# Volume and Euler classes in bounded cohomology of transformation groups

Michael Brandenbursky  Department of Mathematics, Ben Gurion University,
Israel<EMAIL_ADDRESS>and  Michał Marcinkowski  Institute of Mathematics,
Wrocław University, Poland<EMAIL_ADDRESS>
###### Abstract.

Let $M$ be an oriented smooth manifold, and $\operatorname{Homeo}(M,\omega)$
the group of measure preserving homeomorphisms of $M$, where $\omega$ is a
finite measure induced by a volume form. In this paper, we define volume and
Euler classes in bounded cohomology of an infinite dimensional transformation
group $\operatorname{Homeo}_{0}(M,\omega)$ and
$\operatorname{Homeo}_{+}(M,\omega)$ respectively, and in several cases prove
their non-triviality. More precisely, we define:

  * •

Volume classes in
$\operatorname{H}_{b}^{n}(\operatorname{Homeo}_{0}(M,\omega))$ where $M$ is a
hyperbolic manifold of dimension $n$.

  * •

Euler classes in
$\operatorname{H}_{b}^{2}(\operatorname{Homeo}_{+}(S,\omega))$ where $S$ is an
oriented closed hyperbolic surface.

We show that Euler classes have positive norms for any closed hyperbolic
surface and volume classes have positive norms for all hyperbolic surfaces and
certain hyperbolic $3$-manifolds; hence, they are non-trivial.

M.B. was partially supported by the Israel Science Foundation grant 823/23.

M.M. was supported by Opus 2017/27/B/ST1/01467 funded by Narodowe Centrum
Nauki.

##  1\. Introduction

Let $M$ be an oriented connected smooth manifold. Suppose $\omega$ is a finite
measure induced by a volume form on $M$. In [BM22] we defined a homomorphism

$\Gamma_{b}\colon\operatorname{H}_{b}^{\bullet}(\pi_{1}(M))\to\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{0}(M,\omega)),$

where $\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{0}(M,\omega))$ is
the bounded cohomology of a discrete group. The map $\Gamma_{b}$ is a
generalization of a map defined by Gambaudo and Ghys in the quasimorphism
setting [GG04, Section 5].

$\Gamma_{b}$ was used in [BM22] to show that the $3^{rd}$ bounded cohomology
of $\operatorname{Homeo}_{0}(M,\omega)$ is infinite dimensional for many
manifolds $M$. In [Nit] more results concerning bounded cohomology, as well as
standard cohomology, of $\operatorname{Homeo}_{0}(M,\omega)$ were obtained.
Variations of $\Gamma_{b}$ were used in [Kim20] to prove similar results
concerning $\operatorname{Diff}_{0}(S,area)$ where $S$ is a disc, sphere or
torus.

In this paper, we continue this line of research and focus on two important
families of cohomology classes described below. Moreover, we construct an
extension of $\Gamma_{b}$ to a map

$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}\colon\operatorname{H}_{b}^{\bullet}(\mathcal{M}(M,*))\to\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}(M,\omega)),$

where $\mathcal{M}(M,*)$ is the mapping class group of once punctured $M$, see
Section 3.C. This extension is used to define the Euler class on
$\operatorname{Homeo}_{+}(S,\omega)$ for an oriented closed hyperbolic surface
$S$ and we hope that it might be useful to study the cohomology of
$\operatorname{Homeo}(M,\omega)$ for other manifolds $M$.

The bounded volume class. Let $M$ be an oriented hyperbolic manifold and let
$\omega_{h}$ be the volume form induced by the hyperbolic metric. In this
setting $\omega_{h}$ defines a class
$Vol_{M}\in\operatorname{H}_{b}^{n}(\pi_{1}(M))\simeq\operatorname{H}_{b}^{n}(M)$,
see Section 2.B. If $M$ is closed, $Vol_{M}$ is a natural bounded
representative of $[\omega_{h}]\in\operatorname{H}_{dR}^{n}(M)$.

Our main result positively answers the question in Section 5 of [BM22] for
degree 2 and partially for degree 3. Moreover, it may be seen as a successful
attempt to define a volume class in the bounded cohomology of an infinite
dimensional transformation group.

######  Theorem 1.1.

Let $M$ be an oriented manifold of dimension $n$ such that it is either:

  * •

A hyperbolic surface with a non-abelian fundamental group or

  * •

A complete $3$-dimensional hyperbolic manifold that fibers over the circle
with a non-compact fiber.

Suppose a measure $\omega$ is induced by a volume form on $M$ and $\omega$ is
finite. Then the class
$\Gamma_{b}(Vol_{M})\in\operatorname{H}_{b}^{n}(\operatorname{Homeo}_{0}(M,\omega))$
has positive norm and hence is non-trivial.

The class $Vol_{M}\in\operatorname{H}_{b}^{n}(M)$ was considered by Gromov and
Thurston in the proof of the proportionality principle [Gro82, Thu22a].
Moreover, it serves as a rich source for classes in $3^{rd}$ bounded
cohomology of free and surface groups [Som97].

It is worth mentioning that it is not known if the bounded cohomology in
degree $n$ of a non-abelian free group for $n>3$ is non-trivial, hence our
proof works in dimension $2$ and sometimes in dimension $3$ so far. The
problem of non-triviality of $\Gamma_{b}(Vol_{M})$ for higher dimensional
hyperbolic manifolds or closed hyperbolic $3$-manifolds is still open.

The bounded Euler class. Let $S$ be a closed oriented surface and $\omega$ a
measure defined by an area form on $S$. Denote by
$\operatorname{Homeo}_{+}(S,\omega)$ the subgroup of
$\operatorname{Homeo}(S,\omega)$ of orientation preserving homeomorphisms.

Let $*\in S$ be a point in $S$ and $\mathcal{M}_{+}(S,*)$ be the orientation
preserving mapping class group of $S~{}\backslash~{}\\{*\\}$. Recall that
$\pi_{1}(S,*)<\mathcal{M}_{+}(S,*)$ due to the Birman exact sequence. As we
will see in Section 3.C, the map $\Gamma_{b}$ can be extended to a map
$\Gamma^{{\scriptscriptstyle\mathcal{M}_{+}}}_{b}$ such that the following
diagram commutes:

${\operatorname{H}_{b}^{\bullet}(\mathcal{M}_{+}(S,*))}$${\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{+}(S,\omega))}$${\operatorname{H}_{b}^{\bullet}(\pi_{1}(S,*))}$${\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{0}(S,\omega))}$$\scriptstyle{\Gamma_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}}$$\scriptstyle{\Gamma_{b}}$

The vertical arrows are induced by inclusions. Thus at the cost of passing to
a bigger group $\mathcal{M}_{+}(S,*)$, we can generate classes in the group of
$\omega$-preserving homeomorphisms of $S$ not necessarily isotopic to the
identity.

Let $\operatorname{Homeo}_{+}(S^{1})$ be the group of orientation preserving
homeomorphisms of the circle and let
$e_{b}\in\operatorname{H}_{b}^{2}(\operatorname{Homeo}_{+}(S^{1}))$ be the
bounded Euler class. There is a natural map
$\alpha\colon\mathcal{M}_{+}(S,*)\to\operatorname{Homeo}_{+}(S^{1})$ defined
by the action of
$\mathcal{M}_{+}(S,*)\simeq\operatorname{Aut}_{+}(\pi_{1}(S))$ on the Gromov
boundary of $\pi_{1}(S)$. Let $e_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}$
and $e_{b}^{S}$ be the pull-backs of $e_{b}$ to $\mathcal{M}_{+}(S,*)$ and
$\pi_{1}(S)$.

######  Theorem 1.2.

Let $S$ be a closed oriented surface of genus $\geq 2$, and $\omega$ a measure
induced by an area form on $S$. Then the classes
$\Gamma_{b}(e_{b}^{S})\in\operatorname{H}_{b}^{2}(\operatorname{Homeo}_{0}(S,\omega))$
and
$\Gamma_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}(e_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}})\in\operatorname{H}_{b}^{2}(\operatorname{Homeo}_{+}(S,\omega))$
have positive norms and hence are non-trivial.

We emphasise that in Theorem 1.1 and in Theorem 1.2 one can take instead of
$\operatorname{Homeo}_{0}(M,\omega)$ smaller groups like
$\operatorname{Diff}_{0}(M,\omega)$, or $\operatorname{Symp}_{0}(S,\omega)$ or
$\operatorname{Ham}(S)$ whenever $S$ is a hyperbolic surface, and the same
results hold.

On the proof and organization of the paper. The method of the proof is a
refinement (and at the same time a simplification) of the one from [BM22]. In
the case of bounded volume classes, it is based on mapping a non-abelian free
group $F$ to $\pi_{1}(M)$ and to $\operatorname{Homeo}_{0}(M,\omega)$,
restricting $Vol_{M}$ and $\Gamma_{b}(Vol_{M})$ to $F$ and then comparing
these two classes. An identical method is used for Euler classes. This
technique is quite special, since we do not know many subgroups of
$\operatorname{Homeo}_{0}(M,\omega)$. See [MT19, Chapter 4] for a survey on
realizations of groups by diffeomorphisms and homeomorphisms.

The outline of the paper is as follows. In Section 2.A we give basic
definitions of bounded cohomology. In Section 2.B we define two versions of
the volume class: the topological $Vol_{M}\in\operatorname{H}_{b}^{n}(M)$ and
its group version $Vol_{M}^{gp}\in\operatorname{H}_{b}^{n}(\pi_{1}(M))$. The
topological version is well known and we use it in the proofs. We need the
group version because $\Gamma_{b}$ takes classes from the group cohomology. As
expected, $Vol_{M}$ and $Vol^{gp}_{M}$ define the same class under the
canonical identification
$\operatorname{H}_{b}^{n}(M)\simeq\operatorname{H}_{b}^{n}(\pi_{1}(M))$. We
could not find a proof of this fact in the literature, thus for completeness
we provide a detailed argument in Lemma 2.2. In Section 2.C we define the
Euler class in the mapping class group of a punctured surface. In Section 3 we
define the maps $\Gamma_{b}$ and $\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$.
In Section 4 we show that for hyperbolic surfaces and certain $3$-dimensional
hyperbolic manifolds, the class $Vol_{M}$ restricts non-trivially to a free
subgroup of $\pi_{1}(M)$. The same result holds for Euler classes and closed
surfaces of genus $\geq 2$. In Section 5 we give a simpler and at the same
time more general version of a technical Lemma 3.1 from [BM22] and prove the
main theorems. In Section 6 we discuss the case when $\omega$ is the Dirac
measure.

Acknowledgements. MB acknowledges the support of the Israeli Science
Foundation grant 823/23. MB was partially supported by a Humboldt research
fellowship. MM was supported by grant Opus 2017/27/B/ST1/01467 funded by the
Narodowe Centrum Nauki. We thank the Center for Advanced Studies in
Mathematics at Ben Gurion University for supporting the visit of the second
author at BGU. The authors would like to thank Steve Farre for helpful
discussions.

##  2\. Preliminaries

###  2.A. Bounded cohomology.

Let us give the definitions of bounded cohomology of a group and a space.

Let $G$ be a group. The space of bounded $n$-cochains is defined by

$\operatorname{C}^{n}_{b}(G)=\\{c\colon
G^{n+1}\to\mathbb{R}~{}|~{}c~{}\text{is bounded}\\}.$

Let $d_{n}$ be the ordinary coboundary operator
$d_{n}\colon\operatorname{C}^{n}_{b}(G)\to\operatorname{C}^{n+1}_{b}(G)$. The
group $G$ acts on $\operatorname{C}^{n}_{b}(G)$ by

$h(c)(g_{0},\ldots,g_{n})=c(h^{-1}g_{0},\ldots,h^{-1}g_{n})\hskip
11.38092pt\forall c\in\operatorname{C}^{n}_{b}(G),\hskip 11.38092pt\forall
h,g_{0},\ldots,g_{n}\in G.$

Let $\operatorname{C}^{n}_{b}(G)^{G}$ be the space of $G$-invariant cochains.
The bounded cohomology of $G$, denoted by $\operatorname{H}^{\bullet}_{b}(G)$,
is the homology of the cochain complex
$\\{\operatorname{C}^{n}_{b}(G)^{G},d_{n}\\}$. Note that
$\operatorname{C}_{b}^{n}(G)^{G}$ is a subcomplex of the space of all
$G$-invariant cochains, hence we have a map
$\operatorname{H}_{b}^{n}(G)\to\operatorname{H}^{n}(G,\mathbb{R})$ called the
comparison map.

On $\operatorname{C}^{n}_{b}(G)$ we have the supremum norm denoted by
$||\cdot||$. This norm induces a semi-norm on $\operatorname{H}^{n}_{b}(G)$,
i.e., if $C\in\operatorname{H}_{b}^{n}(G)$, then

$||C||=\operatorname{min}\\{||c||~{}|~{}[c]=C\\}.$

Let $M$ be a topological space. A singular simplex is a continuous map from
the standard simplex to $M$. By $\operatorname{C}_{n}(M)$ we denote the space
of singular chains and by
$\operatorname{S}_{n}(M)\subset\operatorname{C}_{n}(M)$ the set of all
singular simplices in $M$. Let $\operatorname{C}^{n}_{b}(M)$ be the set of
linear functions from $\operatorname{C}_{n}(M)$ to the reals that are bounded
on $\operatorname{S}_{n}(M)$. Since $\operatorname{S}_{n}(M)$ generates
$\operatorname{C}_{n}(M)$ one can think of an element in
$\operatorname{C}_{n}^{b}(M)$ as a bounded function
$c\colon\operatorname{S}_{n}(M)\to\mathbb{R}$.

The bounded cohomology of $M$, denoted by $\operatorname{H}^{\bullet}_{b}(M)$,
is the homology of the cochain complex
$\\{\operatorname{C}^{n}_{b}(M),d_{n}\\}$, where
$d_{n}\colon\operatorname{C}^{n}_{b}(M)\to\operatorname{C}^{n+1}_{b}(M)$ is
the standard coboundary operator. Like in the group case, we have the seminorm
and the comparison map. Sometimes it is convenient to work in the universal
cover of $M$. Let us recall that on $\operatorname{C}^{n}_{b}(\widetilde{M})$
we have an action of $G=\pi_{1}(M)$ and $\operatorname{C}^{n}_{b}(M)$ is
naturally isomorphic to $\operatorname{C}^{n}_{b}(\widetilde{M})^{G}$.

We point out that bounded cohomology cannot be defined in terms of simplicial
cochains (given some triangulation of $M$). For example, if the triangulation
is finite, then every simplicial cochain is bounded and we get the standard
cohomology.

In this paper the manifold $M$ is aspherical. In this case
$\operatorname{H}_{b}^{n}(\pi_{1}(M))$ is canonically isometric to
$\operatorname{H}_{b}^{n}(M)$. See [Fri17, Chapter 5] for a relatively
elementary proof of this fact. It is based on an appropriate notion of
resolution for bounded cohomology. Note that by the remarkable Mapping Theorem
of Gromov, the assumption on asphericity of $M$ can be dropped, but the proof
of this fact is much harder [Gro82].

###  2.B. The bounded volume class.

In this paper a hyperbolic manifold is a manifold (with or without boundary)
whose all sectiontal curvatures equal $-1$. In particular we do not assume a
hyperbolic manifold to be complete. Let $M$ be a connected, oriented, and
aspherical hyperbolic $n$-manifold. Below we recall the definition of
$Vol_{M}$ and give a detailed description of its counterpart in the group
cohomology $\operatorname{H}_{b}^{n}(\pi_{1}(M))$, which seems to be less
known. In Lemma 2.2 we show that both versions give the same class under the
canonical identification of $\operatorname{H}_{b}^{n}(M)$ and
$\operatorname{H}_{b}^{n}(\pi_{1}(M))$.

We start with defining the volume class in the cohomology of a group. Let
$Iso_{+}(\mathbb{H}^{n})$ denote the group of orientation preserving
isometries of the hyperbolic $n$-space and let $vol_{h}$ be the hyperbolic
volume form on $\mathbb{H}^{n}$. Fix $*\in\mathbb{H}^{n}$ and for a tuple of
elements $\bar{g}=(g_{0},\ldots,g_{n})$ in $Iso_{+}(\mathbb{H})$ consider the
geodesic simplex $\Delta_{\bar{g}}\subset\mathbb{H}^{n}$ spanned by the points
$g_{0}(*),\ldots,g_{n}(*)$. This simplex can be parametrized using the
barycentric coordinates [Thu22a, Chapter 6]. Therefore we regard
$\Delta_{\bar{g}}$ as a map from the standard simplex to $\mathbb{H}^{n}$. We
define

$v(\bar{g})=\int_{\Delta_{\bar{g}}}vol_{h}.$

Note that $v(\bar{g})$ is the signed volume of $\Delta_{\bar{g}}$. Moreover,
$v$ is an $Iso_{+}(\mathbb{H}^{n})$-invariant cocycle. Since volumes of
geodesic simplices are bounded, $v$ is bounded and we can define:

$Vol=[v]\in\operatorname{H}^{n}_{b}(Iso_{+}(\mathbb{H}^{n})).$

For $G<Iso_{+}(\mathbb{H}^{n})$ we define $v_{G}$ to be the restriction of $v$
to $G$ and $Vol_{G}=[v_{G}]\in\operatorname{H}^{n}_{b}(G)$.

Recall that $M$ is a connected, oriented, and aspherical hyperbolic
$n$-manifold. We allow $M$ to have cusps or a boundary that is not totally
geodesic. Represent $M$ as a quotient $M=X/\pi_{1}(M)$ where
$X\subset\mathbb{H}^{n}$ is contractible and $\pi_{1}(M)$ acts on $X$ by deck
transformations. Note that the action of $\pi_{1}(M)$ on $X$ extends uniquely
to an action on $\mathbb{H}^{n}$. Denote this action by
$\rho\colon\pi_{1}(M)\to Iso_{+}(\mathbb{H}^{n})$. In what follows we usually
do not mention the representation $\rho$ and regard $\pi_{1}(M)$ as a discrete
subgroup of $Iso_{+}(\mathbb{H}^{n})$. To $M$ we associate the class

$Vol_{\rho(\pi_{1}(M))}\in\operatorname{H}_{b}^{n}(\pi_{1}(M)),$

and write $Vol_{M}^{gp}=Vol_{\rho(\pi_{1}(M))}$. The action $\rho$ is well
defined up to conjugacy, thus $Vol_{M}^{gp}$ does not depend on $\rho$.

Let us now describe a singular cocycle that defines the class $Vol_{M}$ in
$\operatorname{H}_{b}^{n}(M)$. Let $\sigma$ be a singular simplex in
$X\subset\mathbb{H}^{n}$. The straightening $str(\sigma)$ is the geodesic
simplex with the same vertices as $\sigma$ and parametrized using the
barycentric coordinates (it is possible that $str(\sigma)$ is not contained in
$X$). We define

$v^{\prime}_{M}(\sigma)=\int_{str(\sigma)}vol_{h}.$

We have that $v^{\prime}_{M}$ is a $\pi_{1}(M)$-invariant cocycle on $X$. It
defines the bounded class $[v^{\prime}_{M}]\in\operatorname{H}_{b}^{n}(M)$
which we denote by $Vol_{M}$.

Hence we associate two classes to $M$: $Vol_{M}$ in the cohomology of the
space $M$ and a class $Vol_{M}^{gp}$ in the group cohomology of $\pi_{1}(M)$.

######  Remark 2.1.

The definition of $Vol_{M}$ can be generalized by integrating, instead of
$vol_{h}$, the pull-back $\tilde{\omega}$ of a closed $k$-form $\omega$ on
$M^{n}$ for $k\leq n$, see [BG88, BFM+24]. For a surface and $2$-forms such
that $\tilde{\omega}=fvol_{h}$ for $f>0$, Theorem 1.1 holds with the same
proof. As well, instead of a hyperbolic metric, one can take a pinched
negatively curved metric on $M$ where one can straighten simplices. This
should generate even more $Vol_{M}$-like classes.

######  Lemma 2.2.

Let $M$ be a connected, oriented and aspherical hyperbolic $n$-dimensional
manifold and let
$r\colon\operatorname{H}_{b}^{n}(\pi_{1}(M))\to\operatorname{H}_{b}^{n}(M)$ be
the canonical isometric isomorphism. Then we have $r(Vol_{M}^{gp})=Vol_{M}$.

###### Proof.

Let $M=X/\pi_{1}(M)$ where $X\subset\mathbb{H}^{n}$. First, we shall show that
without loss of generality, we can assume that $X=\mathbb{H}^{n}$. Consider
$M^{ex}=\mathbb{H}^{n}/\pi_{1}(M)$. Since $X~{}\subset~{}\mathbb{H}^{n}$, we
have that $M$ is a submanifold of $M^{ex}$. Moreover, $M$ is aspherical and,
by the Whitehead theorem, the inclusion $i\colon M\to M^{ex}$ is a homotopy
equivalence. Hence

$i^{*}\colon\operatorname{H}_{b}^{n}(M^{ex})\to\operatorname{H}_{b}^{n}(M)$

is an isometric isomorphism. Moreover, $i^{*}(Vol_{M^{ex}})=Vol_{M}$. Thus
instead of $M$, we can consider $M^{ex}$. In other words, we can assume that
$X=\mathbb{H}^{n}$.

The explicit formula for $r$ can be found in [Fri17, Lemma 5.2] (note that on
the standard cohomology, $r$ is just the map induced by the classifying map).
We shall give a formula for the inverse of $r$.

Let $G=\pi_{1}(M)$ and $*\in\mathbb{H}^{n}$ be a basepoint. Let $\Delta$ be
the map which to each tuple $\bar{g}=(g_{0},\ldots,g_{n})\in G^{n+1}$
associates $\Delta_{\bar{g}}$, the geodesic simplex spanned by the $g_{i}(*)$
and parametrized by barycentric coordinates.

Consider the augmented cochain complexes
$\\{\operatorname{C}^{\bullet}_{b}(G),d\\}$ and
$\\{\operatorname{C}_{b}^{\bullet}(\mathbb{H}^{n})),d\\}$. It means that
$\operatorname{C}^{-1}_{b}(G)=\operatorname{C}_{b}^{-1}(\mathbb{H})=\mathbb{R}$
and in both cases $d_{-1}$ maps a real number to a constant function. The map
$\Delta$ commutes with taking facets. That is, if $\bar{f}\subset\bar{g}$ is
an $n$-tuple in an $(n+1)$-tuple $\bar{g}$, then $\Delta_{\bar{f}}$ is just
$\Delta_{\bar{g}}$ restricted to the corresponding facet. Thus $\Delta$
induces a map of augmented cochain complexes

$\Delta^{*}\colon\\{\operatorname{C}_{b}^{\bullet}(\mathbb{H}^{n})),d\\}\to\\{\operatorname{C}^{\bullet}_{b}(G),d\\}$

given by $\Delta^{*}(c)(\bar{g})=c(\Delta_{\bar{g}})$ and the identity on the
augmentations. The map $\Delta^{*}$ is $G$-invariant and since these
resolutions are relatively injective strong resolutions [Fri17, Lemma 4.12 and
Lemma 5.4], $\Delta^{*}$ induces a map

$\operatorname{H}^{n}_{b}(\Delta)\colon\operatorname{H}_{b}^{n}(M)\to\operatorname{H}_{b}^{n}(\pi_{1}(M))$

which is the inverse of $r$ [Fri17, Theorem 4.15]. Since $r$ is an isometric
isomorphism, the same holds for $\operatorname{H}^{n}_{b}(\Delta)$. Moreover,
it follows directly from the definitions that
$\Delta^{*}(v^{\prime}_{M})=v_{G}$. Thus $r(Vol_{M}^{gp})~{}=~{}Vol_{M}$. ∎

Let $\omega_{h}$ be the hyperbolic volume form on a closed hyperbolic manifold
$M$. Let us point out that $[v^{\prime}_{M}]\in\operatorname{H}_{b}^{n}(M)$
goes to $[\omega_{h}]\in\operatorname{H}_{dR}(M)\simeq H^{n}(M)$ after
applying the comparison map. This can be seen by the straightening of
simplices homotopy, see [Thu22a] or [Fri17, Lemma 8.12].

###  2.C. Euler class in the mapping class group

Let $\operatorname{Homeo}_{+}^{\mathbb{Z}}(\mathbb{R})$ be the set of
orientation preserving homeomorphisms $f$ of $\mathbb{R}$ that are lifts of
maps from $\operatorname{Homeo}_{+}(S^{1})$. That is, $f(x+1)=f(x)+1$. It fits
the central extension

$\mathbb{Z}\xrightarrow{}\operatorname{Homeo}_{+}^{\mathbb{Z}}(\mathbb{R})\xrightarrow{p}\operatorname{Homeo}_{+}(S^{1}).$

The Euler class
$e_{b}\in\operatorname{H}_{b}^{2}(\operatorname{Homeo}_{+}(S^{1}))$ is a
particular bounded class representing the Euler class of this extension
[Fri17, Chapter 10]. Let $S$ be a closed oriented surface of genus $\geq 2$.
On $S$ we fix a hyperbolic metric. Let $\mathcal{M}_{+}(S,*)$ be the subgroup
of $\mathcal{M}(S,*)$ of mapping classes represented by orientation preserving
homeomorphisms. By the Dehn-Nielsen theorem
$\mathcal{M}_{+}(S,*)\simeq\operatorname{Aut}_{+}(\pi_{1}(S,*))$, thus
$\mathcal{M}_{+}(S,*)$ acts on the Gromov boundary $\partial\pi_{1}(S,*)\simeq
S^{1}$. Hence we have a map

$\alpha\colon\mathcal{M}_{+}(S,*)\to\operatorname{Homeo}_{+}(S^{1}).$

In geometric terms, $\alpha$ is described as follows. Let
$f\in\operatorname{Homeo}_{+}(S,*)$ represent an element
$\psi\in\mathcal{M}_{+}(S,*)$. Let $\tilde{*}\in\mathbb{H}^{2}$ be a fixed
preimage of $*$ and let $\tilde{f}$ be the lift of $f$ such that
$\tilde{f}(\tilde{*})=\tilde{*}$. Now $\alpha(\psi)$ is the action of
$\tilde{f}$ on $\partial\mathbb{H}^{2}$ and it does not depend on the choice
of $f$ representing $\psi$. Note that $\alpha$ restricted to
$\pi_{1}(S,*)<\mathcal{M}_{+}(S,*)$ is just the standard action on
$\partial\mathbb{H}^{2}$ by deck transformations.

Let $e_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}=\alpha^{*}(e_{b})$ and
$e_{b}^{S}$ be the restriction of
$e_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}$ to
$\pi_{1}(S,*)<\mathcal{M}_{+}(S,*)$. The class
$e_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}$ was studied in [Che20, JR21].

##  3\. Definitions of $\Gamma_{b}$ and
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$

Suppose $M$ is a connected smooth manifold and $\omega$ is a finite measure
induced by a volume form. In this section, we define the maps $\Gamma_{b}$ and
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ (the latter for compact $M$). We
start with a geometrically motivated definition of $\Gamma_{b}$ by a system of
paths. This definition naturally leads to the definition of
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ by a system of homeomorphisms. To
show that the definitions of $\Gamma_{b}$ and
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ do not depend on the chosen
systems, we use a result from [Nit]. To this end, we rephrase our definitions
in the language of couplings. In the case of $\Gamma_{b}$ the obvious coupling
is given by the universal cover. For the convenience of the reader, we give
the details in Section 3.B. It turns out that
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ can be described in the language
of couplings as well. However, one needs to use a bigger (and therefore
disconnected) cover of $M$, see Section 3.D.

Fix a basepoint $*\in M$. In this section, we assume that $\pi_{1}(M,*)$ is
center free. It holds for all manifolds we are interested in and with this
assumption the construction of $\Gamma_{b}$ and
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ is slightly simpler.

Let $p\colon\widetilde{M}\to M$ be the universal cover of $M$. We view an
element $l\in p^{-1}(x)$ as a homotopy class relative to $\\{*,x\\}$ of a path
connecting $*$ to $x$. The action of $\pi_{1}(M,*)$ on $\widetilde{M}$ is
given by concatenating a loop representing $\gamma\in\pi_{1}(M,*)$ and a path
representing $l\in\widetilde{M}$.

Recall that $\operatorname{Homeo}(M)$ is the group of all homeomorphisms of
$M$ and $\operatorname{Homeo}_{0}(M)$ are those elements of
$\operatorname{Homeo}(M)$ that are isotopic to the identity of $M$. Similarly,
$\operatorname{Homeo}(M,\omega)$ is the group of all homeomorphims of $M$
preserving $\omega$ and $\operatorname{Homeo}_{0}(M,\omega)$ are those
elements of $\operatorname{Homeo}(M,\omega)$ that are isotopic to the identity
of $M$ via $\omega$-preserving maps.

In all these transformation groups, we consider their right group action on
$M$, so that $fg$ acts by $(fg)(x)=g(f(x))$ for all $x\in M$. We are forced to
use this convention, since later in Section 3.C we relate elements of
$\pi_{1}(M,*)$ to homeomorphisms in $\operatorname{Homeo}(M)$ using a push
map, and we want the multiplication in $\pi_{1}(M,*)$ to be compatible with
how we compose homeomorphisms.

Recall that a left action of a group $G$ on a set $X$ is a homomorphism
$G\to\operatorname{Aut}(X)$ and a right action is an antihomomorphism
$G\to\operatorname{Aut}(X)$. On $\operatorname{Aut}(X)$ we compose maps
starting from the right-most. In this way we have a left action of
$\pi_{1}(M,*)$ on $\widetilde{M}$ and a right action of
$\operatorname{Homeo}(M)$ on $M$.

###  3.A. Description of $\Gamma_{b}$

In this paragraph, we give the simplest, in our opinion, description of
$\Gamma_{b}$. A more refined and general definition can be found in [BM22].
Another approach based on coupling of groups is given in [Nit]. Note that the
definition in [Nit] is much more general and works for measurable spaces and
groups of measurable transformations.

Let $s_{x}$ be a path connecting $*$ to $x\in M$. By $[s_{x}]$ we denote the
homotopy class of $s_{x}$ relative to the endpoints $\\{*,x\\}$ and by
$\bar{s}_{x}$ we denote the reverse of $s_{x}$. A system of paths for $M$ is a
function on $M$ of the form $S(x)=[s_{x}]$ where $s_{x}$ is any path
connecting $*$ to $x$.

Assume $S$ is a system of paths. We shall define a map

$\gamma\colon\operatorname{Homeo}_{0}(M,\omega)\times M\to\pi_{1}(M,*).$

Let $S(x)$ be represented by a path $s_{x}$. Fix
$f\in\operatorname{Homeo}_{0}(M,\omega)$ and $f_{t}$ an isotopy connecting
$Id_{M}$ to $f$. We define $\gamma(f,x)\in\pi_{1}(M,*)$ to be the homotopy
class of the loop based at $*$ which is the concatenation of $s_{x}$, the
trajectory $f_{t}(x)$ and $\bar{s}_{f(x)}$. The element $\gamma(f,x)$ does not
depend on the choice of paths representing $S(x)$ and $S(f(x))$. Moreover,
$\gamma(f,x)$ does not depend on the isotopy $f_{t}$. Indeed, it follows from
[BM22, Proposition 3.1] that changing the isotopy $f_{t}$ to another isotopy
connecting $Id_{M}$ and $f$ would change $\gamma(f,x)$ by an element of the
center of $\pi_{1}(M,*)$. Since we assumed that the center of $\pi_{1}(M,*)$
is trivial, $\gamma(f,x)$ is well defined. Note that $\gamma$ depends on the
choice of the system of paths $S$.

It is a simple calculation, that $\gamma$ satisfies a cocycle condition in the
following form:

$\gamma(fg,x)=\gamma(f,x)\gamma(g,f(x)).$

Define $\widetilde{\omega}$ to be the measure on $\widetilde{M}$ induced by
the pull-back of the volume form on $M$ that defines $\omega$. Since elements
of $\widetilde{M}$ are homotopy classes relative to the endpoints of paths in
$M$ starting at $*$, the image $im(S)$ is a subset of $\widetilde{M}$. We say
that $S$ is measurable if $im(S)$ is $\widetilde{\omega}$-measurable. In
Section 3.B we show that measurable systems of paths exist.

Now assume $S$ is a measurable system of paths. For each $n$ we define

$\Gamma_{b}\colon\operatorname{H}^{n}_{b}(\pi_{1}(M,*))\to\operatorname{H}^{n}_{b}(\operatorname{Homeo}_{0}(M,\omega))$

to be the map induced by the following function defined on cochains (called
again $\Gamma_{b}$):

$\Gamma_{b}(c)(f_{0},\ldots,f_{n})=\int_{M}c\big{(}\gamma(f_{0},x),\ldots,\gamma(f_{n},x)\big{)}d\omega(x).$

In Section 3.B we show that the function under the integral is measurable and
that $\Gamma_{b}$ does not depend on the choice of a measurable system of
paths.

######  Remark 3.1.

In [BM22] the group $\operatorname{Homeo}_{0}(M,\omega)$ is defined like in
this paper with an additional assumption that homeomorphisms have compact
support. This additional assumption was needed to define an analog of
$\Gamma_{b}$ for the ordinary cohomology in [BM22, Section 3.4]. In this paper
we deal only with the bounded cohomology, therefore we do not assume that
elements of $\operatorname{Homeo}_{0}(M,\omega)$ are compactly supported.
Moreover, contrary to [BM22], here we do not assume that $M$ carries a
complete Riemannian metric.

######  Example 3.2.

Now we explicitly describe a cocycle representing the element
$\Gamma_{b}(Vol_{M}^{gp})$. Let $M=X/\pi_{1}(M)$ where
$X\subset\mathbb{H}^{n}$ and let $\omega$ be any finite measure induced by a
volume form on $M$. Assume that $\pi_{1}(M)$ has trivial center (equivalently,
$\pi_{1}(M)$ is not abelian, for example $M$ is not a quotient of
$\mathbb{H}^{n}$ by a $\mathbb{Z}$-action). Let $\tilde{*}\in X$ be a lift of
the basepoint $*$ and
$f_{0},\ldots,f_{n}\in\operatorname{Homeo}_{0}(M,\omega)$. For every $x\in M$
we get a tuple of elements $\gamma(f_{0},x),\ldots,\gamma(f_{n},x)$ in
$\pi_{1}(M)$ that act on $X$. Consider the points
$v_{i}(x)=\gamma(f_{i},x)\tilde{*}$ and span a geodesic simplex $\Delta(x)$ in
$X$ with vertices $v_{i}(x)$. Now $\Gamma_{b}(Vol_{M}^{gp})$ is represented by
a cocycle that assigns to $(f_{0},\ldots,f_{n})$ the average signed volume of
$\Delta(x)$ over $M$ with respect to $\omega$.

###  3.B. $\Gamma_{b}$ via coupling

Let us describe the construction of $\Gamma_{b}$ via couplings given in [Nit].
A coupling is a measured space $(X,\mu)$ together with a $\mu$-preserving left
action of a group $\Gamma$ and a commuting $\mu$-preserving right action of a
group $\Lambda$. Suppose that $\Gamma$ is countable, the action of $\Gamma$ is
free and $F\subset X$ is a strict measurable fundamental domain for an action
of $\Gamma$. A map $\chi\colon X\to\Gamma$ is defined by the equation
$\chi(\gamma.x)=\gamma$ for $x\in F$ and $\gamma\in\Gamma$. Note that $\chi$
is measurable, i.e., the preimage of every element in $\Gamma$ is a measurable
set. A function $\bar{\chi}\colon\Lambda\times F\to\Gamma$ given by
$\bar{\chi}(\lambda,x)=\chi(x.\lambda)$ induces a homomorphism
$tr_{\Gamma}^{\Lambda}X\colon\operatorname{H}_{b}^{\bullet}(\Gamma)\to\operatorname{H}_{b}^{\bullet}(\Lambda)$
by the formula [Nit, Section 3]

$tr_{\Gamma}^{\Lambda}(c)(\lambda_{0},\ldots,\lambda_{n})=\int_{F}c\big{(}\bar{\chi}(\lambda_{0},x),\ldots,\bar{\chi}(\lambda_{n},x)\big{)}d\mu(x).$

The map $\bar{\chi}$ is a composition of measurable functions, hence the
function under the integral is measurable. Moreover, $tr_{\Lambda}^{\Gamma}X$
does not depend on the choice of $F$ [Nit, Lemma 3.3].

Let $f\in\operatorname{Homeo}_{0}(M,\omega)$ and suppose $f_{t}$ is an isotopy
connecting the identity of $M$ to $f$. Denote by $\widetilde{f}$ the endpoint
of a lift of $f_{t}$ to $\widetilde{M}$. Note that, again by [BM22,
Proposition 3.1], and by the assumption that $\pi_{1}(M,*)$ has a trivial
center, $\widetilde{f}$ does not depend on the chosen isotopy $f_{t}$.
Therefore we have a right $\widetilde{\omega}$-preserving action of
$\operatorname{Homeo}_{0}(M,\omega)$ on $\widetilde{M}$, given by
$x.f=\widetilde{f}(x)$, $x\in\widetilde{M}$. Note that $x.f$ is represented by
a path connecting $*$ to $x$, and then following the trajectory from $x$ to
$f(x)$ given by $f_{t}(x)$.

Now we consider the following coupling: $X=(\widetilde{M},\widetilde{\omega})$
together with the left action of $\Gamma=\pi_{1}(M,*)$ and the right action of
$\Lambda=\operatorname{Homeo}_{0}(M,\omega)$. For this coupling, we have
$\Gamma_{b}=tr_{\Gamma}^{\Lambda}X$. Indeed, if $S$ is a measurable system of
paths, $im(S)$ is a strict measurable fundamental domain of the
$\pi_{1}(M,*)$-action, and vice versa, every measurable fundamental domain
defines a measurable system of paths. Moreover, let $S$ be a measurable system
of paths and $F=im(S)$. Recall that $p\colon\widetilde{M}\to M$ is the
universal covering map. If $\gamma$ and $\bar{\chi}$ are defined by $S$ and
$F$ respectively, then $\gamma(f,p(x))=\bar{\chi}(f,x)$ for every $x\in F$ and
$f\in\operatorname{Homeo}_{0}(M,\omega)$. Thus $\Gamma_{b}$ and
$tr_{\Gamma}^{\Lambda}X$ coincide. In particular, $\Gamma_{b}$ does not depend
on the choice of a measurable system of paths.

To finish the construction of $\Gamma_{b}$, we shall show that measurable
fundamental domains exist. Suppose $\\{U_{i}\\}_{i\in\mathbb{N}}$ is a cover
of $M$ by simply connected open sets. Let $\widetilde{U}_{i}$ be a
homeomorphic lift of $U_{i}$ to $\widetilde{M}$. Set
$F_{i}=\widetilde{U}_{i}\backslash p^{-1}(\cup_{j=1}^{i-1}U_{j})$. Then
$F=\cup_{j=1}^{\infty}F_{j}$ is a strict measurable fundamental domain.

###  3.C. Description of $\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$.

We describe an extension of $\Gamma_{b}$ to a map that ranges in the bounded
cohomology of $\operatorname{Homeo}(M,\omega)$. It is done by modifying the
notion of a system of paths. In this and the next subsection, we assume that
$M$ is compact.

Fix a point $*\in M$. Let $\operatorname{Homeo}(M,*)$ be the group of
homeomorphisms of $M$ fixing $*$ and $\operatorname{Homeo}_{0}(M,*)$ the
subgroup of homeomorphisms that are isotopic to the identity by $*$-fixing
homeomorphisms. It follows from [EK71, Corollary 1.1], that for compact $M$,
$\operatorname{Homeo}_{0}(M,*)$ is the connected component of the identity in
$\operatorname{Homeo}(M,*)$ equipped with the compact-open topology. Recall,
that in these groups we compose homeomorphisms from left to right. Consider
the mapping class group

$\mathcal{M}(M,*)=\operatorname{Homeo}(M,*)/\operatorname{Homeo}_{0}(M,*).$

Let $\operatorname{Homeo}(M,*\to x)<\operatorname{Homeo}(M)$ be the subset of
homeomorphisms of $M$ that send $*$ to $x$. Suppose
$h_{x}\in\operatorname{Homeo}(M,*\to x)$. Denote by $[h_{x}]$ the connected
component of $h_{x}$ in $\operatorname{Homeo}(M,*\to x)$ with the compact-open
topology. Note that two elements $h_{x}$ and $g_{x}$ in
$\operatorname{Homeo}(M,*\to x)$ are in the same connected component if and
only if there exists an isotopy $f_{t}$ connecting $h_{x}$ to $g_{x}$ such
that $f_{t}(*)=x$ for all $t$. A system of homeomorphisms for $M$ is a
function on $M$ of the form $P(x)=[h_{x}]$, where
$h_{x}\in\operatorname{Homeo}(M,*\to x)$. For example, it can be a point-
pushing map along $s_{x}$, where $S(x)=[s_{x}]$ is a system of paths.

Let $P(x)=[h_{x}]$ be a system of homeomorphisms. For every
$f\in\operatorname{Homeo}(M,\omega)$, we have $h_{x}\circ f\circ
h_{f(x)}^{-1}\in\operatorname{Homeo}(M,*)$. We define a cocycle

$\gamma^{\scriptscriptstyle\mathcal{M}}\colon\operatorname{Homeo}(M,\omega)\times
M\to\mathcal{M}(M,*)$

by $\gamma^{\scriptscriptstyle\mathcal{M}}(f,x)=[h_{x}\circ f\circ
h_{f(x)}^{-1}]$. Note that $\gamma^{\scriptscriptstyle\mathcal{M}}$ does not
depend on the homeomorphisms $h_{x}$ representing $P(x)$ but depends on the
choice of $P$. There is a notion of a measurable system of homeomorphisms, we
discuss it in Section 3.D.

Likewise in Subection 3.A, any measurable system of homeomorphisms $P$ induces
a map

$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}\colon\operatorname{H}_{b}^{n}(\mathcal{M}(M,*))\to\operatorname{H}_{b}^{n}(\operatorname{Homeo}(M,\omega)).$

In Section 3.D we show that $\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ does
not depend on the choice of $P$. Note that
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ generalizes the homomorphism
$\mathcal{G}_{S,1}$ from [BM19] defined only for surfaces $S$ and ranging in
quasimorphisms on $\operatorname{Homeo}(S,\omega)$.

######  Remark 3.3.

If $M$ is oriented, we can consider the oriented version of
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$. Namely, let
$\mathcal{M}_{+}(M,*)$ be the subgroup of $\mathcal{M}(M,*)$ of mapping
classes represented by orientation preserving homeomorphisms and let
$\operatorname{Homeo}_{+}(M,\omega)$ be homeomorphisms preserving $\omega$ and
the orientation. Assuming that for every $x\in M$, $P(x)$ is represented by an
orientation preserving homeomorphism, we have
$\gamma^{\scriptscriptstyle\mathcal{M}}(f,x)\in\mathcal{M}_{+}(M,*)$ for
$f\in\operatorname{Homeo}_{+}(M,\omega)$. Thus we can define

$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}_{+}}\colon\operatorname{H}_{b}^{n}(\mathcal{M}_{+}(M,*))\to\operatorname{H}_{b}^{n}(\operatorname{Homeo}_{+}(M,\omega)).$

This map is used in Theorem 1.2.

Let us now explain in what sense $\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$
extends $\Gamma_{b}$. Note that
$\mathcal{M}(M,*)=\pi_{0}(\operatorname{Homeo}(M,*))$ and
$\mathcal{M}(M)=\pi_{0}(\operatorname{Homeo}(M))$. Consider the fiber bundle

$\operatorname{Homeo}(M,*)\xrightarrow{}\operatorname{Homeo}(M)\xrightarrow{ev}M,$

where $ev(f)=f(*)$. The long exact sequence of the homotopy groups gives

$\pi_{1}(\operatorname{Homeo}(M))\xrightarrow{ev_{1}}\pi_{1}(M,*)\xrightarrow{Pu}\mathcal{M}(M,*)\xrightarrow{F}\mathcal{M}(M).$

For $g\in\pi_{1}(M,*)$, $Pu(g)$ is the mapping class represented by the time-
one map $f_{1}$ of a point-pushing isotopy $\\{f_{t}\\}_{t\in[0,1]}$ along a
loop representing $g$. By [BM22, Proposition 3.1], $im(ev_{1})$ lies in the
center of $\pi_{1}(M,*)$, therefore $ev_{1}$ is trivial and $Pu$ is an
embedding.

Let $[f]\in im(Pu)=ker(F)<\mathcal{M}(M,*)$ and let $f_{t}$ be an isotopy
connecting the identity to $f$ in $\operatorname{Homeo}_{0}(M)$. Let $Tr([f])$
be the element of $\pi_{1}(M,*)$ represented by $f_{t}(*)$. By the triviality
of $ev_{1}$, this map is well defined, and it is the inverse of $Pu$ on
$im(Pu)$.

Let $S$ be any measurable system of paths and $P$ a system of homeomorphims
such that $P(x)=[h_{x}]$ with $h_{x}$ the point-pushing map along a path
$s_{x}$ such that $S(x)=[s_{x}]$. The cocycles $\gamma$ and
$\gamma^{\scriptscriptstyle\mathcal{M}}$ are defined with respect to these
systems. For every $f\in\operatorname{Homeo}_{0}(M,\omega)$ we have
$\gamma(f,x)=Tr(\gamma^{\scriptscriptstyle\mathcal{M}}(f,x))$. Thus
$Pu(\gamma(f,x))=\gamma^{\scriptscriptstyle\mathcal{M}}(f,x)$ and we have a
commutative diagram:

${\operatorname{Homeo}(M,\omega)\times
M}$${\mathcal{M}(M,*)}$${\operatorname{Homeo}_{0}(M,\omega)\times
M}$${\pi_{1}(M,*).}$$\scriptstyle{\gamma^{\scriptscriptstyle\mathcal{M}}}$$\scriptstyle{\gamma}$$\scriptstyle{Pu}$

Thus the following diagram commute:

${\operatorname{H}_{b}^{\bullet}(\mathcal{M}(M,*))}$${\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}(M,\omega))}$${\operatorname{H}_{b}^{\bullet}(\pi_{1}(M,*))}$${\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{0}(M,\omega)).}$$\scriptstyle{Pu^{*}}$$\scriptstyle{\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}}$$\scriptstyle{\Gamma_{b}}$

###  3.D. $\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ via couplings

We shall construct a cover of $M$ on which $\mathcal{M}(M,*)$ acts on the left
and $\operatorname{Homeo}(M,\omega)$ acts on the right.

Recall that $ev\colon\operatorname{Homeo}(M)\to M$ is a fibre bundle defined
by $ev(f)=f(*)$. Note that $\operatorname{Homeo}(M,*\to x)=ev^{-1}(x)$. Denote
by $\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ the set of connected
components of the fibers of $ev$, and by
$q\colon\operatorname{Homeo}(M)\to\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$
the quotient map. On $\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ we
consider the quotient topology. Note that we have $q(f)=q(g)$ if and only if
$f$ can be connected to $g$ via an isotopy preserving $f(*)=g(*)$ at all
times.

The map $ev$ factors via $q$:

${\operatorname{Homeo}(M)}$${M}$${\widetilde{M}^{\scriptscriptstyle\mathcal{M}}}$$\scriptstyle{q}$$\scriptstyle{ev}$$\scriptstyle{p^{\scriptscriptstyle\mathcal{M}}}$

where $p^{\scriptscriptstyle\mathcal{M}}$ is the unique map making the above
diagram commutative.

The evaluation map is a fiber bundle, thus every $x\in M$ has an open
neighborhood $U$ such that $ev^{-1}(U)$ is homeomorphic to
$U\times\operatorname{Homeo}(M,*)$ by a fiber preserving homeomorphism.
Moreover, by the definition of $q$, we have that $q(ev^{-1}(U))$ is the
disjoint union of copies of $U$. On each such a copy
$p^{\scriptscriptstyle\mathcal{M}}$ is a homeomorphism. Thus
$p^{\scriptscriptstyle\mathcal{M}}$ is a covering map.

Denote by $\widetilde{\omega}^{\scriptscriptstyle\mathcal{M}}$ the measure on
$\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ defined by the pull-back of
the volume form on $M$ that defines $\omega$. We shall define the mentioned
left and right actions.

Let $[c]\in\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ be a connected
component of $ev^{-1}(c(*))$ represented by $c\in\operatorname{Homeo}(M)$, and
$[h]\in\mathcal{M}(M,*)$ be a mapping class represented by
$h\in\operatorname{Homeo}(M,*)$. The left action of $\mathcal{M}(M,*)$ on
$\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ is given by
$[h].[c]=[hc]\in\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ (note that we
compose homeomorphisms starting from the left), where $[hc]$ denotes the
connected component of $ev^{-1}(c(*))$ containing $hc$. This action is
transitive on the fibers of $p^{\scriptscriptstyle\mathcal{M}}$ and permutes
the components of $q(ev^{-1}(U))$, where $U$ is as above. Therefore it is a
deck-transformation group of $\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$
and preserves $\widetilde{\omega}^{\scriptscriptstyle\mathcal{M}}$.

Let $[c]\in\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ and
$f\in\operatorname{Homeo}(M,\omega)$. The right action of
$\operatorname{Homeo}(M,\omega)$ on
$\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ is given by $[c].f=[cf]$,
where $[cf]$ is the connected component of $ev^{-1}(f(c(*))$ containing $cf$.
This action covers the $\operatorname{Homeo}(M,\omega)$-action on $M$ and thus
is $\widetilde{\omega}^{\scriptscriptstyle\mathcal{M}}$-preserving.

The above actions define a coupling on
$\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$. Moreover, by construction, if
$P$ is a system of homeomorphisms, $im(P)$ is a subset of
$\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$. We say that a system of
homeomorphisms $P$ is measurable if $im(P)$ is
$\widetilde{\omega}^{\scriptscriptstyle\mathcal{M}}$-measurable.

Every measurable system of homeomorphisms defines a strict fundamental domain
for the $\mathcal{M}(M,*)$-action and vice versa. It follows directly from
definitions (see Section 3.B) that the constructions described in Section 3.C
and [Nit] coincide. In particular,
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ does not depend on the choice of
a measurable system of paths [Nit, Lemma 3.3].

Note that $\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ is disconnected and
contains the universal cover of $M$. Indeed, let $\widetilde{M}_{0}$ be the
subset of $\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ containing classes
that are represented by homeomorphisms isotopic to the identity in
$\operatorname{Homeo}(M)$. The subgroup $\pi_{1}(M,*)<\mathcal{M}(M,*)$ acts
on $\widetilde{M}_{0}$ and the quotient is $M$, thus $\widetilde{M}_{0}$ is
the universal cover of $M$. An explicit isomorphism between
$\widetilde{M}_{0}$ and $\widetilde{M}$ is given by the map
$T\colon\widetilde{M}_{0}\to\widetilde{M}$ where $T([c])$ is the homotopy
class of the path $c_{t}(*)$ traced by any isotopy $c_{t}$ connecting $Id_{M}$
to $c$. By the triviality of
$ev_{1}\colon\pi_{1}(\operatorname{Homeo}(M))\to\pi_{1}(M,*)$, this map is
well defined. Therefore, $\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$
consists of infinitely many copies of $\widetilde{M}$ indexed by the right
cosets of $\pi_{1}(M,*)$ in $\mathcal{M}(M,*)$.

Finally, we show that measurable systems of homeomorphisms exist. It follows
from the existence of measurable systems of paths. Let $S(x)=[s_{x}]$ be a
measurable system of paths and let $P(x)=[h_{x}]$, where $h_{x}$ is a point-
pushing map along $s_{x}$. Using the isomorphism
$T\colon\widetilde{M}_{0}\to\widetilde{M}$ we can regard $im(S)$ as a
$\widetilde{\omega}^{\scriptscriptstyle\mathcal{M}}$-measurable subset of
$\widetilde{M}_{0}$, and under this identification $im(S)=im(P)$. Thus $P$ is
a measurable system of homeomorphisms.

######  Remark 3.4.

Set $\pi=\pi_{1}(M,*)$. The cover
$\widetilde{M}^{\scriptscriptstyle\mathcal{M}}$ is isomorphic to
$(\mathcal{M}(M,*)\times\widetilde{M})/_{\pi}$, where $\pi$ acts on
$\mathcal{M}(M,*)\times\widetilde{M}$ by
$\gamma.(h,x)=(h\gamma^{-1},\gamma.x)$.

##  4\. Restriction to a free subgroup

In this section, we find a free subgroup $F$ of $\pi_{1}(M,*)$ such that
volume and Euler classes restricted to $F$ have positive norms.

###  4.A. Volume class in dimension 2

Let $X$ be a topological space. The $l^{1}$-homology of $X$ is denoted by
$\operatorname{H}^{l_{1}}_{n}(X)$ [Fri17, Chapter 6]. In $l^{1}$-homology we
allow chains to be infinite sums $c=\Sigma_{i}^{\infty}a_{i}\sigma_{i}$, where
$||c||=\Sigma_{i}^{\infty}|a_{i}|<\infty$. As usual, the norm on chains
induces the norm on $\operatorname{H}^{l_{1}}_{n}(X)$. We have a Kronecker
product between $l^{1}$-homology and bounded cohomology:

$\langle\cdot,\cdot\rangle\colon\operatorname{H}_{b}^{n}(X)\times\operatorname{H}_{n}^{l_{1}}(X)\to\mathbf{R}.$

The Kronecker product is defined on the level of chains by $\langle
b,a\rangle=\Sigma_{i}^{\infty}a_{i}b(\sigma_{i})$ where
$a=\Sigma_{i}^{\infty}a_{i}\sigma_{i}$ and $b$ is a bounded cochain. Moreover
we have $|\langle B,A\rangle|\leq||B||||A||$ where
$B\in\operatorname{H}_{b}^{n}(X)$ and $A\in\operatorname{H}_{n}^{l_{1}}(X)$.
The following lemma is a variation of a result obtained in [Mit84]. We do not
assume that $S$ is closed.

######  Lemma 4.1.

Let $S$ be an oriented hyperbolic surface with non-abelian fundamental group.
Then $Vol_{S}$ has a positive norm.

###### Proof.

We shall find $C\in\operatorname{H}_{2}^{l_{1}}(S)$ such that $\langle
Vol_{S},C\rangle\neq 0$. We can assume that $S$ is a quotient of
$\mathbb{H}^{2}$, as in the beginning of the proof of Lemma 2.2. Let
$p\colon\mathbb{H}^{2}\to S$ be the covering map and let $G=\pi_{1}(S)$. Since
$G$ is a surface group or is free, the commutator subgroup $[G,G]$ is non-
abelian and hence contains a hyperbolic element $\gamma\in[G,G]$ [Kat92,
Theorem 2.4.4]. Let $A_{\gamma}\subset\mathbb{H}^{2}$ be the axis of $\gamma$.
The conjugacy class of $\gamma$ is represented by the closed geodesic
$L_{\gamma}=p(A_{\gamma})\subset S$. We regard $L_{\gamma}$ as a map from
$[0,1]$ to $S$. Since $\gamma\in[G,G]$, $\gamma$ is homologically trivial.
Hence there exists a triangulated subsurface $S_{0}\subset S$ such that
$\partial S_{0}=L_{\gamma}$.

The loop $L_{\gamma}$ is the boundary of an $l^{1}$-chain $c_{0}$ whose
simplices are contained in the image of $L_{\gamma}$ [Mit84, Section 3]. Thus
$c=S_{0}-c_{0}$ is an $l^{1}$-cycle. Recall that $Vol_{S}=[v^{\prime}_{S}]$.
The straightening of every simplex in $c_{0}$ is degenerate ($L_{\gamma}$ is a
geodesic), thus we have $\langle v^{\prime}_{S},c_{0}\rangle=0$. We can assume
that the triangulation of $S_{0}$ consists of geodesic simplices, hence
$\langle v^{\prime}_{S},S_{0}\rangle$ equals the hyperbolic volume of $S_{0}$.
Hence if we set $C~{}=~{}[c]$, we obtain $\langle Vol_{S},C\rangle>0$. The
inequality $\langle Vol_{S},C\rangle\leq||Vol_{S}||||C||$ implies that the
norm of $Vol_{S}$ is positive. ∎

The following corollary is stated in the group version terms of the volume
class.

######  Corollary 4.2.

Let $S$ be an oriented hyperbolic surface with non-abelian fundamental group.
There exists an embedding $i\colon F\to\pi_{1}(S)$ of a free non-abelian group
$F$ such that $i^{*}(Vol_{S}^{gp})$ has a positive norm.

###### Proof.

Let $F$ be any free non-abelian subgroup of $S$ (which can be equal to
$\pi_{1}(S)$ if $S$ is not closed). We have $i^{*}(Vol_{S}^{gp})=Vol_{i(F)}$.
By Lemma 2.2 we know that the norm of $Vol_{i(F)}$ is equal to the norm of
$Vol_{S^{\prime}}$ where $S^{\prime}=\mathbb{H}^{2}/i(F)$ and by Lemma 4.1 the
norm of $Vol_{S^{\prime}}$ is positive. ∎

###  4.B. Volume class in dimension 3

For some Kleinian groups, i.e. the discrete subgroups of
$Iso_{+}(\mathbb{H}^{3})$, the volume class was studied in [Som97]. Note that
if $G$ is a torsion-free Kleinian group, then it acts freely and properly
discontinuously on $\mathbb{H}^{3}$. Thus $\mathbb{H}^{3}/G$ is a manifold.

######  Theorem 4.3.

[Som97, Theorem 1] Let $G<Iso_{+}(\mathbb{H}^{3})$ be a torsion-free
topologically tame Kleinian group such that the volume of $M=\mathbb{H}^{3}/G$
is infinite. Then $Vol_{M}$ has a positive norm if and only if $G$ is not
elementary and geometrically infinite.

A Kleinian group $G$ is geometrically finite if $N_{\epsilon}(H(L_{G})/G)$ has
finite volume for some $\epsilon>0$, where $N_{\epsilon}$ is an $\epsilon$
neighborhood and $H(L_{G})$ is the convex closure of the limit set of $G$
[Thu22a, Chapter 8, Definition 8.4.1]. A torsion-free Kleinian group $G$ is
topologically tame if $\mathbb{H}^{3}/G$ is homeomorphic to the interior of a
compact manifold.

Note that every discrete finitely generated non-abelian free subgroup $F$ of
$Iso_{+}(\mathbb{H}^{3})$ is topologically tame [Ago04, CG06], not elementary
and $\mathbb{H}^{3}/F$ has infinite volume (otherwise, by the thick-thin
decomposition, $\mathbb{H}^{3}/F$ would have a cusp and $\mathbb{Z}^{2}$ would
embed in $F$). Let $M$ be an oriented $3$-dimensional hyperbolic manifold.
Then for every free group $F$ in $\pi_{1}(M)<Iso_{+}(\mathbb{H}^{3})$ which is
geometrically infinite, $Vol_{F}$ has positive norm. Below we describe our
main example of such a situation, i.e., manifolds that fiber over the circle
with non-compact fiber.

######  Example 4.4.

Suppose $S$ is a connected oriented surface without boundary and free
fundamental group $F=\pi_{1}(S)$. Let $f\in\operatorname{Diff}^{+}(S)$. The
mapping torus of $f$ is a $3$-dimensional manifold $M_{f}=S\times[0,1]/\sim$
where $(x,0)\sim(f(x),1)$. That is, the boundary components of $S\times[0,1]$
are glued together via $f$. Note that $M_{f}$ fibers over the circle with
fiber $S$, and conversely, every $3$-manifold that fibers over a circle with
fiber $S$ can be constructed in this way. The mapping torus $M_{f}$ is
hyperbolic if and only if $f$ is isotopic to a pseudo-Anosov map [Thu22b]. If
$M_{f}$ is hyperbolic, the hyperbolic structure is unique and we have a unique
class $Vol^{gp}_{M_{f}}\in\operatorname{H}_{b}^{3}(\pi_{1}(M_{f}))$. Now the
inclusion of the fiber $S$ into $M_{f}$ gives an embedding $i\colon
F\to\pi_{1}(M_{f})$ and $i(F)$ is a normal subgroup of $\pi_{1}(M)$. It
follows that the limit set of $i(F)$ is equal to the limit set of $\pi_{1}(M)$
[Thu22a, Chapter 8, Corollary 8.1.3], thus it is the entire sphere at
infinity. Hence $i(F)$ is geometrically infinite. By Theorem 4.3 we have that
$i^{*}(Vol^{gp}_{M_{f}})=Vol_{i(F)}$ has positive norm.

###  4.C. Euler class

We briefly recall basic definitions concerning quasimorphisms [Fri17, Chapter
2]. Let $G$ be a group. A real function $q\colon G\to\mathbb{R}$ is called a
quasimorphism if there exists some $D\in\mathbb{R}$ such that

$|q(ab)-q(a)-q(b)|\leq D$

for any $a,b\in G$. The minimal such $D$ is called the defect of $q$. A
quasimorphism is homogeneous if $q(a^{n})=nq(a)$ for any $a\in G$ and
$n\in\mathbb{Z}$. The non-homogeneous coboundary $dq(a,b)=q(a)-q(ab)+q(b)$ of
$q$ is interpreted as a second bounded cohomology class
$[dq]\in\operatorname{H}_{b}^{2}(G)$. If $q$ is homogeneous and not a
homomorphism, then $[dq]$ is non-trivial and has a positive norm [Fri17,
Corollary 6.7].

######  Lemma 4.5.

Let $S$ be an oriented closed surface of genus $\geq 2$. There exists an
embedding $i\colon F\to\pi_{1}(S)$ of a free non-abelian group $F$ such that
$i^{*}(e_{b}^{S})$ has a positive norm.

###### Proof.

Let $T^{1}S$ be the unit tangent bundle of $S$ and denote by
$q\colon\pi_{1}(T^{1}S)\to\pi_{1}(S)$ the map induced by the projection
$T^{1}S\to S$. We will use the rotation quasimorphism
$Rot\colon\pi_{1}(T^{1}S)\to\mathbb{R}$ defined in [Hub12]. It is a
homogeneous quasimorphism of defect $1$ which trivialises the pull-back
$q^{*}(e_{b}^{S})\in\operatorname{H}_{b}^{2}(\pi_{1}(T^{1}S))$, i.e.,
$q^{*}(e_{b}^{S})=[dRot]$ [Hub12, Theorem 5.9].

Let $a,b\in\pi_{1}(T^{1}S)$ be such that $Rot(ab)\neq Rot(a)+Rot(b)$. Let
$F=\langle q(a),q(b)\rangle<\pi_{1}(S)$. Denote this inclusion by $i\colon
F\to\pi_{1}(S)$ and set $F^{\prime}=q^{-1}(F)$. It follows from the definition
of $a$ and $b$, that $Rot_{|F^{\prime}}$ is a homogeneous quasimorphism that
is not a homomorphism. Thus $q^{*}(e^{S}_{b})_{|F^{\prime}}$ has positive norm
and $i^{*}(e^{S}_{b})$ must have positive norm, since
$q^{*}i^{*}(e^{S}_{b})=q^{*}(e^{S}_{b})_{|F^{\prime}}$. Finally, $F$ must be
free of rank $2$. Indeed, every subgroup of $\pi_{1}(S)$ is free non-abelian,
abelian, or is a surface group. But surface groups are not generated by $2$
elements and abelian groups do not carry a non-trivial class in their second
bounded cohomology. Thus $F$ is free non-abelian of rank $2$. ∎

##  5\. Proof of the theorem

Let $M$ be a manifold with a volume form $\omega$. As usual, $\omega$ denotes
as well the induced measure and we assume that this measure is finite. Suppose
$i\colon F\to\pi_{1}(M)$ is an embedding and consider
$i^{*}\colon\operatorname{H}_{b}^{\bullet}(\pi_{1}(M))\to\operatorname{H}_{b}^{\bullet}(F)$.
Let $\rho\colon F\to\operatorname{Homeo}_{0}(M,\omega)$ be a representation of
$F$ by homeomorphisms. Let
$\rho^{*}\colon\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{0}(M,\omega))\to\operatorname{H}_{b}^{\bullet}(F)$.
Thus we have a not necessarily commutative diagram:

${\operatorname{H}_{b}^{\bullet}(\pi_{1}(M))}$${\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{0}(M,\omega))}$${\operatorname{H}_{b}^{\bullet}(F)}$$\scriptstyle{\Gamma_{b}}$$\scriptstyle{i^{*}}$$\scriptstyle{\rho^{*}}$

Let $\Lambda,\epsilon\in\mathbb{R}$. We say that $\rho$ is an
$(F,\Lambda,\epsilon)$-inverse of $\Gamma_{b}$ if for every
$C\in\operatorname{H}_{b}^{\bullet}(\pi_{1}(M))$ we have

$||\rho^{*}\Gamma_{b}(C)-\Lambda i^{*}(C)||\leq\epsilon||C||.$

######  Lemma 5.1.

Let $M$ be a manifold and $\omega$ a finite measure induced by a volume form
on $M$. Suppose that $i\colon F\to\pi_{1}(M)$ is an embedding of a non-abelian
free group $F$. There exists $\Lambda\in\mathbb{R}$ such that for every
$\epsilon>0$ there exists an $(F,\Lambda,\epsilon)$-inverse of $\Gamma_{b}$.

###### Proof.

Let $\operatorname{dim}(M)=m$. Denote by $B^{m-1}\subset\mathbb{R}^{m-1}$ the
$m-1$ dimensional closed unit ball, and let $S^{1}=\mathbb{R}/\mathbb{Z}$. Let
us fix $\eta\in(0,1)$ and define an isotopy
$P^{t}_{\eta}~{}\in~{}\operatorname{Diff}(S^{1}\times B^{m-1})$ by

$P^{t}_{\eta}(\psi,x)=(\psi+tf(||x||),x)\hskip 11.38092pt\forall(\psi,x)\in
S^{1}\times B^{m-1},$

where $t\in[0,1]$, and $f:[0,1]\to\mathbb{R}$ is a smooth function such that
$f(y)=1$ for $y\leq 1-\eta$ and $f(1)=0$. We call $P^{t}_{\eta}$ the finger-
pushing isotopy and $P^{1}_{\eta}$ the finger-pushing map. Note that
$P^{0}_{\eta}=Id$ and that $P^{1}_{\eta}$ fixes point-wise the boundary of
$S^{1}\times B^{m-1}$ and fixes all points $(\psi,x)$ for which $||x||\leq
1-\eta$. Moreover, $P^{t}_{\eta}$ fixes the boundary of $S^{1}\times B^{m-1}$
for all $t$. Denote by $g_{0}$ be the product of the standard Euclidean
Riemannian metrics on $B^{m-1}$ and $S^{1}$. By the theorem of Fubini, the
measure induced by $g_{0}$ is preserved by the map $P^{t}_{\eta}$ for every
$t\in[0,1]$. Let $a_{1},\ldots,a_{k}$ be generators of $F:=F_{k}$, where
$k>1$. We represent $i(a_{i})$ by a loop $\alpha_{i}$ which is based at $*\in
M$.

Let $B$ be a closed ball in $M$ containing $*$ and set $\Lambda=\omega(B)$.
Suppose $A_{i}$ are closed small tubular neighborhoods of $\alpha_{i}$. Then
$N_{i}=B\cup A_{i}$ is a closed neighborhood of $\alpha_{i}$ which is
diffeomorphic to $S^{1}\times B^{m-1}$. Let
$P^{t}_{\eta}(\alpha_{i})\in\operatorname{Diff}_{0}(M)$ be the isotopy defined
by pulling-back $P^{t}_{\eta}$ via a diffeomorphism $n_{\alpha_{i}}\colon
N_{i}\to S^{1}\times B^{m-1}$ (i.e., we have
$P^{t}_{\eta}(\alpha_{i})=n_{\alpha_{i}}^{-1}\circ P^{t}_{\eta}\circ
n_{\alpha_{i}}$ on $N_{i}$), and extending it by the identity outside $N_{i}$.
Note that the Moser trick [Mos65] allows us to choose $n_{\alpha_{i}}$ such
that $P^{t}_{\eta}(\alpha_{i})$ preserves $\omega$. Let $S_{i}$ be the support
of $P^{1}_{\eta}(\alpha_{i})$. It is a small thickening of the boundary of
$N_{i}$.

The homomorphism $\rho\colon F\to\operatorname{Homeo}_{0}(M,\omega)$ is given
by:

$\rho(a_{i})=P^{1}_{\eta}(\alpha_{i}).$

To simplify the notation, we identify $F$ with its image $i(F)$. Now we
consider the values of $\gamma$ on elements of the form $(\rho(w),x)$, where
$w\in F,x\in M$.

From the description of $\gamma$ in Section 3.A we have:

$\gamma(\rho(w),x)=\begin{cases}e&x\in M-\bigcup_{i=1}^{k}N_{i},\\\ w&x\in
B-\bigcup_{i=1}^{k}S_{i},\\\
?&x\in(\bigcup_{i=1}^{k}A_{i}-B)\cup\bigcup_{i=1}^{k}S_{i}.\end{cases}$

Let $C\in\operatorname{H}_{b}^{n}(M)$ and let $c$ be a bounded cochain
representing $C$. Since any cochain and its anti-symmetrisation define the
same class, we may assume that $c(e,\ldots,e)~{}=~{}0$. Let

$\bar{f}~{}=~{}(f_{0},f_{1},\ldots,f_{n})\in~{}\operatorname{Homeo}_{0}(M,\omega)^{n+1},$

and denote

$\gamma(\bar{f},x)=(\gamma(f_{0},x),\gamma(f_{1},x),\ldots,\gamma(f_{n},x)).$

Let $\overline{w}\in F^{n+1}$. We have:

$\rho^{*}\Gamma_{b}(c)(\overline{w})=\Gamma_{b}(c)(\rho(\overline{w}))=\int_{M}c(\gamma(\rho(\overline{w}),x))d\omega(x).$

Denote $E:=(\bigcup_{i=1}^{k}A_{i}-B)\cup\bigcup_{i=1}^{k}S_{i}$. We obtain

$\displaystyle\rho^{*}\Gamma_{b}(c)(\overline{w})$
$\displaystyle=\int_{B-\bigcup_{i=1}^{k}S_{i}}c(\overline{w})d\omega(x)+\int_{E}c(\gamma(\rho(\overline{w}),x))d\omega(x)$
$\displaystyle=\omega\left(B-\bigcup_{i=1}^{k}S_{i}\right)i^{*}(c)(\overline{w})+\int_{E}c(\gamma(\rho(\overline{w}),x))d\omega(x).$

Let

$c_{res}(\overline{w}):=\int_{E}c(\gamma(\rho(\overline{w}),x))d\omega(x).$

Note that $c_{res}$ represents a class in $\operatorname{H}_{b}^{n}(F)$ and we
can write

$\rho^{*}\Gamma_{b}(c)=\omega(B-\bigcup_{i=1}^{k}S_{i})i^{*}(c)+c_{res},$

and

$||c_{res}||\leq\omega(E)||c||.$

Moreover:

$\displaystyle||\rho^{*}\Gamma_{b}(c)-\omega(B)i^{*}(c)||\leq$
$\displaystyle\leq\omega(\bigcup_{i=1}^{k}S_{i})||i^{*}(c)||+\omega(E)||c||\leq$
$\displaystyle\leq\big{[}\omega(\bigcup_{i=1}^{k}S_{i})+\omega(E)\big{]}||c||\
.$

Now $\omega(S_{i})$ and $\omega(E)$ can be taken to be arbitrarily small by
taking small $\eta$ and small neighborhoods $A_{i}$. The cochain $c$ was any
cochain representing $C$. Thus for any chosen $\epsilon$ we can have:

$||\rho^{*}\Gamma_{b}(C)-\Lambda i^{*}(C)||\leq\epsilon||C||,$

where $\Lambda=\omega(B)$. ∎

######  Theorem 5.2.

Let $M$ be an oriented manifold of dimension $n$ such that it is either:

  * •

A hyperbolic surface with a non-abelian fundamental group or

  * •

A complete hyperbolic $3$-manifold whose fundamental group contains a
geometrically infinite finitely generated free group (e.g. $M$ fibers over the
circle with non-compact fiber).

Let $\omega$ be a volume form on $M$ such that the induced measure is finite.
Then
$\Gamma_{b}(Vol_{M}^{gp})\in\operatorname{H}_{b}^{n}(\operatorname{Homeo}_{0}(M,\omega))$
has positive norm.

###### Proof.

By Corollary 4.2 and Theorem 4.3 in both cases we have a free group $F$ and an
embedding $i\colon F\to\pi_{1}(M)$ such that $i^{*}(Vol_{M}^{gp})$ has
positive norm. Let $\rho\colon F\to\operatorname{Homeo}_{0}(M,\omega_{h})$ be
an $(F,\Lambda,\epsilon)$-inverse of $\Gamma_{b}$ with $\epsilon$ satisfying

$0<\Lambda||i^{*}(Vol_{M}^{gp})||-\epsilon||Vol_{M}^{gp}||.$

We have

$\Lambda||i^{*}(Vol_{M}^{gp})||-||\rho^{*}\Gamma_{b}(Vol_{M}^{gp})||\leq||\Lambda
i^{*}(Vol_{M}^{gp})-\rho^{*}\Gamma_{b}(Vol_{M}^{gp})||\leq\epsilon||Vol_{M}^{gp}||.$

Thus

$0<\Lambda||i^{*}(Vol_{M}^{gp})||-\epsilon||Vol_{M}^{gp}||\leq||\rho^{*}\Gamma_{b}(Vol_{M}^{gp})||.$

Since $\rho^{*}$ is a contraction, $\Gamma_{b}(Vol_{M}^{gp})$ must have
positive norm. ∎

######  Theorem 5.3.

Let $S$ be an oriented closed surface of genus $\geq 2$ and $\omega$ a measure
induced by an area form on $S$. Then the classes
$\Gamma_{b}(e_{b}^{S})\in\operatorname{H}_{b}^{2}(\operatorname{Homeo}_{0}(S,\omega))$
and
$\Gamma_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}(e_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}})\in\operatorname{H}_{b}^{2}(\operatorname{Homeo}(S,\omega))$
have positive norms.

###### Proof.

The proof for $e_{b}^{S}$ is the same as in Theorem 5.2 using Lemma 4.5. Note
that elements of $\operatorname{Homeo}(S,\omega)$ automatically preserve the
orientation of $S$. Positivity of the norm of
$e_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}$ follows from the commutative
diagram

${\operatorname{H}_{b}^{2}(\mathcal{M}_{+}(S,*))}$${\operatorname{H}_{b}^{2}(\operatorname{Homeo}(S,\omega))}$${\operatorname{H}_{b}^{2}(\pi_{1}(S,*))}$${\operatorname{H}_{b}^{2}(\operatorname{Homeo}_{0}(S,\omega)),}$$\scriptstyle{Pu^{*}}$$\scriptstyle{\Gamma_{b}^{{\scriptscriptstyle\mathcal{M}_{+}}}}$$\scriptstyle{\Gamma_{b}}$

where $Pu\colon\pi_{1}(S,*)\to\mathcal{M}_{+}(S,*)$ is the injection from the
Birman exact sequence. ∎

##  6\. Dirac measure

The constructions of $\Gamma_{b}$ and
$\Gamma_{b}^{\scriptscriptstyle\mathcal{M}}$ are flexible and admit more
variants. First of all, one does not need to restrict to measures coming from
a volume form. What is needed, is a measure with a cocycle for which the
integral in the definition is well-defined. Moreover, one can relax the
definition of an isotopy. For example, it is not necessary to assume that
isotopy preserves the measure at all times. Isotopy might be as well
substituted by homotopy.

In this short section, we discuss the (somewhat degenerate) case where the
measure is the Dirac measure and isotopies do not preserve the measure. In
this case, $\Gamma_{b}$ is induced by a homomorphism. Let $M$ be a manifold
and $*\in M$ a basepoint. We assume that the center of $\pi_{1}(M,*)$ is
trivial (what we really need to assume is the triviality of $ev_{1}$, and even
in the case where it is not, one could substitute $\pi_{1}(M,*)$ with the
quotient $\pi_{1}(M,*)/im(ev_{1})$). By $*$ we denote as well the Dirac
measure centered on $*$.

Let $G$ be the subgroup of $\operatorname{Homeo}_{0}(M)$ of all homeomorphisms
$f$ preserving $*$. Thus an element of $G$ is isotopic to the identity by an
isotopy that can move $*$. Suppose $S$ is a system of paths. As in Section
3.A, we get a cocycle:

$\gamma\colon G\times M\to\pi_{1}(M,*)$

and a map

$\Gamma_{b}\colon\operatorname{H}_{b}^{\bullet}(\pi_{1}(M,*))\to\operatorname{H}_{b}^{\bullet}(G).$

Note that on $\widetilde{M}$ we can consider the counting measure on the orbit
$p^{-1}(*)$. With such a measure $\widetilde{M}$ defines a coupling and every
$S$ is measurable. Moreover, $\Gamma_{b}$ does not depend on $S$ [Nit, Lemma
3.3].

Recall that we have a homomorphism

$Tr\colon G\to\pi_{1}(M,*)$

defined in the following way: $Tr(f)$ is the homotopy class of the loop
$f_{t}(*)$, where $f_{t}$ is any isotopy between $Id_{M}$ and $f$.

It is straightforward to see that $\Gamma_{b}=Tr^{*}$, the map induced on
bounded cohomology by $Tr$. Note that if we started with the group
$\operatorname{Homeo}_{0}(M,*)$ (isotopies preserve $*$ at all times), instead
of $G$, then
$\Gamma_{b}\colon\operatorname{H}_{b}^{\bullet}(\pi_{1}(M,*))\to\operatorname{H}_{b}^{\bullet}(\operatorname{Homeo}_{0}(M,*))$
would be trivial in positive degrees. Indeed, in this case $\Gamma_{b}(c)$ is
a constant cocycle, and constant cocycles in positive degrees represent
trivial classes.

Suppose that a non-abelian free group $F$ embeds in $\pi_{1}(M,*)$. The
representations $\rho$ constructed in Lemma 5.1 are homomorphisms. Since
$\gamma(\rho(w),*)=w$ for every $w\in F$, the following diagram commutes:

${\operatorname{H}_{b}^{\bullet}(\pi_{1}(M,*))}$${\operatorname{H}^{\bullet}_{b}(G)}$${\operatorname{H}^{\bullet}_{b}(F)}$$\scriptstyle{\Gamma_{b}}$$\scriptstyle{i^{*}}$$\scriptstyle{\rho^{*}}$

Thus $\rho$ is a $(F,1,0)$-inverse of $\Gamma_{b}$. It follows that Theorem
5.2 and Theorem 5.3 hold as well for the Dirac measure. Similarly, Theorem A
and Theorem B from [BM22] hold with $\mathcal{T}_{M}=G$, i.e., $Tr^{*}$ has
the image of dimension continuum in degree $2$ and $3$ (if $M$ satisfies the
assumptions of Theorems A and B).

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