Local rigidity of aspherical three-manifolds
type: Item
instance of: scholarly article
title: Local rigidity of aspherical three-manifolds
zbMATH Open document ID: 1255.57016
author: Pierre Derbez
publication date: 2012-08-09T00:00:00Z
review text: For each aspherical oriented \(3\)-manifold \(M\), the author constructs a \(2\)-dimensional class in the \(l_1\)-topology of \(M\) whose norm combined with the Gromov simplicial volume of \(M\) gives a characterisation of the non-zero degree maps from \(M\) to \(N\) which are homotopic to a covering map (here, \(N\) refers to some closed orientable irreducible \(3\)-manifold). It is then shown how to apply this result in order to characterise maps of degree \(1\) which are homotopic to a homeomorphism in terms of isometries between the bounded cohomology groups of \(M\) and \(N\).
reviewed by: Bernd O. Stratmann
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 51H20
zbMATH DE Number: 6064521
zbMATH Keywords: aspherical 3-manifolds
zbMATH Keywords: bounded cohomology
zbMATH Keywords: topological rigidity
MaRDI profile type: Publication
OpenAlex ID: W2313150443
cites work: Lower bounds on volumes of hyperbolic Haken 3-manifolds
cites work: Surfaces et cohomologie bornée. (Surfaces and bounded cohomology)
cites work: Non-zero degree maps and surface bundles over \(S^ 1\)
cites work: The Godbillon-Vey Invariant of a Transversely Homogeneous Foliation
cites work: Volumes in Seifert space
cites work: Topological rigidity and Gromov simplicial volume
cites work: Finiteness of mapping degrees and \(\text{PSL}(2,R\mathbf)\)-volume on graph manifolds
cites work: Bounded classes in the cohomology of manifolds
cites work: Volume and bounded cohomology
cites work: Q4114528
cites work: Seifert fibered spaces in 3-manifolds
cites work: Homotopy equivalences of 3-manifolds with boundaries
cites work: Topological rigidity for non-aspherical manifolds
cites work: 3-manifolds with(out) metrics of nonpositive curvature
cites work: Q4382044
cites work: Bounded Cohomology of Certain Groups of Homeomorphisms
cites work: 3-Manifolds Fibering Over S 1
cites work: A Calculus for Plumbing Applied to the Topology of Complex Surface Singularities and Degenerating Complex Curves
cites work: Maps between Seifert fibered spaces of infinite \(\pi_ 1\)
cites work: A rigidity theorem for Haken manifolds
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
cites work: Q3709485
cites work: The existence of maps of nonzero degree between aspherical 3-manifolds
cites work: The \(\pi_ 1\)-injectivity of self-maps of nonzero degree on 3-manifolds
cites work: Any 3-manifold 1-dominates at most finitely many geometric 3-manifolds
DOI: 10.5802/AIF.2708
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 6064521