Finiteness results for Heegaard surfaces in surgered manifolds

Abstract

We demonstrate that for all but a finite number of Dehn fillings on a cusped manifold, the core of the attached solid torus is isotopic into every Heegaard surface for the filled manifold. Furthermore, if the cusped manifold does not contain a closed, non-peripheral, incompressible surface, then after excluding the aforementioned set and those filled manifolds containing incompressible surfaces (also a finite set) every other manifold obtained by Dehn filling contains at most a finite number of Heegaard surfaces that are not Heegaard surfaces for the cusped manifold. It follows that these manifolds contain a finite number of Heegaard surfaces of bounded genera. For each cusped manifold, the excluded manifolds are contained in a finite set that can be determined algorithmically.