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Some remarks on acyclicity in bounded cohomology

Marco Moraschini, George Raptis

Abstract

We show that a surjective homomorphism $\varphi \colon Γ\to K$ of (discrete) groups induces an isomorphism $H^\bullet_b(K; V) \to H^\bullet_b(Γ; \varphi^{-1} V)$ 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 $Γ$ has trivial bounded cohomology with respect to all dual normed trivial $Γ$-modules.

Some remarks on acyclicity in bounded cohomology

Abstract

We show that a surjective homomorphism of (discrete) groups induces an isomorphism in bounded cohomology for all dual normed -modules if and only if the kernel of 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 -generated Banach -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 -generated Banach modules. The main new input is the proof of the fact that every boundedly acyclic group has trivial bounded cohomology with respect to all dual normed trivial -modules.

Paper Structure

This paper contains 7 theorems, 10 equations.

Key Result

Theorem 1

Let $f \colon X \to Y$ be a map between based path-connected spacesFollowing the conventions of moraschiniraptis, we restrict to topological spaces that admit a universal covering., let $F$ denote its homotopy fiber, and let $n \geq 0$ be an integer or $n = \infty$. We denote by $f_* \colon \pi_1(X)

Theorems & Definitions (10)

  • Theorem 1: see moraschiniraptis
  • Proposition 2
  • Lemma 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof : Proofs of Theorem \ref{['thm:main:groups']} and Theorem \ref{['main:thm']}
  • Corollary 6
  • Remark 7
  • Corollary 8