Complex surface singularities with integral homology sphere links

Abstract

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities of "splice type", generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. 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. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Sigma should be a 4-manifold canonically associated to Sigma. 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 also point out recent counterexamples to some overly optimistic earlier conjectures in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann. 326(2003) 75--93].

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper18.abs.html

References in corpus (6)

• Seiberg--Witten invariants and surface singularities
• Complete intersection singularities of splice type as universal abelian covers
• Seiberg-Witten invariants and surface singularities II. Singularities with good $\C^*$-action
• Seiberg-Witten invariants and surface singularities III. Splicings and cyclic covers
• Links and analytic invariants of superisolated singularities
• The link of {f(x,y)+z^n=0} and Zariski's Conjecture

Cited by in corpus (17)

• Complete intersection singularities of splice type as universal abelian covers
• The geometric genus of splice-quotient singularities
• On the Casson Invariant Conjecture of Neumann--Wahl
• Surgery formula for Seiberg--Witten invariants of negative definite plumbed 3-manifolds
• Generalized Monodromy Conjecture in dimension two
• Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones
• The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses
• Analytic singularities supported by a specific integral homology sphere link
• Rational blow-downs and smoothings of surface singularities
• Kulikov singularities
• The multiplicity of abelian covers of splice quotient singularities
• Links and analytic invariants of superisolated singularities
• Determining When The Universal Abelian Cover of a Graph Manifold is a Rationla Homology Sphere
• Constructing Universal Abelian Covers of Graph Manifolds
• On the topology of surface singularities {z^n=f(x,y)}, for f irreducible
• Open book decompositions of ${\mathbb{S}}^5$ and real singularities
• Local tropicalizations of splice type surface singularities