Some remarks on acyclicity in bounded cohomology

TL;DR

This paper characterizes when surjective group homomorphisms induce isomorphisms in bounded cohomology, showing it occurs precisely when the kernel is boundedly acyclic, and extends this to maps between spaces.

Contribution

It provides a new characterization of boundedly acyclic groups and maps that induce bounded cohomology isomorphisms for all dual normed modules.

Findings

Boundedly acyclic groups have trivial bounded cohomology with all dual normed trivial modules.

Surjective homomorphisms induce cohomology isomorphisms iff their kernels are boundedly acyclic.

Characterization of maps between spaces that induce bounded cohomology isomorphisms.

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.