Some remarks on acyclicity in bounded cohomology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910686696701952 |
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| author | Moraschini, Marco Raptis, George |
| author_facet | Moraschini, Marco Raptis, George |
| contents | We show that a surjective homomorphism $φ\colon Γ\to K$ of (discrete) groups induces an isomorphism $H^\bullet_b(K; V) \to H^\bullet_b(Γ; φ^{-1} V)$ 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 $\mathbb{R}$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_03761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some remarks on acyclicity in bounded cohomology Moraschini, Marco Raptis, George Algebraic Topology Functional Analysis We show that a surjective homomorphism $φ\colon Γ\to K$ of (discrete) groups induces an isomorphism $H^\bullet_b(K; V) \to H^\bullet_b(Γ; φ^{-1} V)$ 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 $\mathbb{R}$-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. |
| title | Some remarks on acyclicity in bounded cohomology |
| topic | Algebraic Topology Functional Analysis |
| url | https://arxiv.org/abs/2411.03761 |