#### 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.

Notes on file formats
Walter D Neumann, Jonathan Wahl

Department of Mathematics, Barnard College, Columbia University

New York, NY 10027, USA

and

Department of Mathematics,
The University of North Carolina

Chapel Hill, NC 27599-3250, USA

Email: neumann@math.columbia.edu, jmwahl@email.unc.edu

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