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