Vol. 228, No. 2, 2006

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ISSN: 0030-8730
Proximity in the curve complex: boundary reduction and bicompressible surfaces

Martin Scharlemann

Vol. 228 (2006), No. 2, 325–348

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.

Heegaard splitting, strongly irreducible, handlebody, weakly incompressible
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M50
Received: 28 March 2005
Accepted: 12 May 2006
Published: 1 December 2006
Martin Scharlemann
Mathematics Department
University of California at Santa Barbara
Santa Barbara, CA 93106
United States