Georgia Institute of Technology College of Sciences

An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the story is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is “finitely many”? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on a quantitative version of the “how many?” question.

We will define transverse surgery, and study its effects on open books, the Heegaard Floer contact invariant, and tightness. We show that surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than 2g-1. We also give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

A well known result of Giroux tells us that isotopy classes ofcontact structures on a closed three manifold are in one to onecorrespondence with stabilization classes of open book decompositions ofthe manifold. We will introduce a characterization of tightness of acontact structure in terms of corresponding open book decompositions, andshow how this can be used to resolve the question of whether tightness ispreserved under Legendrian surgery.

In joint work with Joan Birman and Bill Menasco, we describe a new finite set of geodesics connecting two given vertices of the curve complex. As an application, we give an effective algorithm for distance in the curve complex.