The Topology of the Normalization of Complex Surface Germs

Let (X, p) be a reduced complex surface germ and let LX be its well defined link. If (X, p) is normal at p, D. Mumford [7] shows that (X, p) is smooth if and only if LX is simply connected. Moreover, if p is an isolated singular point, LX is a three dimensional Waldhausen graph manifold. Then, the Plumbing Calculus of W. Neumann [8] shows that the homeomorphism class of LX determines a unique plumbing in normal form and consequently, determines the topology of the good minimal resolution of (X, p). Here, we do not assume that X is normal at p, and so, the singular locus (Σ, p) of (X, p) can be one dimensional. We describe the topology of the singular link LX and we show that the homeomorphism class of LX (Theorem 3.0.10) determines the homeomorphism class of the normalization and consequently the plumbing of the minimal good resolution of (X, p). Moreover, in Proposition 4.0.11, we obtain the following generalization of the D. Mumford theorem [7]: Let ν : (X , p) → (X, p) be the normalization of an irreducible germ of complex surface (X, p). If the link LX of (X, p) is simply connected then ν is a homeomorphism and (X , p) is a smooth germ of surface