Constructing Universal Abelian Covers of Graph Manifolds

TL;DR

This paper presents an algorithm to explicitly construct the universal abelian cover of a rational homology sphere graph manifold using its splice diagram, advancing the understanding of the manifold's topological invariants.

Contribution

It transforms the theoretical proof that splice diagrams determine the universal abelian cover into a practical algorithm for construction.

Findings

Algorithm successfully constructs universal abelian covers from splice diagrams.

Provides explicit method for topological invariant computation.

Enhances computational tools in 3-manifold topology.

Abstract

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. It was shown earlier that the splice diagram determines the universal abelian cover of the manifold. We will in this article turn the proof of this in to an algorithm to explicitly construct the universal abelian cover from the splice diagram.