Stabilizing Heegaard splittings of toroidal 3-manifolds

Abstract

We show that after one stabilization, a strongly irreducible Heegaard splitting of suitably large genus of a graph manifold is isotopic to an amalgamation along a modified version of the system of canonical tori in the JSJ decomposition. As a corollary, two strongly irreducible Heegaard splittings of a graph manifold of suitably large genus are isotopic after at most one stabilization of the higher genus splitting.

Stabilization, amalgamation and curves of intersection of Heegaard splittings

“…Rubinstein and Scharlemann have shown that if M is non-Haken, then two Heegaard splittings of M of genus g and g 0 , respectively, with g g 0 are isotopic after at most 7g C 5g 0 9 stabilizations of the larger genus splitting [19]. In previous work [7], the author showed that under mild assumptions, two Heegaard splittings of genus g obtained by Dehn twisting along a JSJ torus in M are isotopic after at most 4g 4 stabilizations. In neither case have these bounds been shown to be sharp.…”

“…This is called the Stabilization Problem. Several examples are known where only one stabilization (of the larger genus splitting) is needed to achieve isotopy (see eg Derby-Talbot [6], Hagiwara [8], Schultens [23] and Sedgwick [25]). Recently, Bachman [2] and independently Hass, Thompson and Thurston [10] as well as Johnson [13; 14] have shown that the necessary number of stabilizations can be much greater than one, in fact as large as g , where g denotes the genera of the initial Heegaard splittings.…”

“…Thus V [ S W and P [ † Q are isotopic after at most s C s 0 1 stabilizations. This strategy is utilized by the author in [6], where it is shown that s D s 0 D 1 for strongly irreducible Heegaard splittings of totally oriented graph manifolds (assuming that one of the splittings has genus at least as large as a minimal genus amalgamation along the JSJ tori of M ). More specifically, if ‚ denotes the canonical system of JSJ tori of M , then it is shown that a strongly irreducible Heegaard splitting V [ S W of M and an amalgamation along ‚, P [ † Q, become isotopic after at most one stabilization of the larger genus splitting.…”

“…The main theorem in [7] states that if F is a torus and V [ S W intersects F in 2k simple closed curves essential on both surfaces, then V [ S W is isotopic to an amalgamation along F after at most k stabilizations. This theorem can be seen to be a special case of Theorem 1.1, by first applying the techniques of Lemma 4.6 in [7] to construct an isotopy of S with a surface that intersects F in k inessential simple closed curves.…”

“…This theorem can be seen to be a special case of Theorem 1.1, by first applying the techniques of Lemma 4.6 in [7] to construct an isotopy of S with a surface that intersects F in k inessential simple closed curves.…”