Some remarks on acyclicity in bounded cohomology

Abstract

We show that a surjective homomorphism $\varphi \colon \Gamma \to K$ of (discrete) groups induces an isomorphism $H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \varphi^{-1} V)$ in bounded cohomology for all dual normed $K$-modules $V$ if and only if the kernel of $\varphi$ is boundedly acyclic. This complements a previous result by the authors that characterized this class of group homomorphisms as bounded cohomology equivalences with respect to $\mathbb{R}$-generated Banach $K$-modules. We deduce a characterization of the class of maps between path-connected spaces that induce isomorphisms in bounded cohomology with respect to coefficients in all dual normed modules, complementing the corresponding result shown previously in terms of $\mathbb{R}$-generated Banach modules. The main new input is the proof of the fact that every boundedly acyclic group $\Gamma$ has trivial bounded cohomology with respect to all dual normed trivial $\Gamma$-modules.