On the degrees of branched coverings over links
An irreducible 3-manifold \(M\) is geometrisable if there is a (possibly empty) family \(J\) of incompressible tori such that \(M\setminus N(J)\) is a disjoint union of manifolds which are either Seifert fibered or hyperbolic. A \(JSJ\) decomposition of \(M\) is the decomposition corresponding to a minimal such family. When a manifold is geometrisable, its JSJ graph is the dual graph of the \(JSJ\) decomposition, whose vertices correspond to the pieces of the decomposition and the edges correspond to the tori of the chosen minimal family. It was recently proved that for all geometrisable manifolds \(M'\), except for those which are covered by a (surface)\(\times \mathbb S^1\) or a torus bundle over \(\mathbb S^1\), the degree of a finite covering \(\pi :M\to M'\) is determined by the topological type of \(M\) [see \textit{S. Wang} and \textit{Y.Q. Wu}, Proc. Lond. Math. Soc. 68, 203-224 (1994; Zbl 0841.57028)] and [\textit{F. Yu} and \textit{S. Wang}, Comment. Math. Helv. 74, 238-247 (1999; Zbl 0930.57011)]. In this note, the author studies the analogous question about branched coverings of a link. The main result is that if \(M\) and \(M'\) are compact 3-manifolds and \(L\) is a nontrivial link in \(M'\) whose exterior is irreducible and its JSJ graph is a tree, then there is at most one prime number \(d>3\) (\(d\geq 3\) if \(M'\) is irreducible) for which \(M\) is a \(d\)--fold covering of \(M'\) branched along \(L\). In particular, if \(M'\) is a rational homology sphere and \(L\subset M'\) is a nontrivial prime link with irreducible exterior, then any two branched coverings of \((M',L)\) with different prime degrees \(>3\) (\(\geq 3\) if \(M'\) is irreducible) are nonhomeomorphic.
scientific article; zbMATH DE number 1907788
scientific article