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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.
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