Emmanuel Giroux

On the stable equivalence of open books in three-manifolds

We show that two open books in a given closed, oriented three-manifold admit isotopic stabilizations, where the stabilization is made by successive plumbings with Hopf bands, if and only if their associated plane fields are homologous. Since this condition is automatically fulfilled in an integral homology sphere, the theorem implies a conjecture of J Harer, namely, that any fibered link in the three-sphere can be obtained from the unknot by a sequence of plumbings and deplumbings of Hopf bands. The proof presented here involves contact geometry in an essential way. 57M50, 57R17; 57M25, 57R52

Existence of Lefschetz fibrations on Stein and Weinstein domains

We show that every Stein or Weinstein domain may be presented (up to deformation) as a Lefschetz fibration over the disk. The proof is an application of Donaldsons quantitative transversality techniques.