Explicit Dehn filling and Heegaard splittings

Abstract

We prove an explicit, quantitative criterion that ensures the Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic 3-manifold X, and a Dehn filling whose meridian and longitude curves are longer than 2π(2g - 1), we show that every genus g Heegaard splitting of the filled manifold is isotopic to a splitting of the original manifold X. The analogous statement holds for fillings of multiple boundary tori. This gives an effective version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.