Stabilizing Heegaard splittings of toroidal 3-manifolds

Abstract

Let $T$ be a separating incompressible torus in a 3-manifold $M$. Assuming that a genus $g$ Heegaard splitting $V{\cup }_{S}W$ can be positioned nicely with respect to $T$ (e.g. $V{\cup }_{S}W$ is strongly irreducible), we obtain an upper bound on the number of stabilizations required for $V{\cup }_{S}W$ 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 $4g-4$. As a corollary, this establishes an upper bound on the number of stabilizations required for $V{\cup }_{S}W$ and any Heegaard splitting obtained by a Dehn twist of $V{\cup }_{S}W$ along $T$ to become isotopic.

21 pages, 18 figures. Version for publication. Generalization of the main theorem and minor changes in style and format

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