Let $M$ be a closed, connected, orientable 3-manifold with a Heegaard splitting $(F,H\_1,H\_2)$. Haken's Lemma states that if $M$ is reducible, i.e. if there is an embedded essential sphere in $M$, then there is an embedded essential sphere $S$ in $M$ such that $S\cap F$ is a simple closed curve, and hence $S\cap H\_i$ are disks.


Is there a version of this lemma for essential tori embedded in $M$? In other words, can one say that if there is an embedded essential torus in $M$, then there is one which cuts each handlebody $H\_i$ in an annulus?



Haken's Lemma for Essential Tori
Yes. If the splitting surface $F$ is strongly irreducible then, after an isotopy you can assume that $T$ meets each of $H\_1$ and $H\_2$ in collections of disjoint incompressible annuli. This has been independently proved by Kobayashi, Thompson, and Hempel. For references, please see the bibliography of Hempel's [paper](https://arxiv.org/abs/math/9712220) "3-manifolds as viewed from the curve complex".


