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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?

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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 "3-manifolds as viewed from the curve complex".

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  • $\begingroup$ Thanks @Sam. Why did you edit your post as "$T$ meets each of $H_1$ and $H_2$ in collections of disjoint incompressible annuli"? I just checked Hempel's paper, and he proves that each intersection is an essential annulus in Lemma 3.6. $\endgroup$ – Mustafa Nov 2 '16 at 0:56
  • $\begingroup$ By the way, I might need this fact for Heegaard splittings which are not strongly irreducible. I will check the proof to see if we can drop the strong irreducibility assumption. It's still nice to see a partial result on that though. $\endgroup$ – Mustafa Nov 2 '16 at 2:07
  • $\begingroup$ I usually reserve the term "essential" for surfaces that are incompressible and boundary incompressible. I think that getting rid of strong irreducibility will be difficult... $\endgroup$ – Sam Nead Nov 2 '16 at 8:47
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    $\begingroup$ One way to try to get something for weakly reducible splittings is to untelescope (a la Scharlemann-Thompson) apply the Kobayashi/Thompson/Hempel result and then amalgamate back to the original splitting. It might happen that the essential annuli obstruct the amalgamation, but maybe in your situation you can show that doesn't happen? $\endgroup$ – Scott Taylor Nov 2 '16 at 15:05
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    $\begingroup$ @Mustafa I guess there would be counterexample for your question, since it is so strong. But it is more interesting to consider what these essential annuli looks like in these two compression bodies. $\endgroup$ – yanqing Nov 15 '16 at 8:02

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