Links, two-handles, and four-manifolds

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitely many smooth closed 4-manifolds constructed from some Kirby diagram with bounded number of crossings, discs, and strands, or from some Turaev special shadow with bounded number of vertices. (These are the 4-dimensional analogues of Heegaard diagrams and special spines for 3-manifolds.) We therefore get two filtrations on the set of all smooth closed 4-manifolds with finite sets. The two filtrations are equivalent after linear rescalings, and their cardinality grows at least as n^{c*n}.Comment: 23 pages, 9 figures. Final versio

Hyperbolic three-manifolds that embed geodesically

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Whitehead and the Borromean links complements. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold.Comment: 11 pages, 6 figures. Minor corrections from the previous version

Complexity of 3-manifolds

We give a summary of known results on Matveev's complexity of compact 3-manifolds. The only relevant new result is the classification of all closed orientable irreducible 3-manifolds of complexity 10.Comment: 26 pages, 7 figures, minor correction

AdS/CFT Correspondence and Type 0 String Theory

We review some applications of Type 0 string theory in the context of the AdS/CFT correspondence.Comment: 6 pages, LaTeX + JHEP.cls, talk presented at the TMR conference ``Quantum aspects of gauge theories, supersymmetry and unification'', Paris, September 1-7, 199