Unstabilized weakly reducible Heegaard splittings (Q2631637)

This paper gives a sufficient condition for weakly reducible Heegaard splittings to be unstabilized and to be uncritical, and also gives a sufficient condition for irreducible Heegaard splittings to be critical. Let \(M\) be a compact orientable 3-manifold and \(F\) a surface in \(M\). We say that \(F\) is almost incompressible if \(F\) is incompressible in \(M\) except for \([\partial F]\), where \([\partial F]\) is the isotopy class of \(\partial F\) in \(F\). If \(F\) is bicompressible, then the (Hempel) distance \(d(F)\) can be defined by using its curve complex as in [\textit{J. Hempel}, Topology 40, No. 3, 631--657 (2001; Zbl 0985.57014)]. We say that a bicompressible surface \(F\) is almost strongly irreducible if \(F\) is strongly irreducible except for \([\partial F]\). Let \(M=V \cup_S W\) be a Heegaard splitting of \(M\). Suppose that there is an essential disk \(D\) in \(V\) such that \(\partial D\) cuts \(S\) into an almost incompressible surface \(F\) and an almost strongly irreducible surface \(S'\). Then the author proves the following: i) the Heegaard splitting is unstabilized and uncritical if \(d(S')\ge 5\), and ii) the Heegaard splitting is critical if \(M\) is irreducible and there are essential disks \(D_V\subset V\) and \(D_W\subset W\) with \([\partial D_V]\ne [\partial D]\), \(D_W\cap D \ne \emptyset\) and \(D_V \cap D_W =\emptyset\).