Splice Diagram Singularities and The Universal Abelian Cover of Graph Orbifolds

TL;DR

This paper generalizes the concept of splice diagram singularities and the universal abelian cover from rational homology spheres to graph orbifolds, broadening the applicability of these mathematical structures.

Contribution

It extends the orbifold congruence condition to graph orbifolds and demonstrates that this condition ensures the link is the universal abelian cover, generalizing previous results.

Findings

Orbifold congruence condition implies universal abelian cover

Any two node splice diagrams satisfying the semigroup condition can be realized by a graph orbifold

Generalization from rational homology spheres to graph orbifolds

Abstract

Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They were also able to show that under an additional hypothesis on M called the congruence condition, the link of the splice diagram singularity is the universal abelian cover of M. In this article we generalize the congruence condition to the class of orbifolds called graph orbifold. We show that under a small additional hypothesis, this orbifold congruence condition implies that the link or the splice diagram equations is the universal abelian cover. We also show that any two node splice diagram satisfying the semigroup condition, is the splice diagram of a graph orbifold satisfying the orbifold congruence condition.