\(L^ 2\)-topological invariants of 3-manifolds
For a large class of compact 3-manifolds \(M\), the authors compute the \(L^2\)-Betti numbers \(b_k(M)\) and give estimation for the Novikov-Shubin invariants \(\alpha_k(M)\). It turns out that in these cases the \(b_k(M)\) are rational and the \(\alpha_k(M)\) are positive. Thus, basic conjectures concerning these invariants are confirmed in dimension 3. In the first half of the paper, the authors study properties of the \(b_k\) and the \(\alpha_k\) in arbitrary dimensions. They work in the general setting of Hilbert modules over finite von Neumann algebras. The main result in this part is a set of inequalities for the \(\alpha_k\) in situations of Mayer-Vietoris type. These estimates are then repeatedly used, namely to study connected sums, to extend results known for closed manifolds to compact manifolds \(M\) with boundary by looking at the double of \(M\), and to study decompositions of 3-manifolds. The paper closes with a very useful section, where important conjectures in the field are stated and commented upon.
scientific article; zbMATH DE number 746068
scientific article