#### Volume 9, issue 2 (2005)

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Complete intersection singularities of splice type as universal abelian covers

### Walter D Neumann and Jonathan Wahl

Geometry & Topology 9 (2005) 699–755
 arXiv: math.AG/0407287
##### Abstract

It has long been known that every quasi-homogeneous normal complex surface singularity with $ℚ$–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 $ℚ$–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 $ℚ$–homology sphere links. As quotients of complete intersections, they are necessarily $ℚ$–Gorenstein, and many $ℚ$–Gorenstein singularities with $ℚ$–homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with $ℚ$–homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture.

##### Keywords
surface singularity, Gorenstein singularity, rational homology sphere, complete intersection singularity, abelian cover
##### Mathematical Subject Classification 2000
Primary: 32S50, 14B05
Secondary: 57M25, 57N10