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Suppose N is a compressible
boundary component of a compact irreducible orientable 3-manifold M, and
(Q,∂Q) ⊂ (M,∂M) is an orientable properly embedded essential surface in M, some
essential component of which is incident to N and no component is a disk. Let 𝒱 and
𝒬 denote respectively the sets of vertices in the curve complex for N represented by
boundaries of compressing disks and by boundary components of Q. We prove that, if
Q is essential in M, then d(𝒱,𝒬) ≤ 1 − χ(Q).
Hartshorn showed that an incompressible surface in a closed 3-manifold puts a
limit on the distance of any Heegaard splitting. An augmented version of
our result leads to a version of Hartshorn’s theorem for merely compact
3-manifolds.
Our main result is: If a properly embedded connected surface Q is incident to N,
and Q is separating and compresses on both its sides, but not by way of
disjoint disks, then either d(𝒱,𝒬) ≤ 1 − χ(Q), or Q is obtained from two
nested connected incompressible boundary-parallel surfaces by a vertical
tubing.
Forthcoming work with M. Tomova will show how an augmented version of this
theorem leads to the same conclusion as Hartshorn’s theorem, not from an essential
surface, but from an alternate Heegaard surface. That is, if Q is a Heegaard splitting
of a compact M then no other Heegaard splitting has distance greater than twice the
genus of Q.
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