Tori and Heegaard splittings

Abstract

In Studies in modern topology (1968) 39–98 Prentice Hall, Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the “interesting” surface increases from zero, more complicated phenomena occur. Kobayashi (Osaka J. Math. 24 (1987) 173–215) showed that if a 3-manifold $M^{3}$ contains an essential torus $T$, then it contains one which can be isotoped to intersect a (strongly irreducible) Heegaard splitting surface $F$ in a collection of simple closed curves which are essential in $T$ and in $F$. In general, there is no global bound on the number of curves in this collection. We show that given a 3-manifold $M$, a minimal genus, strongly irreducible Heegaard surface $F$ for $M$, and an essential torus $T$, we can either restrict the number of curves of intersection of $T$ with $F$ (to four), find a different essential surface and minimal genus Heegaard splitting with at most four essential curves of intersection, find a thinner decomposition of $M$, or produce a small Seifert-fibered piece of $M$.