Splice diagram determining singularity links and universal abelian covers

TL;DR

This paper establishes a numerical criterion based on splice diagrams to identify singularity links among rational homology sphere graph manifolds and proves that identical splice diagrams imply homeomorphic universal abelian covers, extending to orbifolds.

Contribution

It introduces a sufficient numerical condition on splice diagrams for a manifold to be a singularity link and shows that identical splice diagrams imply homeomorphic universal abelian covers, generalizing to orbifolds.

Findings

Numerical condition characterizes singularity links.

Identical splice diagrams imply homeomorphic universal abelian covers.

Extension of concepts to orbifolds.

Abstract

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their universal abelian covers are homeomorphic. To prove the last theorem we have to generalize our notions to orbifolds.