Abstract
Given a rational homology 3-sphere M whose splice diagram \(\varGamma (M)\) satisfies the semigroup condition, Neumann and Wahl define a complete intersection surface singularity called a splice diagram singularity. Under an additional hypothesis on M called the congruence condition they show that the link of this singularity is the universal abelian cover of M. They ask if this still holds if the congruence condition fails. In this article we generalize the congruence condition to orientable graph orbifolds. We show that under a small additional hypothesis this orbifold congruence condition implies that the link of the splice diagram singularity is the universal abelian cover. By showing that any two-node splice diagram satisfying the semigroup condition is the splice diagram of an orbifold satisfying the orbifold congruence condition, we answer the question of Neumann and Wahl affirmatively for two-node diagrams. However, examples show this approach to their question no longer works for three nodes.
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Pedersen, H.M. Splice diagram singularities and the universal abelian cover of graph orbifolds. Geom Dedicata 195, 283–305 (2018). https://doi.org/10.1007/s10711-017-0290-5
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DOI: https://doi.org/10.1007/s10711-017-0290-5