Strong Haken via thin position (Q6655973)

Let \(M\) be a compact, connected, orientable \(3\)-manifold. It is a classical result of Haken that if \(M\) is reducible, every Heegaard splitting of \(M\) is reducible [\textit{W. Haken}, in: Stud. in Math. 5 (Studies Modern Topol.), 39--98 (1968; Zbl 0194.24902)]. In other words, if \(H\) is a Heegaard surface for \(M\), there exists an essential sphere in \(M\) that intersects \(H\) in a single circle. A similar statement holds when \(M\) is \(\partial\)-reducible [\textit{A. J. Casson} and \textit{C. McA. Gordon}, Topology Appl. 27, 275--283 (1987; Zbl 0632.57010)]. In [Algebr. Geom. Topol. 24, No. 2, 717--753 (2024; Zbl 1544.57012)], \textit{M. Scharlemann} refined these results by showing that if \(\mathcal{S}\) is the union of pairwise disjoint properly embedded essential spheres and disks in \(M\), then \(H\) can be isotoped so that \(H\) intersects each component of \(\mathcal{S}\) in a single circle. In the present paper, new proofs of these results are given using thin position.\N\NHere are details of the argument. Suppose for simplicity that \(\mathcal{S}\) is the union of essential spheres. First, we thin \(H\) to obtain a generalized Heegaard surface \(\mathcal{H}\). If \(\mathcal{H}\) cannot be thinned anymore, \(\mathcal{H}\) is said to be locally thin. The locally thin condition guarantees that \(\mathcal{H}\) intersects \(\mathcal{S}\) in circles that are inessential in \(\mathcal{H}\). An operation on generalized Heegaard surfaces, called juggling, is introduced to remove such intersections. The original Heegaard surface \(H\) is recovered by attaching tubes to \(\mathcal{H}\), and this can be done carefully so that \(H\) intersects each component of \(\mathcal{S}\) in a single circle as desired.\N\NAs another application of thin position, the following is proved: If \(H\) is a weakly reducible Heegaard surface for \(M\), then \(M\) contains an incompressible surface \(F\) that is in a good position with respect to \(H\). This is a reformulation of Casson-Gordon's theorem and Hayashi-Shimokawa's theorem [\textit{C. Hayashi} and \textit{K. Shimokawa}, Pac. J. Math. 197, No. 2, 301--324 (2001; Zbl 1050.57016)] in the mold of Haken's lemma.