Manifolds of even dimension with amenable fundamental group (Q1842233)

The main topic of the paper is to investigate the relation between the Euler characteristic and the fundamental group \(\pi\) of a closed 4-manifold \(X\). The paper deals with the question, when \(\chi(X)\geq 0\) resp. \(\chi(X)=0\) holds and when \(X\) is aspherical, i.e. all higher homotopy groups of \(X\) vanish. The main results are: 1. If \(\pi\) is infinite amenable, then \(\chi(X) \geq 0\); 2. If \(\pi\) is infinite amenable and \(\chi(X)=0\), then \(H_2 (\widetilde X) \cong H^2 (\pi; \mathbb{Z}\pi)\); 3. If \(H^1(\pi; \mathbb{Z}\pi)\) and \(H^2(\pi; \mathbb{Z}\pi)\) vanish and \(\chi(X)=0\), then \(X\) is aspherical; The proof uses the reduced and unreduced \(L^2\)-cohomology of the universal covering with the \(\pi\)-action by deck transformations. Some of the results are also formulated for \(2n\)-dimensional closed manifolds for \(n\geq 3\) under appropriate connectivity conditions.