Weakly reducible Heegaard splittings of Seifert fibered spaces
The paper under review shows that a weakly reducible Heegaard splitting of an exceptional Seifert fibered space is reducible. This together with Theorem 2.6 in [\textit{Y. Moriah} and \textit{J. Schultens}, Topology 37, No. 5, 1089-1112 (1998; Zbl 0926.57016)] shows that an irreducible Heegaard splitting of an orientable Seifert fibered space over an orientable surface is either horizontal or vertical. A handlebody is a connected orientable \(3\)-(dimensional) manifold obtained by attaching \(1\)-handles to \(0\)-handles. A disc \(D\) in a handlebody \(V\) is called essential if its boundary loop \(\partial D\) is contained in \(\partial V\) and does not bound a disc in \(\partial V\). Every closed orientable \(3\)-manifold has a decomposition into two handlebodies \(W_1\), \(W_2\), called a Heegaard splitting. A Heegaard splitting is called reducible (weakly reducible respectively) if \(W_i\) contains an essential disc \(D_i\) for \(i=1\) and \(2\) such that \(\partial D_1 = \partial D_2\) (\(\partial D_1 \cap \partial D_2 = \emptyset\) respectively). A Seifert fibered space \(M\) is a \(3\)-manifold which admits a foliation of codimension \(2\) with all leaves being circles, called fibers. A fiber \(c\) is called regular if there is a regular neighbourhood \(N\) of \(c \cong S^1\) such that \(N\) is homeomorphic to a solid torus \(\cong D^2 \times S^1\) and that \(p \times S^1\) is a fiber for every point \(p\) in the \(2\)-disc \(D_2\). Otherwise, a fiber is called exceptional. When by identifying all the fibers to points we obtain a surface \(F\), we say that \(M\) is a Seifert fibered space over \(F\). An orientable Seifert fibered space is called exceptional if it is fibered over the \(2\)-sphere \(S^2\) with three exceptional fibers and rational Euler number \(0\). The proof of the theorem is based on the fact that every exceptional Seifert fibered space fibers as an \(S\)-bundle over \(S^1\) so that \(M \cong S \times [0,1] / \sim_\phi\), where \(\phi\) is a periodic automorphism on the surface \(S\) and \(x \times \{ 1 \} \sim_\phi \phi(x) \times \{ 0 \}\). The author shows that a weakly reducible Heegaard splitting is given by a surface obtained from horizontal surfaces \(S \times \{ 0, p_1, \cdots, p_k, 1 \}\), \(( 0 < p_1 \leq p_k < 1)\) by tubing along vertical lines, one between each adjacent pair of horizontal surfaces. She finds a simple loop \(\gamma\) on \(S\) such that \(\phi^i(\gamma)\) intersects \(\gamma\) in a single point. She takes a vertical annulus between \(S \times p_k\) and \(S \times \{ 1 \}\) so that it contains \(\phi^{i-1} (\gamma) \subset S \times \{ 1 \}\) and the vertical line, and another one between \(S \times \{ 0 \}\) and \(S \times \{ p_1 \}\) so that it contains \(\gamma \subset S \times \{ 0 \}\) and the vertical line along which we tubed. These annuli are cut by vertical lines into discs showing the splitting is reducible.
scientific article; zbMATH DE number 1395065