Geometric Structures on Branched Covers over Universal Links

A number of recent results are presented which bear on the question of what geometric information can be gleaned from the representation of a three-manifold as a branched cover over a fixed universal link. Results about Seifert-fibered manifolds, graph manifolds and hyperbolic manifolds are discussed. Section 0 Introduction. Closed, orientable three-manifolds admit a variety of universal constructions, that is, constructions by which all manifolds of that class are obtainable, e.g., Heegaard diagrams, surgery diagrams, etc. One of the difficulties faced in three-manifold topology is the decision as to which of the universal constructions is most likely to yield a solution to a particular problem. In this paper, we present several recent results which come from the use of universal links to work on the Thurston Geometrization Conjecture. More specifically, we present a structure theorem for nonpositively curved Euclidean cone manifolds without vertices, which allows us to deduce which geometries are possible for the pieces of the torus decomposition of certain branched covers, and, in fact, to construct the characteristic submanifold for such covers (in this context, “curvature” refers to a combinatorial condition on the branching indices of a branched cover). We also obtain results concerning hyperbolic structures on negatively curved hyperbolic cone manifolds. Cone manifolds are the natural geometric structure to consider in connection with branched covers, since a cone metric on the base space of a branched covering map may be lifted to the cover. We also present the negative result that, at least for some universal links, a geometric structure on the cover is not always reflected in the branched covering map itself. More specifically, there exist hyperbolic manifolds, all of whose cone manifold structures arising from branched covering maps over a fixed universal link (the Borromean rings) have some positive curvature. Section 1 Universal Links. We begin by defining branched covers as well as fixing the notation we will use subsequently 1991 Mathematics Subject Classification. Primary: 57M12, Secondary: 57R15, 57M25.