Amenability and acyclicity in bounded cohomology

Johnson's characterization of amenable groups states that a discrete group $\Gamma$ is amenable if and only if $H_b^{n \geq 1}(\Gamma; V) = 0$ for all dual normed $\R[\Gamma]$-modules $V$. In this paper, we extend the previous result to homomorphisms by proving the converse of the \emph{Mapping Theorem}: a surjective group homomorphism $\phi \colon \Gamma \to K$ has amenable kernel $H$ if and only if the induced inflation map $H^\bullet_b(K; V^H) \to H^\bullet_b(\Gamma; V)$ is an isometric isomorphism for every dual normed $\R[\Gamma]$-module $V$. In addition, we obtain an analogous characterization for the (smaller) class of surjective group homomorphisms $\phi \colon \Gamma \to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for \emph{all} Banach $\Gamma$-modules. Finally, we also prove a characterization of the (larger) class of \emph{boundedly acyclic} homomorphisms, that is, the class of group homomorphisms $\phi \colon \Gamma \to K$ for which the restriction maps in bounded cohomology $H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \phi^{-1}V)$ are isomorphisms for a suitable family of dual normed $\R[K]$-modules $V$ including the trivial $\R[K]$-module $\R$. We then extend the first and third results to topological spaces and obtain characterizations of \emph{amenable} maps and \emph{boundedly acyclic} maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.