2.1. 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

\begin{equation*}\operatorname {C}^n_b(G) = \{c\ :\ G^{n+1} \to \mathbb {R}\; |\; c\; \text {is bounded}\}.\end{equation*}

Let $d_n$ be the ordinary coboundary operator $d_n\ :\ \operatorname{C}^n_b(G) \to \operatorname{C}^{n+1}_b(G)$ . The group $G$ acts on $\operatorname{C}^n_b(G)$ by

\begin{equation*}h(c)(g_0,\ldots, g_n) = c(h^{-1}g_0,\ldots, h^{-1}g_n) \hspace {0.4cm} \forall c \in \operatorname {C}^n_b(G), \hspace {0.4cm} \forall h, g_0,\ldots, g_n \in G.\end{equation*}

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 $|\hspace{-1px}|\hspace{-3px}\cdot \hspace{-3px}|\hspace{-1px}|$ . This norm induces a semi-norm on $\operatorname{H}^n_b(G)$ ; that is, if $C \in \operatorname{H}_b^n(G)$ , then

\begin{equation*}|\hspace {-1px}| C |\hspace {-1px}| = \operatorname {min}\{|\hspace {-1px}| c |\hspace {-1px}|\; |\; [c] = C\}.\end{equation*}

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 $c\ :\ \operatorname{S}_n(M) \to \mathbb{R}$ .

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 $d_n\ :\ \operatorname{C}^n_b(M) \to \operatorname{C}^{n+1}_b(M)$ is the standard coboundary operator. Like in the group case, we have the semi-norm 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 [Reference Frigerio9, 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 [Reference Gromov11].

2.2. 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 [Reference Thurston21, Chapter 6]. Therefore, we regard $\Delta _{\bar{g}}$ as a map from the standard simplex to $\mathbb{H}^n$ . We define

\begin{equation*}v(\bar {g}) = \int _{\Delta _{\bar {g}}} vol_h.\end{equation*}

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:

\begin{equation*}Vol = [v] \in \operatorname {H}^n_b(Iso_+(\mathbb {H}^n)).\end{equation*}

For $G \lt Iso_+(\mathbb{H}^n)$ , 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 $\rho\ :\ \pi _1(M) \to Iso_+(\mathbb{H}^n)$ . 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

\begin{equation*}Vol_{\rho (\pi _1(M))} \in \operatorname {H}_b^n(\pi _1(M)),\end{equation*}

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

\begin{equation*}v^{\prime}_M(\sigma ) = \int _{str(\sigma )} vol_h.\end{equation*}

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 $\pi _1(M)$ .

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 $i\ :\ M \to M^{ex}$ is a homotopy equivalence. Hence

\begin{equation*}i^*\ :\ \operatorname {H}_b^n(M^{ex}) \to \operatorname {H}_b^n(M)\end{equation*}

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 [Reference Frigerio9, 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

\begin{equation*}\Delta ^*\ :\ \{\operatorname {C}_b^\bullet (\mathbb {H}^n)),d\} \to \{\operatorname {C}^\bullet _b(G),d\}\end{equation*}

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 [Reference Frigerio9, Lemmas 4.12 and Lemma 5.4], $\Delta ^*$ induces a map

\begin{equation*}\operatorname {H}^n_b(\Delta )\ :\ \operatorname {H}_b^n(M) \to \operatorname {H}_b^n(\pi _1(M))\end{equation*}

which is the inverse of $r$ [Reference Frigerio9, 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 [Reference Thurston21] or [Reference Frigerio9, Lemma 8.12].

2.3. 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

\begin{equation*} \mathbb {Z} \xrightarrow {} \operatorname {Homeo}_+^{\mathbb {Z}}(\mathbb {R}) \xrightarrow {p} \operatorname {Homeo}_+(S^1). \end{equation*}

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 [Reference Frigerio9, 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

\begin{equation*} \alpha\ :\ \mathcal {M}_+(S,*) \to \operatorname {Homeo}_+(S^1). \end{equation*}

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 $\pi _1(S,*) \lt \mathcal{M}_+(S,*)$ 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 $\pi _1(S,*) \lt \mathcal{M}_+(S,*)$ . The class $e_b^{{\scriptscriptstyle \mathcal{M}_+}}$ was studied in [Reference Chen7, Reference Jekel and Rolland13].

3.1. Description of $\Gamma _{\hspace{-2px}b}$

In this paragraph, we give the simplest, in our opinion, description of $\Gamma _{\hspace{-2px}b}$ . A more refined and general definition can be found in [Reference Brandenbursky and Marcinkowski5]. Another approach based on coupling of groups is given in [Reference Nitsche19]. Note that the definition in [Reference Nitsche19] 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

\begin{equation*} \gamma\ :\ \operatorname {Homeo}_0\!(M,\omega ) \times M \to \pi _1(M,*).\end{equation*}

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 [Reference Brandenbursky and Marcinkowski5, 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:

\begin{equation*}\gamma (fg,x) = \gamma (f,x)\gamma (g,f(x)).\end{equation*}

Define $\widetilde{\omega }$ to be the measure on $\widetilde{M}$ induced by the pullback 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 3.2, we show that measurable systems of paths exist.

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

\begin{equation*}\Gamma _{\hspace {-2px}b}\ :\ \operatorname {H}^n_b(\pi _1(M,*)) \to \operatorname {H}^n_b(\operatorname {Homeo}_0\!(M,\omega ))\end{equation*}

to be the map induced by the following function defined on cochains (called again $\Gamma _{\hspace{-2px}b}$ ):

\begin{equation*}\Gamma _{\hspace {-2px}b}(c)(f_0,\ldots, f_n) = \int _M c\big (\gamma (f_0,x),\ldots, \gamma (f_n,x)\big ) d\omega (x) .\end{equation*}

In Section 3.2, 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.

3.2. $\Gamma _b$ via coupling

Let us describe the construction of $\Gamma _b$ via couplings given in [Reference Nitsche19]. 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 $\chi\ :\ X \to \Gamma$ is defined by the equation $\chi (\gamma .x)=\gamma$ for $x \in F$ and $\gamma \in \Gamma$ . Note that $\chi$ is measurable; that is, the preimage of every element in $\Gamma$ is a measurable set. A function $\bar{\chi }\ :\ \Lambda \times F \to \Gamma$ given by $\bar{\chi }(\lambda, x) = \chi (x.\lambda )$ induces a homomorphism $tr_\Gamma ^\Lambda X\ :\ \operatorname{H}_b^\bullet (\Gamma ) \to \operatorname{H}_b^\bullet (\Lambda )$ by the formula [Reference Nitsche19, Section 3]

\begin{equation*}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).\end{equation*}

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 $F$ [Reference Nitsche19, 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 [Reference Brandenbursky and Marcinkowski5, 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 $\Gamma _{\hspace{-2px}b} = tr_\Gamma ^\Lambda X$ . 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 $p\ :\ \widetilde{M} \to M$ 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 $\Gamma _{\hspace{-2px}b}$ and $tr_\Gamma ^\Lambda X$ coincide. In particular, $\Gamma _{\hspace{-2px}b}$ does not depend on the choice of a measurable system of paths.

To finish the construction of $\Gamma _{\hspace{-2px}b}$ , 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.

3.3. 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 [Reference Edwards and Kirby8, Corollary 1.1], which 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

\begin{equation*}\mathcal {M}(M,*) = \operatorname {Homeo}(M,*)/\operatorname {Homeo}_0(M,*).\end{equation*}

Let $\operatorname{Homeo}(M,*\to x) \lt \operatorname{Homeo}(M)$ 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

\begin{equation*} \gamma ^{\scriptscriptstyle \mathcal {M}}\ :\ \operatorname {Homeo}\!(M,\omega ) \times M \to \mathcal {M}(M,*) \end{equation*}

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 3.4.

Likewise in Section 3.1, any measurable system of homeomorphisms $P$ induces a map

\begin{equation*} \Gamma _b^{\scriptscriptstyle \mathcal {M}}\ :\ \operatorname {H}_b^n(\mathcal {M}(M,*)) \to \operatorname {H}_b^n(\operatorname {Homeo}\!(M,\omega )). \end{equation*}

In Section 3.4, we show that $\Gamma _{\hspace{-2px}b}^{\scriptscriptstyle \mathcal{M}}$ does not depend on the choice of $P$ . Note that $\Gamma _b^{\scriptscriptstyle \mathcal{M}}$ generalizes the homomorphism $\mathcal{G}_{S,1}$ from [Reference Brandenbursky and Marcinkowski4] 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

\begin{equation*} \Gamma _b^{\scriptscriptstyle \mathcal {M}_+}\ :\ \operatorname {H}_b^n(\mathcal {M}_+(M,*)) \to \operatorname {H}_b^n(\operatorname {Homeo}_+\!(M,\omega )). \end{equation*}

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

\begin{equation*}\operatorname {Homeo}(M,*) \xrightarrow {} \operatorname {Homeo}(M) \xrightarrow {ev} M,\end{equation*}

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

\begin{equation*} \pi _1(\operatorname {Homeo}(M)) \xrightarrow {ev_1} \pi _1(M,*) \xrightarrow {Pu} \mathcal {M}(M,*) \xrightarrow {F} \mathcal {M}(M). \end{equation*}

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 [Reference Brandenbursky and Marcinkowski5, 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 $[f] \in im(Pu) = ker(F) \lt \mathcal{M}(M,*)$ , 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:

Thus the following diagram commute:

3.4. $\Gamma _{\hspace{-2px}b}^{\scriptscriptstyle \mathcal{M}}$ 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 $ev\ :\ \operatorname{Homeo}(M) \to M$ is a fiber 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 $q\ :\ \operatorname{Homeo}(M) \to \widetilde{M}^{\scriptscriptstyle \mathcal{M}}$ 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 $q$ :

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 pullback 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 3.2) that the constructions described in Section 3.3 and [Reference Nitsche19] coincide. In particular, $\Gamma _{\hspace{-2px}b}^{\scriptscriptstyle \mathcal{M}}$ does not depend on the choice of a measurable system of paths [Reference Nitsche19, 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 $\pi _1(M,*) \lt \mathcal{M}(M,*)$ 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 $T\ :\ \widetilde{M}_0 \to \widetilde{M}$ , 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 $ev_1\ :\ \pi _1(\operatorname{Homeo}(M)) \to \pi _1(M,*)$ , 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 $T\ :\ \widetilde{M}_0 \to \widetilde{M}$ , 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.

4.1. Volume class in dimension 2

Let $X$ be a topological space. The $l^1$ -homology of $X$ is denoted by $\operatorname{H}^{l_1}_n(X)$ [Reference Frigerio9, Chapter 6]. In $l^1$ -homology, we allow chains to be infinite sums $c = \Sigma _i^\infty a_i\sigma _i$ , where $|\hspace{-1px}| c |\hspace{-1px}| = \Sigma _i^\infty |a_i| \lt \infty$ . 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:

\begin{equation*}\langle \cdot, \cdot \rangle\ :\ \operatorname {H}_b^n(X) \times \operatorname {H}_n^{l_1}(X) \to \mathbf{R}.\end{equation*}

The Kronecker product is defined on the level of chains by $\langle b, a \rangle = \Sigma _i^\infty a_ib(\sigma _i)$ , where $a = \Sigma _i^\infty a_i\sigma _i$ and $b$ is a bounded cochain. Moreover, we have $| \langle B, A \rangle | \leq |\hspace{-1px}| B |\hspace{-1px}| |\hspace{-1px}| A |\hspace{-1px}|$ , 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 [Reference Mitsumatsu16]. We do not assume that $S$ is closed.

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 $p\ :\ \mathbb{H}^2\to S$ 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 $\gamma \in [G,G]$ [Reference Katok14, 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 $L_\gamma$ [Reference Mitsumatsu16, 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 $\langle Vol_S, C \rangle \gt 0$ . The inequality $\langle Vol_S, C \rangle \leq |\hspace{-1px}| Vol_S |\hspace{-1px}| |\hspace{-1px}| C |\hspace{-1px}|$ implies that the norm of $Vol_S$ is positive.

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

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.

4.2. Volume class in dimension 3

For some Kleinian groups, that is, the discrete subgroups of $Iso_+(\mathbb{H}^3)$ , the volume class was studied in [Reference Soma20]. 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.

A Kleinian group $G$ is geometrically finite if $N_\epsilon (H(L_G)/G)$ has finite volume for some $\epsilon \gt 0$ , where $N_\epsilon$ is an $\epsilon$ neighborhood and $H(L_G)$ is the convex closure of the limit set of $G$ [Reference Thurston21, 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 [Reference Agol1, Reference Calegari and Gabai6], 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 $\pi _1(M) \lt Iso_+(\mathbb{H}^3)$ which is geometrically infinite, $Vol_F$ has positive norm. Below we describe our main example of such a situation, that is, manifolds that fiber over the circle with non-compact fiber.

4.3. Euler class

We briefly recall basic definitions concerning quasimorphisms [Reference Frigerio9, Chapter 2]. Let $G$ be a group. A real function $q\ :\ G \to \mathbb{R}$ is called a quasimorphism if there exists some $D \in \mathbb{R}$ such that

\begin{equation*}|q(ab)-q(a)-q(b)| \leq D\end{equation*}

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 nonhomogeneous 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 [Reference Frigerio9, Corollary 6.7].

Proof. Let $T^1S$ be the unit tangent bundle of $S$ and denote by $q\ :\ \pi _1(T^1S) \to \pi _1(S)$ the map induced by the projection $T^1S \to S$ . We will use the rotation quasimorphism $Rot\ :\ \pi _1(T^1S) \to \mathbb{R}$ defined in [Reference Huber12]. It is a homogeneous quasimorphism of defect $1$ , which trivializes the pullback $q^*(e_b^S) \in \operatorname{H}_b^2(\pi _1(T^1S))$ , that is, $q^*(e_b^S) = [d Rot]$ [Reference Huber12, Theorem 5.9].

Let $a,b \in \pi _1(T^1S)$ be such that $Rot(ab) \neq Rot(a)+Rot(b)$ . Let $F = \langle q(a),q(b) \rangle \lt \pi _1(S)$ . Denote this inclusion by $i\ :\ F \to \pi _1(S)$ , 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 $2$ .