Transverse Heegaard splittings (Q1363620)

This is the second paper in a series of papers of the authors on Heegaard splittings. Among the sequence results in the first one [\textit{H. Rubinstein} and \textit{M. Scharlemann}, Topology 35, No. 4, 1005-1026 (1996; Zbl 0858.57020)] of the authors, the key one is that any two Heegaard surfaces of an irreducible non-Haken orientable 3-manifold may be isotoped so that they intersect in a nonempty collection of simple closed curves, each of which is essential in both surfaces. In the paper under review, the authors describe an analog to the theorem that applies to the Haken case. From [\textit{M. Scharlemann} and \textit{A. Thompson}, Contemp. Math. 164, 231-238 (1994; Zbl 0818.57013)], any irreducible Heegaard splitting \(M=A\cup_P B\) can be broken up into a series of strongly irreducible splittings, called untelescoping. That is, one can begin with the handle structure determined by \(A\cup_P B\) and rearrange the order of the 1- and 2-handles, so that ultimately, \[ M=M_0\cup_{F_1}M_1\cup_{F_2}\cdots\cup_{F_m}M_m. \] The 1- and 2-handles which occur in \(M_i\) provide it with a strongly irreducible splitting \(A_i\cup_{P_i} B_i\), with \(\partial_- A_i=F_i\) and \(\partial_- B_{i-1} =F_i\) for \(1\leq i\leq m\); \(\partial_- A_0 =\partial_- M\); and \(\partial_- B_m=\partial_+ M\), which satisfy some technical conditions. Suppose the irreducible 3-manifold \(M\) has two Heegaard splittings \(A\cup_P B\) and \(X\cup_Q Y\), as above, which have the respective untelescopings \[ M=M_0\cup_{F_1}M_1\cup_{F_2}\cdots\cup_{F_m}M_m \] and \[ M=N_0\cup_{G_1}N_1\cup_{G_2}\cdots\cup_{G_n}N_n. \] Here \(A_i\cup_{P_i} B_i\) and \(X_j\cup_{Q_j} Y_j\) are as above. Let \(\partial_- N=\partial_- X_0=\partial_- X\) and \(\partial_+ N=\partial_- Y_n =\partial_- Y\), and let \(F_0=\partial_- M\), \(F_{m+1}=\partial_+ M\), \(G_0 =\partial_- N\) and \(G_{n+1}=\partial_+ N\). Finally, define \(P'=\sum_{0\leq i\leq m}{P_i}\), \(F=\sum_{1\leq i\leq m}\), \(P^+ =P'\cup F\), \(Q'=\sum_{0\leq i\leq n}{Q_i}\), \(G=\sum_{1\leq i\leq n}\), and \(Q^+ =Q'\cup G\). Here is the main result of the article: Suppose that \(A\cup_P B\) and \(X\cup_Q Y\) are two irreducible Heegaard splittings of the same irreducible compact orientable 3-manifold \(M\), and that \(P^+\) and \(Q^+\) are surfaces (described above) coming from untelescopings of \(A\cup_P B\) and \(X\cup_Q Y\), respectively. Then \(P^+\) and \(Q^+\) can be isotoped so that they are in general position and each curve of intersection is essential in both surfaces. Each \(P_i\) \((0\leq i\leq m)\) and each \(F_i\) \((1\leq i\leq m)\) intersects \(Q^+\) nontrivially, and each \(Q_j\) \((0\leq j\leq n)\) and each \(G_j\) \((1\leq j\leq n)\) intersects \(P^+\) nontrivially.