A derived functor approach to bounded cohomology

Bühler, Theo

doi:10.1016/j.crma.2008.04.003

Cited by 2 publications

(4 citation statements)

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“…We note that the characterization in (3) states that a map f : X → Y is amenable if and only if f is fiberwise amenable, that is, if and only if the homotopy fiber F is amenable. Using the Mapping Theorem, Theorem A will be obtained from an analogous characterization of amenable homomorphisms in the context of discrete groups (see Definition 2.4.6 and Theorem 3.1.3).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…If we ignore the metric structure, bounded cohomology theory of groups can be described as a (universal) δ-functor in the classical sense of homological algebra. A foundational approach to bounded cohomology based on this observation is developed by Bühler [4,3]. In this subsection, we recall the main algebraic properties of bounded cohomology from the viewpoint of homological algebra.…”

Section: Induction Modulesmentioning

confidence: 99%

“…Proof. We include a few comments for the convenience of the reader and refer to Bühler [4,3] and Monod [20] for detailed proofs. (1) follows easily from the fact that C 0 b (Γ; V ) Γ ∼ = V Γ .…”

Section: Induction Modulesmentioning

confidence: 99%

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Amenability and Acyclicity in Bounded Cohomology Theory

Moraschini¹,

Raptis²

2021

Preprint

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Johnson's characterization of amenable groups states that a discrete group Γ is amenable if and only if H n≥1 b (Γ; V ) = 0 for all dual normed R[Γ]-modules V . In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective homomorphism ϕ : Γ → K has amenable kernel H if and only if the induced inflation mapIn addition, we obtain an analogous characterization for the (smaller) class of surjective homomorphisms ϕ : Γ → K with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all normed R[Γ]-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms ϕ : Γ → K, for which the restriction maps in bounded cohomology HWe then extend the first and third results to 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.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Induction Modulesmentioning

confidence: 99%

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Amenability and Acyclicity in Bounded Cohomology Theory

Moraschini¹,

Raptis²

2021

Preprint

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“…Gromov further developed bounded cohomology and studied its relation with the (Riemannian) volume of manifolds [13]. A more algebraic approach to bounded cohomology was subsequently developed by Brooks [7], Ivanov [17], Noskov [33], Monod [28] [29], and Bühler [11].…”

Section: Bulletin Of the Manifold Atlas 2011mentioning

confidence: 99%

Simplicial volume

2017

Bounded Cohomology of Discrete Groups

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Simplicial volume is a homotopy invariant of oriented closed connected manifolds introduced by Gromov in his proof of Mostow rigidity. On the one hand, the simplicial volume of a Riemannian manifold encodes non-trivial information about the Riemannian volume; on the other hand, simplicial volume can be described in terms of a certain functional analytic version of homological algebra (bounded cohomology). In this article we survey important properties and applications of simplicial volume as well as useful techniques for working with simplicial volume. 53C23, 57R99 1. Definition and history Simplicial volume is a homotopy invariant of oriented closed connected manifolds that was introduced by Gromov in his proof of Mostow rigidity [31][13]. Intuitively, simplicial volume measures how difficult it is to describe the manifold in question in terms of simplices (with real coefficients): Definition 1.1. (Simplicial volume) Let M be an oriented closed connected manifold of dimension n. Then the simplicial volume of M (also called the Gromov norm of M) is defined as M := [M ] 1 = inf |c| 1 c ∈ C n (M ; R) is a fundamental cycle of M ∈ R ≥0 , where [M ] ∈ H n (M ; R) is the fundamental class of M with real coefficients. • Here, | • | 1 denotes the 1-norm on the singular chain complex C * (• ; R) with real coefficients induced from the (unordered) basis given by all singular simplices, i.e.: for a topological space X and a chain c = k j=0 a j • σ j ∈ C * (X; R) (in reduced form), the 1-norm of c is given by |c| 1 := k j=0 |a j |. • Moreover, • 1 denotes the 1-semi-norm on singular homology H * (• ; R) with real coefficients, which is induced by | • | 1. More explicitly, if X is a topological space and α ∈ H * (X; R), then α 1 := inf |c| 1 c ∈ C * (X; R) is a cycle representing α. Convention 1.2. In the following, if not explicitly stated otherwise, all manifolds are topological manifolds and are of non-zero dimension.

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A derived functor approach to bounded cohomology

Cited by 2 publications

References 10 publications

Amenability and Acyclicity in Bounded Cohomology Theory

Amenability and Acyclicity in Bounded Cohomology Theory

Simplicial volume

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