We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in Q and inQ(Z[t, t−1]) respectively, where Q(Z[t, t−1]) denotes the quotient field of Z[t, t−1]. It is known that the modulo-Z linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-Z[t, ...

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(χ) inequality. We prove a general CAT(χ) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT(χ) metric space for the completion to be CAT(χ). We use this to prove t...

We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.

Given a knot K in S 3 and a positive integer p, there is a unique p-fold cyclic connected cover X v --, S 3 K, and this can be completed to a branched cover M e --* S 3. When p is prime, the homology group H1 (M e) is torsion and was one of the earliest knot invariants (predating the Alexander polynomial). It was used by Alexander and Briggs [A-B] to distinguish knots up to 8 crossings and all ...

Abstract We completely determine finite abelian regular branched covers of the 2-sphere $S^{2}$ with property that each homeomorphism preserving branching set can be lifted.

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. We also fin...