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Proximity in the curve complex: boundary reduction and bicompressible surfaces. (English) Zbl 1127.57010
J. Hempel [Topology 40, No. 3, 631–657 (2001; Zbl 0985.57014)] defined the distance of a Heegaard splitting $$M=V\cup_FW$$ to be the distance in the curve complex $$\mathcal{C}(F)$$ between a loop bounding a disk in $$V$$ and one bounding a disk in $$W$$. This notion and its many variants have far-reaching applications and are currently a topic of intense investigation. A prototypical result was K. Hartshorn’s theorem [Pac. J. Math. 204, No. 1, 61–75 (2002; Zbl 1065.57021)], which says that when $$M$$ is closed and contains an incompressible surface of genus $$g$$, any Heegaard splitting has distance at most $$2g$$.
The paper under review contains a version of this in the case when $$M$$ is bounded, based on the following result: Suppose that $$Q$$ is an orientable properly imbedded essential (incompressible and some component not boundary-parallel) surface in $$M$$, having no disk components, and some essential component of $$Q$$ meets a compressible boundary component $$N$$. Let $$\mathcal{V}$$ and $$\mathcal{Q}$$ be the sets of vertices in $$\mathcal{C}(N)$$ consisting of boundaries of compressing disks for $$N$$ and boundary components of $$Q$$, respectively. Then the distance from $$\mathcal{V}$$ to $$\mathcal{Q}$$ is at most $$1-\chi(Q)$$. This result implies a version of Hartshorn’s theorem for the case when $$M$$ has boundary: If $$Q$$ is a properly imbedded incompressible surface in $$M$$, then the distance of any Heegaard splitting of $$M$$ is at most $$2-\chi(Q)$$.
The previous results may be somewhat expected, perhaps even known (but if so, the author’s complete and well-motivated proofs are nonetheless welcome). The main result of the paper, however, is a significant extension to the case when $$Q$$ is bicompressible (has compressions on both sides) but weakly incompressible (the boundaries of any two compression disks on opposite sides must meet). Assuming that $$Q$$ is also connected and separating, the author shows that either (i) the distance from $$\mathcal{V}$$ to $$\mathcal{Q}$$ is at most $$1-\chi(Q)$$, or (ii) $$Q$$ is obtained from two nested connected boundary-parallel surfaces by connecting them with a vertical tube. In the latter case, there exists for any $$n$$ a $$Q$$ for which the distance is more than $$n$$.

##### MSC:
 57M99 General low-dimensional topology 57N10 Topology of general $$3$$-manifolds (MSC2010)
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