Complete Intersection Singularities of Splice Type as Universal Abelian Covers

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 “splicequotient 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.