Stabilizing Heegaard splittings of toroidal 3-manifolds

Let T be a separating incompressible torus in a 3-manifold M. Assuming that a genus g Heegaard splitting V ∪ S W can be positioned nicely with respect to T (e.g., V ∪ S W is strongly irreducible), we obtain an upper bound on the number of stabi-lizations required for V ∪ S W to become isotopic to a Heegaard splitting which is an amalgamation along T. In particular, if T is a canonical torus in the JSJ decomposition of M, then the number of necessary stabilizations is at most 4 g − 4 . As a corollary, this establishes an upper bound on the number of stabilizations required for V ∪ S W and any Heegaard splitting obtained by a Dehn twist of V ∪ S W along T to become isotopic.