Complex Surface Singularities with Integral Homology Sphere Links

The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. We study a large class of such complete intersections, those of “splice type,” and boldly conjecture that all Gorenstein singularities with homology sphere links are equisingular deformations of singularities of this type. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We prove the CIC itself for a special class of splice type singularities; this result includes all previously proven cases of the Conjecture. We show also that a singularity with homology sphere link is of splice type (up to equisingular deformation) if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. In [22] we formulated the Casson Invariant Conjecture. Let (X, o) be an isolated complete intersection surface singularity whose link Σ is an integral homology 3-sphere. Then the Casson invariant λ(Σ) is one-eighth the signature of the Milnor fiber of X. The interest of this conjecture is its suggestion that the Milnor fiber is a “natural” 4-manifold which is attached to its boundary Σ, and for which the signature computes the Casson invariant exactly (and not just mod 2). Specifically, it implies that for a complete intersection singularity whose link is a homology sphere, analytic invariants like the Milnor number and geometric genus are determined by the link. (Such results are known to be false for general hypersurface singularities.) In [22] we verified the Casson Invariant Conjecture for Brieskorn-Pham complete intersections by direct computation. It was a challenge to find other examples; having done so, the conjecture was verified in these cases, with the serious work being calculation of the signature. The topological types of normal surface singularities with homology sphere link may be classified, and form a rich class ([7]); they are obtained by repeatedly splicing together the links Σ(p1, · · · , pn) of Brieskorn-Pham complete intersections along naturally occurring knots. But it is completely unclear which topological types may be realized by complete intersection (or even Gorenstein) singularities. In a parallel paper [25], we describe how “most” homology sphere singularity links do, in fact, arise as links of complete intersection singularities, and we give explicit equations. These equations, which we call “splice type,” generalize the