Geometry & Topology, Vol. 9 (2005) Paper no. 18, pages 757--811.

Complex surface singularities with integral homology sphere links

Walter D Neumann, Jonathan Wahl

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].

Keywords. Casson invariant, integral homology sphere, surface singularity, complete intersection singularity, monomial curve, plane curve singularity

AMS subject classification. Primary: 14B05, 14H20. Secondary: 32S50, 57M25, 57N10.

DOI: 10.2140/gt.2005.9.757

E-print: arXiv:math.AG/0301165

Submitted to GT on 24 May 2004. (Revised 18 April 2005.) Paper accepted 6 March 2005. Paper published 24 April 2005.

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Walter D Neumann, Jonathan Wahl
Department of Mathematics, Barnard College, Columbia University
New York, NY 10027, USA
Department of Mathematics, The University of North Carolina
Chapel Hill, NC 27599-3250, USA

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