Celebratio Mathematica

W. D. Neumann: “A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves,” Trans. Am. Math. Soc. 268 : 2 (1981), pp. 299–344. MR 632532 Zbl 0546.57002 article

W. D. Neumann: “Abelian covers of quasihomogeneous surface singularities,” pp. 233–243 in Singularities (Arcata, CA, 20 July–7 August 1981), part 2. Edited by P. Orlik. Proceedings of Symposia in Pure Mathematics 40. American Mathematical Society (Providence, RI), 1983. MR 713252 Zbl 0519.32010 incollection

D. Eisenbud and W. Neumann: Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematics Studies 110. Princeton University Press, 1985. MR 817982 Zbl 0628.57002 book

W. Neumann and J. Wahl: “Casson invariant of links of singularities,” Comment. Math. Helv. 65 : 1 (1990), pp. 58–78. MR 1036128 Zbl 0704.57007 article

W. D. Neumann and J. Wahl: “Universal abelian covers of quotient-cusps,” Math. Ann. 326 : 1 (2003), pp. 75–93. MR 1981612 Zbl 1032.14010 ArXiv math/0101251 article

W. D. Neumann and J. Wahl: “Complete intersection singularities of splice type as universal abelian covers,” Geom. Topol. 9 : 2 (2005), pp. 699–755. MR 2140991 Zbl 1087.32017 ArXiv math/0407287 article

It has long been known that every quasi-homogeneous normal complex surface singularity with \( \mathbb{Q} \)-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called “splice type singularities,” which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with \( \mathbb{Q} \)-homology sphere links, called “splice-quotient singularities.” According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with \( \mathbb{Q} \)-homology sphere links. As quotients of complete intersections, they are necessarily \( \mathbb{Q} \)-Gorenstein, and many \( \mathbb{Q} \)-Gorenstein singularities with \( \mathbb{Q} \)-homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with \( \mathbb{Q} \)-homology sphere links are splice-quotients. A recent preprint of T. Okuma presents confirmation of this conjecture.

W. D. Neumann and J. Wahl: “Complex surface singularities with integral homology sphere links,” Geom. Topol. 9 : 2 (2005), pp. 757–811. MR 2140992 Zbl 1087.32018 ArXiv math/0301165 article

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 \( \Sigma \) 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.

W. D. Neumann and J. Wahl: “The end curve theorem for normal complex surface singularities,” J. Eur. Math. Soc. 12 : 2 (2010), pp. 471–503. MR 2608949 Zbl 1204.32019 ArXiv 0804.4644 article

We prove the “End Curve Theorem,” which states that a normal surface singularity \( (X,o) \) with rational homology sphere link \( \Sigma \) is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree.

An “end curve function” is an analytic function \( (X,o)\to (\mathbb{C},0) \) whose zero set intersects \( \Sigma \) in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf.

A “splice quotient singularity” \( (X,o) \) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in \( \mathbb{C}^t \), where \( t \) is the number of leaves in the resolution graph for \( (X,o) \), together with an explicit description of the covering transformation group.

Among the immediate consequences of the End Curve Theorem are the previously known results: \( (X,o) \) is a splice quotient if it is weighted homogeneous [Neumann 1981], or rational or minimally elliptic [Okuma 2004].

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Walter D. Neumann

Splice diagrams and splice-quotient surface singularities

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