turkmath.org

A classical theorem of Lickorish and Wallace says that every three-manifold can be obtained by a Dehn surgery on a link, but this surgery representation is not unique. In fact the whole machinery of Kirby calculus would give infinitely many different surgery pictures of the same three-manifold. A natural question here is whether one can always use the tools of Kirby calculus to make the number of link components in a surgery representation of a three-manifold to be one. This question is especially subtle if the three-manifold is an irreducible homology sphere. In fact, until recently the only known obstruction to being surgery on a knot was found 20 years ago by Auckly who used a hard gauge theoretical result of Taubes. Recently in a joint work with J. Hom and T. Lidman, we found a more elementary obstruction which uses Heegaard Floer homology. In this talk, I’ll mention the essentials of this work and give infinitely many examples of Brieskorn spheres which are not surgery on a knot.