Bounded classes in the cohomology of manifolds
type: Item
instance of: scholarly article
title: Bounded classes in the cohomology of manifolds
zbMATH Open document ID: 1035.57012
DOI: 10.1023/A:1019677231286
author: Koji Fujiwara
author: Teruhiko Soma
published in: Geometriae Dedicata
publication date: 2002-09-18T00:00:00Z
review text: For a topological space \(X\), the bounded cohomology \(H^*_b (X)\) is defined as the cohomology of the subcomplex of the real singular cochain complex consisting of the maps to \(\mathbb{R}\) that are bounded on the simplices. Let \(BH^* (X)\) denote the image of the natural map from \(H^*_b (X)\) to the ordinary cohomology \(H^* (X)\). Elements of \(BH^* (X)\) are called bounded classes in \(H^* (X)\). In the paper under review, the authors study the bounded classes in the cohomology of manifolds. It is proved that bounded classes in \(H^{n-1} (M)\) for a complete, noncompact, \(n\)-dimensional (\(n\geq 3\)) hyperbolic manifold of finite volume are precisely those cohomology classes which can be represented by compactly supported, closed \((n-1)\)-forms on \(M\). Then they give a necessary and sufficient condition for a Haken manifold \(M\) to satisfy \(BH^m (M)=H^m (M)\) for all \(m>1\).
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 1802406
zbMATH Keywords: bounded cohomology
zbMATH Keywords: Haken manifolds
zbMATH Keywords: Kleinian groups
zbMATH Keywords: hyperbolic manifold
reviewed by: Mustafa Korkmaz
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1802406