
10.2. Bounded singular cochains as relatively injective modules.Henceforth, we assume that X is a path connected topological space admit-
ting the universal covering X, we denote by G the fundamental group of X,and we fix an identification of G with the group of covering automorphisms
of X. Just as we did in Subsection 7.4, we endow C•b (X,R) with the struc-
ture of a normed R[G]-module. Our arguments are based on the obviousbut fundamental isometric identification
C•b (X,R) ∼= C•
b (X,R)G ,
which induces a canonical isometric identification
H•b (X,R) ∼= H•(C•
b (X,R)G) .
As a consequence, in order to prove the isomorphism H•b (X,R) ∼= H•
b (G,R)
it is sufficient to show that the complex C•b (X,R) provides a relatively
injective strong resolution of R (some more care is needed to prove thatthis isomorphism is also isometric). The relative injectivity of the modules
Cnb (X,R) can be easily deduced from the argument described in the proofof Lemma 7.6, which applies verbatim in the context of bounded singularcochains:
37
Lemma 10.1. For every n ∈ N, the bounded cochain module Cnb (X,R) isrelatively injective.
Therefore, in order to show that the bounded cohomology of X is isomor-phic to the bounded cohomology of G we need to show that the (augmented)
complex C•b (X,R) provides a strong resolution of R. We will show that this
is the case if X is contractible. In the general case, it is not even true
that C•b (X,R) is acyclic (see Remark 10.10). However, in the case when
R = R a deep result by Ivanov shows that the complex C•b (X,R) indeed
provides a strong resolution of R. A sketch of Ivanov’s proof will be givenin Subsection 10.4.
Before going on, we point out that we always have a norm non-increasingmap from the bounded cohomology of G to the bounded cohomology of X:
Lemma 10.2. Let us endow the complexes C•b (G,R) and C•
b (X,R) withthe obvious augmentations. Then, there exists a norm non-increasing chainmap
r• : C•b (G,R)→ C•
b (X,R)
extending the identity of R.
Proof. Let us choose a fundamental region R for the action of G on X . We
consider the map r0 : S0(X) = X → G taking a point x to the unique g ∈ G
such that x ∈ g(R). For n ≥ 1, we define rn : Sn(X) → Gn+1 by setting
rn(s) = (r0(s(e0)), . . . , rn(s(en))), where s ∈ Sn(X) and s(ei) is the i-th
vertex of s. Finally, we extend rn to Cn(X,R) by R-linearity and definern as the dual map of rn. Since rn takes any single simplex to a single(n+1)-tuple, it is readily seen that rn is norm non-increasing (in particular,it takes bounded cochains into bounded cochains). The fact that rn is aG-equivariant chain map is obvious. �
The following result will prove useful later:
Lemma 10.3. Let Y be a path connected topological space. If Y is aspher-ical, then the augmented complex C•
b (Y,R) admits a contracting homotopy(so it is a strong resolution of R). If πi(Y ) = 0 for every i ≤ n, then thenthere exists a partial contracting homotopy
R C0b (Y,R)
k0
oo C1b (Y,R)
k1
oo . . .k2
oo Cnb (Y,R)kn
oo Cn+1b (Y,R)
kn+1
oo
(where we require that the equality δm−1km + km+1δm = IdCm
b(Y,R) holds for
every m ≤ n, and δ−1 = ε is the usual augmentation).
Proof. For every −1 ≤ m ≤ n we construct a map Tm : Cm(Y,R) →Cm+1(Y,R) such that dm+1Tm + Tm−1dm = IdCm(Y,R), where we under-stand that C−1(Y,R) = R and d0 : C0(Y,R)→ R is the augmentation mapd0(∑riyi) =
∑ri. Let us fix a point y0 ∈ Y , and define T−1 : R→ C0(Y,R)
by setting T−1(r) = ry0. For m ≥ 0 we define Tm as the R-linear extension
38
of a map Tm : Sm(Y ) → Sm+1(Y ) having the following property: for everys ∈ Sm(Y ), the 0-th vertex of Tm(s) is equal to y0, and has s as opposite face.We proceed by induction, and suppose that Ti has been defined for every−1 ≤ i ≤ m. Take s ∈ Sm(Y ). Then, using the fact that πm(Y ) = 0 and theproperties of Tm−1, it is not difficult to show that a simplex s′ ∈ Sm+1(Y )exists which satisfies both the equality dm+1s
′ = s − Tm−1(dms) and theadditional requirement described above. We set Tm+1(s) = s′, and we aredone.
Since Tm−1 sends any single simplex to a single simplex, its dual mapkm sends bounded cochains into bounded cochains, and has operator normequal to one. Therefore, the maps km : Cmb (Y,R)→ Cm−1
b (Y,R), m ≤ n+1,provide the desired (partial) contracting homotopy. �
10.3. The aspherical case. We are now ready to show that, under the as-sumption thatX is aspherical, the bounded cohomology ofX is isometricallyisomorphic to the bounded cohomology of G for any ring of coefficients:
Theorem 10.4. Let X be an aspherical space, i.e. a path connected topo-logical space such that πn(X) = 0 for every n ≥ 2. Then Hn
b (X,R) isisometrically isomorphic to Hn
b (G,R) for every n ∈ N.
Proof. Lemmas 10.1 and 10.3 imply that the complex
0 −→ Rε−→ C0
b (X,R)δ0−→ C1
b (X,R)δ1−→ . . .
provides a relatively injective strong resolution of R as a trivial R[G]-module,soHn
b (X,R) is canonically isomorphic toHnb (G,R) for every n ∈ N. The fact
the the isomorphism Hnb (X,R) ∼= Hn
b (G,R) is isometric is a consequence ofCorollary 8.9 and Lemma 10.2. �
Remark 10.5. Let S1 ∨ S1 be the wedge of two copies of the circle. ThenTheorem 10.4 implies thatH2
b (S1∨S1,R) ∼= H2
b (F2,R) is infinite-dimensional,while H2
b (S1) ∼= H2
b (Z) = 0. This shows that bounded cohomology of spacescannot satisfy any Mayer-Vietoris principle.
10.4. Ivanov’s contracting homotopy. We now come back to the general
case, i.e. we do not assume that X is contractible. In order to show thatHnb (X,R) is isometrically isomorphic to Hn
b (G,R) we need to prove that
the complex of singular bounded cochains on X provides a strong resolutionof R. In the case when R = Z, this is false in general, since the complex
C•b (X,Z) may be even non-exact (see Remark 10.10). On the other hand,
a deep result by Ivanov ensures that C•b (X,R) indeed provides a strong
resolution of R.Ivanov’s argument makes use of sophisticated techniques from algebraic
topology, which work under the assumption that X, whence X , is a count-able CW-complex (but see Remark 10.9). We begin with the following:
Lemma 10.6 ([Iva87], Theorem 2.2). Let p : Z → Y be a principal H-bundle, where H is a topological group. Then there exists a chain map
39
A• : C•b (Z,R) → C•
b (Y,R) such that ‖An‖ ≤ 1 for every n ∈ N and A• ◦ p•
is the identity of C•b (Y,R).
Proof. For ϕ ∈ Cnb (Z,R), s ∈ Sn(Y ), the value A(ϕ)(s) is obtained bysuitably averaging the value of ϕ(s′), where s′ ranges over the set
Ps = {s′ ∈ Sn(Z) | p ◦ s′ = s} .
More precisely, let Kn = F (∆n,H) be the space of continuous functionsfrom the standard n-simplex to H, and define on Kn the operation given bypointwise multiplication of functions. With this structure, Kn is an abeliangroup, so it admits an invariant mean µn. Observe that the permutationgroup Sn+1 acts on ∆n via affine transformations. This action induces anaction on Kn, whence on ℓ∞(Kn), and on the space of means on Kn, sothere is an obvious notion of Sn+1-invariant mean on Kn. By averagingover the action of Sn+1, the space of Kn-invariant means may be retractedonto the spaceMn of Sn+1-invariant Kn-invariant means on Kn, which, inparticular, is non-empty. Finally, observe that any affine identification of∆n−1 with a face of ∆n induces a mapMn →Mn−1. Since elements ofMn
are Sn+1-invariant, this map does not depend on the chosen identification.Therefore, we get a sequence of maps
M0 M1oo M2
oo M3oo . . .oo
Recall now that the Banach-Alaouglu Theorem implies that every Mn iscompact. This easily implies that there exists a sequence {µn} of meanssuch that µn ∈ Mn and µn 7→ µn−1 under the map Mn →Mn−1. We saythat such a sequence in compatible.
Let us now fix s ∈ Sn(Y ), and observe that there is a bijection betweenPs and Kn. This bijection is uniquely determined up to the choice of anelement in Ps, up to left multiplication by an element of Kn. In particular,if f ∈ ℓ∞(Ps), and µ is a left invariant mean on Kn, then there is a well-defined value µ(f).
Let us now choose a compatible sequence of means µ0, . . . , µn, . . .. Wedefine the operator An by setting
An(ϕ)(s) = µn(ϕ|Ps) for every ϕ ∈ Cnb (Z), s ∈ Sn(Y ) .
Since the sequence {µn} is compatible, the sequence of maps A• is a chainmap. The inequality ‖An‖ ≤ 1 is obvious, and the fact that A• ◦ p• isthe identity of C•
b (Y,R) may be deduced from the behaviour of means onconstant functions. �
Theorem 10.7 ([Iva87]). Let X be a path connected CW-complex with uni-
versal covering X. Then the (augmented) complex C•b (X,R) provides a rel-
atively injective strong resolution of R.
Proof. We only sketch Ivanov’s argument, referring the reader to [Iva87] forfull details. Building on results by Dold and Thom [DT58], Ivanov constructs
40
an infinite tower of bundles
X1 X2p1
oo X3p2
oo . . .oo Xnoo . . .oo
where X1 = X, πi(Xm) = 0 for every i ≤ m, πi(Xm) = πi(X) for everyi > m and each map pm : Xm+1 → Xm is a principal Hm-bundle for sometopological connected abelian group Hm, which has the homotopy type of aK(πn+1(X), n).
By Lemma 10.3, for every n we may construct a partial contracting ho-motopy
R C0b (Xn,R)
k0n
oo C1b (Xn,R)
k1n
oo . . .k2
noo Cn+1
b (Xn,R) .kn+1
noo
Moreover, Lemma 10.6 implies that for every m ∈ N the chain mapp•m : C•
b (Xm,R)→ C•b (Xm+1,R) admits a left inverse chain mapA•
m : C•b (Xm+1,R)→
C•b (Xm,R) which is norm non-increasing. This allows us to define a partial
contracting homotopy
R C0b (X,R)
k0oo C1
b (X,R)k1
oo . . .k2oo Cn+1
b (X,R)kn+1
oo . . .oo
via the formula
ki = Ai−11 ◦ . . . ◦ Ai−1
n−1 ◦ kin ◦ p
in−1 ◦ . . . ◦ p
i2 ◦ p
i1 for every i ≤ n+ 1 .
The existence of such a partial homotopy is sufficient for all our applications.In order to construct a complete contracting homotopy a further argumentis needed, for which we refer the reader to [Iva87]. �
10.5. Gromov’s Theorem. The discussion in the preceding subsection im-plies the following:
Theorem 10.8 ([Gro82, Iva87]). Let X be a countable CW-complex. ThenHnb (X,R) is canonically isomorphic to Hn
b (G,R).
Proof. Observe that, if X is a countable CW-complex, then G = π1(X) is
countable, so X is also a countable CW-complex. Therefore, Lemma 10.1Theorem 10.7 imply that the complex
0 −→ Rε−→ C0
b (X,R)δ0−→ C1
b (X,R)δ1−→ . . .
provides a relatively injective strong resolution of R as a trivial R[G]-module,so Hn
b (X,R) is canonically isomorphic to Hnb (G,R) for every n ∈ N. The
fact the the isomorphismHnb (X,R) ∼= Hn
b (G,R) is isometric is a consequenceof Corollary 8.9 and Lemma 10.2. �
Remark 10.9. Theo Bulher has communicated to the author that Ivanov’sargument may be generalized to show that C•
b (Y,R) is a strong resolution ofR whenever X is a simply connected topological space. As a consequence,Theorem 10.8 holds even when the assumption that X is a countable CW-complex is replaced by the weaker condition that X is path connected andadmits a universal covering.
41
Remark 10.10. Theorem 10.8 does not hold for bounded cohomology withinteger coefficients. In fact, if X is any topological space, just as in the proofof Proposition 4.9 the short exact sequence 0→ Z→ R→ R/Z→ 0 inducesan exact sequence
Hnb (X,R) −→ Hn(X,R/Z) −→ Hn+1
b (X,Z) −→ Hn+1b (X,R) .
IfX is a simply connected CW-complex, then Hnb (X,R) = 0 for every n ≥ 1,
so we have
Hnb (X,Z) ∼= Hn−1
b (X,R/Z) for every n ≥ 2.
For example, in the case of the 2-dimensional sphere we have H3b (S
2,Z) ∼=H2(S2,R/Z) ∼= R/Z.
11. ℓ1-homology and duality
Suppose that V is any Z[G]-module, and let Cn(G,Z) be the free Z-module admitting the set Gn+1 as a basis. The diagonal action of G ontoGn+1 defined by g · (g0, . . . , gn) = (gg0, . . . , ggn) extends to a an actionof G on Cn(G,Z), which, therefore, is endowed with the structure of aZ[G]-module. Moreover, the elements of the form (1, g1, g1g2, . . . , g1 · · · gn)freely generate Cn(G,Z) as a Z[G]-module, so Cn(G,Z) is a projective Z[G]-module. The map
Gn+1 → Cn−1(G,Z) (g0, . . . , gn)→n∑
i=0
(g0, . . . , gi, . . . , gn)
extends to a Z[G]-map dn : Cn(G,Z) → Cn−1(G,Z), and it is immediate tocheck that ker dn = Im dn+1 = 0 for every n ≥ 1. Moreover, together withthe augmentation map
ε : C0(G,V )→ Z , ε
∑
g∈G
agg
=
∑
g∈G
ag ,
the complex (C•(G,Z), d•) provides a projective resolution of Z over Z[G]:
0←− Zε←− C0(G,Z)
d1←− C1(G,Z)d2←− . . .
dn←− Cn(G,Z)dn+1
←− . . .
Let now V be a Z[G]-module. Then there is an obvious Z[G]-isomorphismbetween the module HomZ(Cn(G,Z), V ) and the cochain module
Cn(G,V ) = {f : Gn+1 → V }
(which was introduced above in the case when V is a vector space). Thisshows that the definition of group cohomology with coefficients in an R[G]-module given in the previous section just specializes the general definitionof group cohomology with coefficients in a Z[G]-module.
Theorem 11.1. AAA
42
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Dipartimento di Matematica, Universita di Pisa, Largo B. Pontecorvo 5,56127 Pisa, Italy
E-mail address: [email protected]
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