Heegaard structures of manifolds in the Dehn filling space

We prove that after Dehn filling an incompressible torus in the boundary of an a-cylindrical 3-manifold the Heegaard genus degenerates by at most one for all but finitely many fillings. We do so by proving that for all but finitely many fillings the core of the attached solid torus can be isotoped into the minimal Heegaard surface of the filled manifold, we say that these manifolds are good. For these fillings, after stabilizing the Heegaard surface once, it becomes a Heegaard surface of the original manifold. We show that any two Heegaard surfaces in different fillings, into which the core is not isotopic, can be isotoped to intersect essentially. Using this, a bound on the distance between fillings containing such surfaces is given in terms of the genera of the Heegaard surfaces.