I will report the on joint work with Andras Nemethi concerning the topological nature of certain analytic invariants of isolated surface singularities. The geometric genus p_g is such an invariant, and the question is whether one can express it in terms of topological invariants of the link of the singularity. For this to happen the link has to be a rational homology sphere (QHS), and the complex structure of the singularity must satisfy some mild rigidity condition (Gorenstein, or complete intersection etc.) A decade ago Fintushel-Stern, Neumann-Wahl gave an affirmative answer for many complete intersection singularities whose links are INTEGRAL homology spheres (IHS) by proving a formula relating the Casson invariant of the link to the signature of the Milnor fiber. This signature differs from p_g by a topological quantity. The obvious change Casson ---> Casson-Walker , when going from IHS to QHS links leads to false conclusions. Even worse, most singularities are not smoothable, and in these cases there is no Milnor fiber to speak of. In my talk I will explain how to resolve this conundrum by relying on the Seiberg-Witten invariants. |