  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.
