A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves

Any graph-manifold can be obtained by plumbing according to some plumbing graph Γ Γ . A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity.