Persistence of Heegaard structures under Dehn filling

It is well known that a Heegaard surface may destabilize after Dehn filling, reducing the genus by one or more. This phenomenon is classified according to whether or not the core of the attached solid torus is isotopic into the destabilized surface. When it is, the destabilized surface will be a Heegaard surface for infinitely many fillings, arranged along a destabilization line in the Dehn surgery space. Here we demonstrate that a destabilization line corresponds to a slope bounding an essential surface. Such slopes are known to be finite in number and therefore so is the number of destabilization lines. ly this result to study Heegaard genus. In particular we prove, using purely topological techniques, that if X is any a-cylindrical manifold, then there are an infinite number of Dehn fillings on X which produce a manifold of the same genus as X.