Some remarks on acyclicity in bounded cohomology

Abstract

We show that a surjective homomorphism $\phi :\Gamma \to K$ of (discrete) groups induces an isomorphism $H_b^{\bullet }(K;V)\to H_b^{\bullet }(\Gamma ;{\phi }^{-1}V)$ in bounded cohomology for all dual normed $K$-modules $V$ if and only if the kernel of $\phi$ 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.

arXiv comment6 pages. To appear in Rev. Mat. Iberoam