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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 \(n\).

57M99 General low-dimensional topology
57N10 Topology of general \(3\)-manifolds (MSC2010)
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