On the degrees of branched coverings over links

Let M and M′ be 3-manifolds and L a link in M′. We prove that, under certain conditions, the degree of a branched covering $\text{π}\text{:M→(M′,L)}$ is determined by the topological types of M and (M′,L). To cite this article: A.M. Salgueiro, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On regular homotopy of branched coverings of the sphere

Let Mⁿ be an orientable closed n-manifold and f, g: MⁿSⁿ branched coverings of the n-sphere Sⁿ. It is a theorem of H. Hopf that if f and g have the same degree then f and g are homotopic. Our interest is to find out whether f and g are then also regular homotopic, that is to say whether there is a level preserving branched covering H: Mⁿ×ISⁿ×I such that H₀=f and H₁=g. If n=2 or if n=3 and M³ is homeomorphic to S³ the answer to this question is affirmative. For some M³ not homeomorphic to S³ there are however counterexamples.     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

Lins-Mandel manifolds as branched coverings of S3

The Lins-Mandel manifolds ℓ(b, 1, t, 1) are proved to bé 2-fold coverings of S³ branched over a link. This permits to prove some conjectured properties of these spaces.     

On $ \pi $-hyperbolic knots and branched coverings

We prove that, for any given , a -hyperbolic knot is determined by its 2-fold and n-fold cyclic branched coverings. We also prove that a -hyperbolic knot which is not determined by its m-fold and n-fold cyclic branched coverings, , must have genus . Received: December 14, 1998.     

On deformations and the local Torelli problem of cyclic branched coverings

Mathematische Annalen -     

On \pi-hyperbolic knots with the same 2-fold branched coverings

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation if is homeomorphic to . It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related. Received: 3 August 1998 / in final form: 17 June 1999     

Cyclic branched coverings of lens spaces

Some infinite family is constructed of orientable three-dimensional closed manifoldsM _n (p, q), where n ≥ 2, p ≥ 3, 0 < q < p, and (p, q) = 1, such that M _n (p, q) is an n-fold cyclic covering of the lens space L(p, q) branched over a two-component link.     

Representing links in 3-manifolds¶ by branched coverings of S3

We introduce a planar coloured-diagram representation of links in 3-manifolds given as branched coverings of the 3-sphere. We also prove an equivalence theorem based on local moves and the existence of a universal configuration for such representation. As an application we give unified proofs of two different results on existence of fibered links in 3-manifolds. Received: 7 April 1997     

-algebras associated with branched coverings

In this note we analyze the -algebra associated with a branched covering both as a groupoid -algebra and as a Cuntz-Pimsner algebra. We determine conditions when the algebra is simple and purely infinite. We indicate how to compute the K-theory of several examples, including one related to rational maps on the Riemann sphere.