Normal complex surface singularities with rational homology disk smoothings

In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk D 4 (called a rational homology disk smoothing). Combining with earlier results, this theorem then provides a complete classification of resolution graphs of normal surface singularities with a rational homology disk smoothing, verifying a conjecture of J. Wahl regarding such singularities. Indeed, together with a recent result of J. Fowler we get the complete list of normal surface singularities which admit rational homology disk smoothings. 1. Introduction Let (X,0) be a (germ of a) normal complex surface singularity. A smoothing of (X,0) is a flat surjective map π: (X,0) → (�,0), where (X,0) has an isolated 3-dimensional singularity and � = {t ∈ C | |t| < ǫ}, such that (π −1 (0),0) is isomorphic to (X,0) and π −1 (t) is smooth for every t ∈ �\{0}. Assume that (X,0) is embedded in (C N ,0). Then there exists an embedding of (X,0) in (C N × �,0) such that the map π is induced by the projection C N × � → � to the second factor. The Milnor fiber M of a smoothing π of (X,0) is defined by the intersection of a fiber π −1 (t) (t 6 0) near the origin with a small ball about the origin, that is, M = π −1 (t) ∩ Bδ(0) (0 < |t| ≪ δ ≪ ǫ).