Amenability and acyclicity in bounded cohomology

Johnson's characterization of amenable groups states that a discrete group Gamma is amenable if and only if Hbn >= 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 mapping theorem: a surjective group homomorphism phi:Gamma -> K has amenable kernel H if and only if the induced inflation map Hb center dot(K;VH)-> Hb center dot(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:Gamma -> K with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all Banach Gamma-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms, that is, the class of group homomorphisms phi:Gamma -> K for which the restriction maps in bounded cohomology Hb center dot(K;V)-> Hb center dot(Gamma;phi-1V) 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 amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.