Bounded classes in the cohomology of manifolds
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 \(m>1\).
scientific article; zbMATH DE number 1802406