Volume 9, issue 2 (2005)

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Complex surface singularities with integral homology sphere links

Walter D Neumann and Jonathan Wahl

Geometry & Topology 9 (2005) 757–811

arXiv: math.AG/0301165


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 a previous paper 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 Σ should be a 4–manifold canonically associated to Σ. 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.

Casson invariant, integral homology sphere, surface singularity, complete intersection singularity, monomial curve, plane curve singularity
Mathematical Subject Classification 2000
Primary: 14B05, 14H20
Secondary: 32S50, 57M25, 57N10
Forward citations
Received: 24 May 2004
Revised: 18 April 2005
Accepted: 6 March 2005
Published: 28 April 2005
Proposed: Robion Kirby
Seconded: Ronald Fintushel, Ronald Stern
Walter D Neumann
Department of Mathematics
Barnard College
Columbia University
New York
New York 10027
Jonathan Wahl
Department of Mathematics
The University of North Carolina
Chapel Hill
North Carolina 27599-3250