2024/11/06 by Marco Moraschini, Moraschini, Marco, George Raptis +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2411.03761
openalex publication_date 2024/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We show that a surjective homomorphism φ\colon Γ→ K of (discrete) groups induces an isomorphism H^\bulletb(K; V) → H^\bulletb(Γ; φ-1 V) in bounded cohomology for all dual normed K-modules V if and only if the kernel of φ 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 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 ℝ-generated Banach modules. The main new input is the proof of the fact that every boundedly acyclic group Γ has trivial bounded cohomology with respect to all dual normed trivial Γ-modules.