  We construct a spectral sequence for L2-type cohomology groups of discrete
measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer
conjecture for aspherical manifolds with poly-surface fundamental groups. More
generally, we obtain a permanence result for the Hopf-Singer conjecture under
taking fiber bundles whose base space is an aspherical manifold with
poly-surface fundamental group. As further sample applications of the spectral
sequence, we obtain new vanishing theorems and explicit computations of
L2-Betti numbers of groups and manifolds and obstructions to the existence of
normal subrelations in measured equivalence relations.
